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    <title>Forem: Melvin Cheah</title>
    <description>The latest articles on Forem by Melvin Cheah (@montmont20z).</description>
    <link>https://forem.com/montmont20z</link>
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      <title>Build Your First 3D Rotating Cube from Scratch (No Libraries)</title>
      <dc:creator>Melvin Cheah</dc:creator>
      <pubDate>Wed, 21 Jan 2026 13:00:00 +0000</pubDate>
      <link>https://forem.com/montmont20z/build-your-first-3d-rotating-cube-from-scratch-no-libraries-kli</link>
      <guid>https://forem.com/montmont20z/build-your-first-3d-rotating-cube-from-scratch-no-libraries-kli</guid>
      <description>&lt;h2&gt;
  
  
  Build Your First 3D Rotating Cube from Scratch (No Libraries)
&lt;/h2&gt;

&lt;h2&gt;
  
  
  What You'll Build
&lt;/h2&gt;

&lt;p&gt;&lt;strong&gt;Want to understand how 3D graphics really work?&lt;/strong&gt; Most tutorials jump straight &lt;br&gt;
into Three.js or WebGL, but the fundamentals get lost. &lt;/p&gt;

&lt;p&gt;By the end of this tutorial, you'll understand the core concepts of 3D graphics by building a rotating cube from scratch using just JavaScript and &lt;a href="https://developer.mozilla.org/en-US/docs/Web/API/Canvas_API" rel="noopener noreferrer"&gt;HTML Canvas&lt;/a&gt; -- no Three.js, no WebGL, just pure fundamentals. &lt;/p&gt;

&lt;p&gt;You'll learn: &lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;How 3D coordinate systems work &lt;/li&gt;
&lt;li&gt;How to convert between coordinate spaces &lt;/li&gt;
&lt;li&gt;How rotation matrices transform 3D objects &lt;/li&gt;
&lt;li&gt;How perspective projection creates depth &lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;Prerequisites&lt;/strong&gt;: Basic JavaScript knowledge. That's it! &lt;br&gt;
&lt;strong&gt;Time needed&lt;/strong&gt;: 30-45 minutes &lt;br&gt;
Let's dive in!&lt;/p&gt;
&lt;h2&gt;
  
  
  Step 1: Draw a square
&lt;/h2&gt;

&lt;p&gt;Open your favourite IDE. I use visual studio code in this case. &lt;br&gt;
Create a file name &lt;code&gt;index.html&lt;/code&gt; and write these codes.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight html"&gt;&lt;code&gt;&lt;span class="nt"&gt;&amp;lt;canvas&lt;/span&gt; &lt;span class="na"&gt;id=&lt;/span&gt;&lt;span class="s"&gt;"canvas"&lt;/span&gt;&lt;span class="nt"&gt;&amp;gt;&amp;lt;/canvas&amp;gt;&lt;/span&gt;
&lt;span class="nt"&gt;&amp;lt;script &lt;/span&gt;&lt;span class="na"&gt;src=&lt;/span&gt;&lt;span class="s"&gt;"cube.js"&lt;/span&gt;&lt;span class="nt"&gt;&amp;gt;&amp;lt;/script&amp;gt;&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;We have defined a canvas element and a script element. The canvas element is where we will draw the Cube. We will load the file &lt;code&gt;cube.js&lt;/code&gt; using the script element. &lt;/p&gt;

&lt;p&gt;Then create a another new file name &lt;code&gt;cube.js&lt;/code&gt;&lt;br&gt;
Then initialize multiple variable&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;BACKGROUND&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="s2"&gt;#404040&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;
&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;FOREGROUND&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="s2"&gt;#FF5050&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;
&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;width&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;800&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;height&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;800&lt;/span&gt;

&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;getContext&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="s2"&gt;2d&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;We have created a grey background color and a orange foreground color for the cube . Then change the canvas's width and height to 800px. Then declare the canvas context.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;(){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;fillStyle&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="nx"&gt;BACKGROUND&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;fillRect&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;width&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;height&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;

&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;create a new function call &lt;strong&gt;clear()&lt;/strong&gt;. This is use to fill up the background with gray color. &lt;br&gt;
Open &lt;code&gt;index.html&lt;/code&gt; you should be able to see the following&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fmdxj25q2yt5r79angi1w.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fmdxj25q2yt5r79angi1w.png" alt="a gray background" width="800" height="790"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;A blank gray canvas. &lt;br&gt;
Now let's define the vertices we will use for our square&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;100&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;100&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// top left&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;400&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;100&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// top right&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;400&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;400&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// bottom right&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;100&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;400&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// bottom left&lt;/span&gt;
&lt;span class="p"&gt;]&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Create a draw line function&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;p1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;p2&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;strokeStyle&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;FOREGROUND&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="c1"&gt;// ctx.lineWidth = 5.0 // (optional) make the line thicker&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;beginPath&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;moveTo&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;p1&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;p1&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;lineTo&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;p2&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;p2&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;stroke&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Then call the drawLine() function&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;This is what you should see at this point. &lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F54v240n5ehhjvhtqfjam.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F54v240n5ehhjvhtqfjam.png" alt="square at top left corner" width="800" height="772"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Full code until now&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;BACKGROUND&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="s2"&gt;#404040&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;
&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;FOREGROUND&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="s2"&gt;#FF5050&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;
&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;width&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;800&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;height&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;800&lt;/span&gt;

&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;getContext&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="s2"&gt;2d&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;

&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;(){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;fillStyle&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="nx"&gt;BACKGROUND&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;fillRect&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;width&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;height&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;

&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;100&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;100&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// top left&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;400&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;100&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// top right&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;400&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;400&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// bottom right&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;100&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;400&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// bottom left&lt;/span&gt;
&lt;span class="p"&gt;]&lt;/span&gt;

&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;p1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;p2&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;strokeStyle&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;FOREGROUND&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="c1"&gt;// ctx.lineWidth = 5.0&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;beginPath&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;moveTo&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;p1&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;p1&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;lineTo&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;p2&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;p2&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;stroke&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;

&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h2&gt;
  
  
  Step 2: Draw a cube
&lt;/h2&gt;

&lt;p&gt;Now, in order to draw a cube we need to first normalize the coordinate system as JavaScript's canvas use the &lt;strong&gt;World coordinate system&lt;/strong&gt;.&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Where &lt;code&gt;0,0&lt;/code&gt; is the top left corner. &lt;/li&gt;
&lt;li&gt;X value increase as you move &lt;strong&gt;horizontally to the right&lt;/strong&gt;. &lt;/li&gt;
&lt;li&gt;Y value increase as you move &lt;strong&gt;vertically downward&lt;/strong&gt;.
&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fv6oasdgqw9hn6c5fnuj4.png" alt="a square showing concept of canvas height and width" width="800" height="676"&gt;
&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Because most mathematic formula assumes you are using the &lt;strong&gt;Normalized Device Coordinates&lt;/strong&gt; &lt;br&gt;
We need to change from &lt;strong&gt;Normalized Device Coordinates&lt;/strong&gt; back to &lt;strong&gt;World coordinate system&lt;/strong&gt;.&lt;br&gt;
&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fdqzwj7cq6360cupd0hg9.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fdqzwj7cq6360cupd0hg9.png" alt="Normalize world coordinate" width="800" height="533"&gt;&lt;/a&gt;&lt;br&gt;
&lt;strong&gt;Normalized Device Coordinates&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt; Where the center is (0, 0)&lt;/li&gt;
&lt;li&gt; All axis's value range from -1 to 1
&lt;/li&gt;
&lt;/ul&gt;
&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="c1"&gt;// NDC (-1..1) -&amp;gt; canvas pixels&lt;/span&gt;
&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="c1"&gt;// translate from -1 .. 1 =&amp;gt; 0..2 =&amp;gt; 0..w/h&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;      &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;/&lt;/span&gt; &lt;span class="mi"&gt;2&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;width&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;/&lt;/span&gt; &lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;height&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;


&lt;p&gt;So this function translate normalize coordinate to the coordinates system canvas uses. &lt;br&gt;
Let's break it down.&lt;/p&gt;

&lt;p&gt;The goal is translate coordinate with (-1 to 1)  to (0,0 to w,h)&lt;br&gt;
Step 1:&lt;br&gt;
&lt;code&gt;point.x + 1&lt;/code&gt;&lt;br&gt;
Translate (-1 to 1) to (0 to 2)&lt;br&gt;
Step 2:&lt;br&gt;
&lt;code&gt;(point.x + 1)/2&lt;/code&gt;&lt;br&gt;
Translate (0 to 2) to (0 to 1)&lt;br&gt;
Step 3:&lt;br&gt;
&lt;code&gt;(point.x + 1) / 2 * canvas.width&lt;/code&gt;&lt;br&gt;
Translate (0 to 1) to (0 to canvas.width)&lt;/p&gt;

&lt;p&gt;For the y-axis, canvas coordinates increase downward (opposite of mathematical convention), so we flip it:&lt;br&gt;
&lt;code&gt;(1 - (point.y + 1) / 2)&lt;/code&gt;&lt;/p&gt;

&lt;p&gt;Then update these code&lt;br&gt;
change from&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;100&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;100&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// top left&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;400&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;100&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// top right&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;400&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;400&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// bottom right&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;100&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mi"&gt;400&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// bottom left&lt;/span&gt;
&lt;span class="p"&gt;]&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;To&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// top left&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// top right&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// bottom right&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// bottom left&lt;/span&gt;
&lt;span class="p"&gt;]&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;






&lt;p&gt;change from&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;To&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;]),&lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;]))&lt;/span&gt;
&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;]),&lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;]))&lt;/span&gt;
&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;]),&lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;]))&lt;/span&gt;
&lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;]),&lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;]))&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Obvious we should use for loop for these&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="k"&gt;for &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;&amp;lt;&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="o"&gt;++&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;]),&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[(&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="o"&gt;+&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;%&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Result:&lt;br&gt;
&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F99czp9ivq5xyzh62ntnb.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F99czp9ivq5xyzh62ntnb.png" alt="Square at center" width="800" height="770"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;As you can see, the square move to the center because we use &lt;strong&gt;Normalized Device Coordinates&lt;/strong&gt; and (0,0) is the center of the canvas.  &lt;/p&gt;

&lt;p&gt;Now add more vertices and change the vertices from 2D to 3D&lt;br&gt;
Change from&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// top left&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// top right&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// bottom right&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="c1"&gt;// bottom left&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;To&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="p"&gt;];&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Now you have a 3D object, but it still seems 2D&lt;br&gt;
&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F0l2v3uy0g7u33qw2mduc.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F0l2v3uy0g7u33qw2mduc.png" alt="A cube that looks like square" width="800" height="782"&gt;&lt;/a&gt;&lt;br&gt;
That is why we need to rotate the cube to see all its side&lt;/p&gt;

&lt;p&gt;Full code until now:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;BACKGROUND&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="s2"&gt;#404040&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;
&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;FOREGROUND&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="s2"&gt;#FF5050&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;
&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;width&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;800&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;height&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;800&lt;/span&gt;

&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;getContext&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="s2"&gt;2d&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;

&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;(){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;fillStyle&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="nx"&gt;BACKGROUND&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;fillRect&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;width&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;height&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;

&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="p"&gt;]&lt;/span&gt;

&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;p1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;p2&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;strokeStyle&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;FOREGROUND&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="c1"&gt;// ctx.lineWidth = 5.0&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;beginPath&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;moveTo&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;p1&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;p1&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;lineTo&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;p2&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;p2&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;stroke&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;

&lt;span class="c1"&gt;// NDC (-1..1) -&amp;gt; canvas pixels&lt;/span&gt;
&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="c1"&gt;// translate from -1 .. 1 =&amp;gt; 0..2 =&amp;gt; 0..w/h&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;/&lt;/span&gt; &lt;span class="mi"&gt;2&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;width&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;/&lt;/span&gt; &lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;height&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;

&lt;span class="k"&gt;for &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;&amp;lt;&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="o"&gt;++&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;]),&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[(&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="o"&gt;+&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;%&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h2&gt;
  
  
  Step 3: Rotate the cube
&lt;/h2&gt;

&lt;p&gt;To rotate a cube or any 3D object you would use these formula&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fagyl8ct6mjhnjuje6mfw.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fagyl8ct6mjhnjuje6mfw.png" alt="Matrix Rotate in Y-axis" width="154" height="128"&gt;&lt;/a&gt; Rotate in Y-axis&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F50f90eqimgax8yqhar3q.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F50f90eqimgax8yqhar3q.png" alt="Matrix Rotate in X-axis" width="153" height="128"&gt;&lt;/a&gt; Rotate in X-axis&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F3gfk11ddkc7fa9l319f2.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F3gfk11ddkc7fa9l319f2.png" alt="Matrix Rotate in Z-axis" width="153" height="128"&gt;&lt;/a&gt; Rotate in Z -axis&lt;/p&gt;

&lt;p&gt;In this case, we will try to rotate the cube in Y-axis&lt;br&gt;
Now try to derive x, y, z coordinates from the formula&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Futfrtilrdopw93p0s4jo.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Futfrtilrdopw93p0s4jo.png" alt="derive the rotation function" width="408" height="133"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;This is how we derive the rotation function. Add this to the code&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;rotateY&lt;/span&gt;&lt;span class="p"&gt;({&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;x&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;cos&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="nx"&gt;z&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;sin&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;x&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;sin&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="nx"&gt;z&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;cos&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Add vertices indices&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;verticeIndexes&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;
    &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// front face &lt;/span&gt;
    &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;4&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;6&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;7&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// back face &lt;/span&gt;
    &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;4&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// top face &lt;/span&gt;
    &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;7&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;6&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// bottom face &lt;/span&gt;
    &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;7&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;4&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// right face &lt;/span&gt;
    &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;6&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// left face&lt;/span&gt;
&lt;span class="p"&gt;]&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;A cube has 8 vertices/corner and 6 faces. We use vertices index because we can reuse vertex without the need of repeatedly defined the same vertex.  This is the standard of all modern graphics API as we can save memory and performance by reducing vertex count. &lt;/p&gt;

&lt;p&gt;Then add this new function called &lt;code&gt;frame()&lt;/code&gt;. We are going to start animating the rotation.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;FPS&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;60&lt;/span&gt;
&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;dt&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;FPS&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;

&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;(){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt; &lt;span class="o"&gt;+=&lt;/span&gt; &lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;PI&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nx"&gt;dt&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="c1"&gt;// increment angle in each frame &lt;/span&gt;
    &lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
    &lt;span class="c1"&gt;// Draw cube&lt;/span&gt;
    &lt;span class="k"&gt;for &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;&amp;lt;&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="o"&gt;++&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;rotateY&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;)),&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;rotateY&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[(&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="o"&gt;+&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;%&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;setTimeout&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1000&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;FPS&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="nf"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Notice this is a recursion function. &lt;code&gt;setTimeout(frame, 1000/FPS)&lt;/code&gt;  For every &lt;code&gt;1000/60 = 16ms&lt;/code&gt; 16 milliseconds, the frame function will be called once, making this a loop that will run forever. &lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Note&lt;/strong&gt;: We're using &lt;code&gt;setTimeout&lt;/code&gt; for simplicity, but for production code, &lt;code&gt;requestAnimationFrame&lt;/code&gt; is preferred as it syncs with the browser's refresh rate for smoother animations.&lt;/p&gt;

&lt;p&gt;&lt;code&gt;angle += Math.PI * 0.5 * dt;&lt;/code&gt;&lt;br&gt;
Notice this line, for every seconds, the cube will rotate 1/4 of a rotation. To make every second 1 rotation, you can use &lt;code&gt;angle += Math.PI * 2 * dt&lt;/code&gt;&lt;br&gt;
2π equal 1 rotation.&lt;/p&gt;

&lt;p&gt;Result:&lt;br&gt;
&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F69equoytlja981efty6k.gif" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F69equoytlja981efty6k.gif" alt="Rotating cube" width="480" height="478"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Notice the shape of the cube is not exactly correct. I will give you a small challenge here, try find out why the shape of the cube is not correct.&lt;br&gt;
&lt;strong&gt;Hint&lt;/strong&gt;: Look at how we're connecting the vertices. We're just connecting them in sequence (0→1→2→3→4→5→6→7→0), which doesn't actually form a proper cube structure!&lt;/p&gt;

&lt;p&gt;Solution:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="p"&gt;];&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;The order of the vertices we previously declare is incorrect. A simple solution is to reorder the vertices. But a more robust solution is to use the &lt;code&gt;verticeIndex&lt;/code&gt; we previously created. &lt;/p&gt;

&lt;p&gt;Update the &lt;code&gt;frame()&lt;/code&gt; function&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;(){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt; &lt;span class="o"&gt;+=&lt;/span&gt; &lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;PI&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nx"&gt;dt&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;for &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;face&lt;/span&gt; &lt;span class="k"&gt;of&lt;/span&gt; &lt;span class="nx"&gt;verticeIndexes&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt; &lt;span class="c1"&gt;// Loop through each face&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;for &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;&amp;lt;&lt;/span&gt; &lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="o"&gt;++&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt; &lt;span class="c1"&gt;// Loop through vertices in this face&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="c1"&gt;// Connect vertices to form edges of this face&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;point1&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;]];&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;point2&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;[(&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="o"&gt;+&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;%&lt;/span&gt; &lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="p"&gt;];&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;rotateY&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;)),&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;rotateY&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point2&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;setTimeout&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1000&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;FPS&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;The problem wasn't the vertex positions themselves - it was that we were connecting vertices sequentially (0→1→2→...→7), which creates a tangled mess instead of proper cube faces. By using &lt;code&gt;verticeIndexes&lt;/code&gt;, we explicitly define which vertices form each face, giving us proper cube geometry.&lt;/p&gt;

&lt;p&gt;Result:&lt;br&gt;
&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fcxx0blwudheh03e8l0d9.gif" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fcxx0blwudheh03e8l0d9.gif" alt="Rotating Cube Completed" width="480" height="468"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Full code:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;BACKGROUND&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="s2"&gt;#404040&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;
&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;FOREGROUND&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="s2"&gt;#FF5050&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;
&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;width&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;800&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;height&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;800&lt;/span&gt;

&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;getContext&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="s2"&gt;2d&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;

&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;(){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;fillStyle&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="nx"&gt;BACKGROUND&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;fillRect&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;width&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;height&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;

&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="p"&gt;]&lt;/span&gt;

&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;verticeIndexes&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;
    &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// front face &lt;/span&gt;
    &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;4&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;6&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;7&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// back face &lt;/span&gt;
    &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;4&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// top face &lt;/span&gt;
    &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;7&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;6&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// bottom face &lt;/span&gt;
    &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;7&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;4&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// right face &lt;/span&gt;
    &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;6&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// left face&lt;/span&gt;
&lt;span class="p"&gt;]&lt;/span&gt;

&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;p1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;p2&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;strokeStyle&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;FOREGROUND&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="c1"&gt;// ctx.lineWidth = 5.0&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;beginPath&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;moveTo&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;p1&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;p1&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;lineTo&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;p2&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;p2&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;stroke&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;

&lt;span class="c1"&gt;// NDC (-1..1) -&amp;gt; canvas pixels&lt;/span&gt;
&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="c1"&gt;// translate from -1 .. 1 =&amp;gt; 0..2 =&amp;gt; 0..w/h&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;/&lt;/span&gt; &lt;span class="mi"&gt;2&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;width&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;/&lt;/span&gt; &lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;height&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;


&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;rotateY&lt;/span&gt;&lt;span class="p"&gt;({&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;x&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;cos&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="nx"&gt;z&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;sin&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;x&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;sin&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="nx"&gt;z&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;cos&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;

&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;FPS&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;60&lt;/span&gt;
&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;dt&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;FPS&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;

&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;(){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt; &lt;span class="o"&gt;+=&lt;/span&gt; &lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;PI&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nx"&gt;dt&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;for &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;face&lt;/span&gt; &lt;span class="k"&gt;of&lt;/span&gt; &lt;span class="nx"&gt;verticeIndexes&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;for &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;&amp;lt;&lt;/span&gt; &lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="o"&gt;++&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;point1&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;]];&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;point2&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;[(&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="o"&gt;+&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;%&lt;/span&gt; &lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="p"&gt;];&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;perspectiveProject&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;rotateY&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;))),&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;perspectiveProject&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;rotateY&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point2&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;)))&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;setTimeout&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1000&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;FPS&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="nf"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h2&gt;
  
  
  Step 4: Apply perspective projection
&lt;/h2&gt;

&lt;p&gt;Now the cube we created is not particularly realistic, it is because it is using orthographic projection implicitly, which means that the object in the distance would not become smaller. To make the object in the distance smaller, we need to apply a perspective projection using these formula.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;x = x/z
y = y/z
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;You are right, just as simple as these 2 lines. &lt;br&gt;
This is because objects farther from the camera (larger z values) should be smaller. Therefore it gets divided by a bigger number, making them smaller on screen. This mimics how our eyes see the world!&lt;br&gt;
Lets implement it.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;perspectiveProject&lt;/span&gt;&lt;span class="p"&gt;({&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;}){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;This function could work, but when &lt;code&gt;z = 0&lt;/code&gt; 💥things could go very bad. That is why I opted for this solution.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;perspectiveProject&lt;/span&gt;&lt;span class="p"&gt;({&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;}){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;eps&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="nx"&gt;e&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mi"&gt;6&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;if &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt; &lt;span class="o"&gt;&amp;lt;=&lt;/span&gt; &lt;span class="nx"&gt;eps&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;eps&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;eps&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;};&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Here we use a very small number &lt;code&gt;eps&lt;/code&gt; to prevent division by zero&lt;/p&gt;

&lt;p&gt;Finally apply the perspective projection.&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;(){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt; &lt;span class="o"&gt;+=&lt;/span&gt; &lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;PI&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nx"&gt;dt&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;for &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;face&lt;/span&gt; &lt;span class="k"&gt;of&lt;/span&gt; &lt;span class="nx"&gt;verticeIndexes&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;for &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;&amp;lt;&lt;/span&gt; &lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="o"&gt;++&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;point1&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;]];&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;point2&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;[(&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="o"&gt;+&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;%&lt;/span&gt; &lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="p"&gt;];&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;perspectiveProject&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;rotateY&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;))),&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;perspectiveProject&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;rotateY&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point2&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;)))&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;setTimeout&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1000&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;FPS&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="nf"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Result:&lt;br&gt;
&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F2mzq0l71wenaqrdsb2yl.gif" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F2mzq0l71wenaqrdsb2yl.gif" alt="Rotating Perspective cube" width="480" height="448"&gt;&lt;/a&gt;&lt;br&gt;
You will notice this weird line fly around. This is because the cube is too close to the camera. We need to move the cube further away (translate z).&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;translateZ&lt;/span&gt;&lt;span class="p"&gt;({&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="nx"&gt;dz&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="na"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="o"&gt;+&lt;/span&gt;&lt;span class="nx"&gt;dz&lt;/span&gt;&lt;span class="p"&gt;};&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Add this translate Z function. Apply to the &lt;code&gt;frame()&lt;/code&gt; function&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;(){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt; &lt;span class="o"&gt;+=&lt;/span&gt; &lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;PI&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nx"&gt;dt&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;for &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;face&lt;/span&gt; &lt;span class="k"&gt;of&lt;/span&gt; &lt;span class="nx"&gt;verticeIndexes&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;for &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;&amp;lt;&lt;/span&gt; &lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="o"&gt;++&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;point1&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;]];&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;point2&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;[(&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="o"&gt;+&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;%&lt;/span&gt; &lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="p"&gt;];&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;dz&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;2&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;perspectiveProject&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;translateZ&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;rotateY&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt;&lt;span class="nx"&gt;dz&lt;/span&gt;&lt;span class="p"&gt;))),&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;perspectiveProject&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;translateZ&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;rotateY&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point2&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt; &lt;span class="nx"&gt;dz&lt;/span&gt;&lt;span class="p"&gt;)))&lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;setTimeout&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1000&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;FPS&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="nf"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;code&gt;const dz = 2&lt;/code&gt;&lt;br&gt;
We translate the cube 2 units away from the camera (at z=0). This ensures &lt;br&gt;
all vertices have positive z values after rotation, preventing division issues &lt;br&gt;
and keeping the cube in front of the camera.&lt;/p&gt;

&lt;p&gt;And finally we are done:&lt;br&gt;
&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fsu3fvcgfod97m4aj8rxp.gif" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fsu3fvcgfod97m4aj8rxp.gif" alt="Rotating Perspective Cube Correct" width="480" height="462"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Final Code:&lt;br&gt;
👜 &lt;a href="https://github.com/Montmont20z/JavaScript-3D-Cube" rel="noopener noreferrer"&gt;https://github.com/Montmont20z/JavaScript-3D-Cube&lt;/a&gt;&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight javascript"&gt;&lt;code&gt;&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;BACKGROUND&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="s2"&gt;#404040&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;
&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;FOREGROUND&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="s2"&gt;#FF5050&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;
&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;width&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;800&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;height&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;800&lt;/span&gt;

&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;getContext&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="s2"&gt;2d&lt;/span&gt;&lt;span class="dl"&gt;"&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;

&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;(){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;fillStyle&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="nx"&gt;BACKGROUND&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;fillRect&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;width&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;height&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;

&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nl"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mf"&gt;0.5&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt;
&lt;span class="p"&gt;]&lt;/span&gt;

&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;verticeIndexes&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// front face&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;4&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;6&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;7&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// back face&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;4&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// top face&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;7&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;6&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// bottom face&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;7&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;4&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// right face&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;6&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="c1"&gt;// left face&lt;/span&gt;
&lt;span class="p"&gt;]&lt;/span&gt;

&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;p1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;p2&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;strokeStyle&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;FOREGROUND&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="c1"&gt;// ctx.lineWidth = 5.0&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;beginPath&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;moveTo&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;p1&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;p1&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;lineTo&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;p2&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;p2&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;ctx&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;stroke&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;

&lt;span class="c1"&gt;// NDC (-1..1) -&amp;gt; canvas pixels&lt;/span&gt;
&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="c1"&gt;// translate from -1 .. 1 =&amp;gt; 0..2 =&amp;gt; 0..w/h&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;/&lt;/span&gt; &lt;span class="mi"&gt;2&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;width&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;/&lt;/span&gt; &lt;span class="mi"&gt;2&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nx"&gt;canvas&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;height&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;

&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;perspectiveProject&lt;/span&gt;&lt;span class="p"&gt;({&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;}){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;eps&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="nx"&gt;e&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mi"&gt;6&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;if &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt; &lt;span class="o"&gt;&amp;lt;=&lt;/span&gt; &lt;span class="nx"&gt;eps&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;eps&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;eps&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;};&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;y&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;

&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;rotateY&lt;/span&gt;&lt;span class="p"&gt;({&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;x&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;x&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;cos&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="nx"&gt;z&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;sin&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="na"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;x&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;sin&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="nx"&gt;z&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nf"&gt;cos&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;

&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;translateZ&lt;/span&gt;&lt;span class="p"&gt;({&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="p"&gt;},&lt;/span&gt; &lt;span class="nx"&gt;dz&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;&lt;span class="nx"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="nx"&gt;y&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="na"&gt;z&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt; &lt;span class="nx"&gt;z&lt;/span&gt;&lt;span class="o"&gt;+&lt;/span&gt;&lt;span class="nx"&gt;dz&lt;/span&gt;&lt;span class="p"&gt;};&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;

&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;FPS&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;60&lt;/span&gt;
&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;dt&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;FPS&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;

&lt;span class="kd"&gt;function&lt;/span&gt; &lt;span class="nf"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;(){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt; &lt;span class="o"&gt;+=&lt;/span&gt; &lt;span class="nb"&gt;Math&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;PI&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="mf"&gt;0.5&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="nx"&gt;dt&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;clear&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;for &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;face&lt;/span&gt; &lt;span class="k"&gt;of&lt;/span&gt; &lt;span class="nx"&gt;verticeIndexes&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="k"&gt;for &lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="kd"&gt;let&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="nx"&gt;i&lt;/span&gt; &lt;span class="o"&gt;&amp;lt;&lt;/span&gt; &lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt; &lt;span class="o"&gt;++&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;){&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;point1&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="p"&gt;]];&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;point2&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nx"&gt;vertices&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;[(&lt;/span&gt;&lt;span class="nx"&gt;i&lt;/span&gt;&lt;span class="o"&gt;+&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;%&lt;/span&gt; &lt;span class="nx"&gt;face&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="nx"&gt;length&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="p"&gt;];&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="kd"&gt;const&lt;/span&gt; &lt;span class="nx"&gt;dz&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;2&lt;/span&gt;

&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;drawLine&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;perspectiveProject&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;translateZ&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;rotateY&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt;&lt;span class="nx"&gt;dz&lt;/span&gt;&lt;span class="p"&gt;))),&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;normalizeCoordinate&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;perspectiveProject&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;translateZ&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nf"&gt;rotateY&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;point2&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nx"&gt;angle&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt; &lt;span class="nx"&gt;dz&lt;/span&gt;&lt;span class="p"&gt;)))&lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="err"&gt; &lt;/span&gt; &lt;span class="err"&gt; &lt;/span&gt; &lt;span class="nf"&gt;setTimeout&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nx"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1000&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="nx"&gt;FPS&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="nf"&gt;frame&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h2&gt;
  
  
  Troubleshooting
&lt;/h2&gt;

&lt;p&gt;&lt;strong&gt;Canvas is blank&lt;/strong&gt;: Make sure you're opening &lt;code&gt;index.html&lt;/code&gt; in a browser, not just viewing the file.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;"canvas is not defined" error&lt;/strong&gt;: The canvas variable is automatically created by the browser when you give an element &lt;code&gt;id="canvas"&lt;/code&gt;. Make sure your HTML has this.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Cube looks distorted&lt;/strong&gt;: Check that your &lt;code&gt;verticeIndexes&lt;/code&gt; array matches the vertex order exactly. The indices must correspond to the correct vertex positions.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Weird flickering lines&lt;/strong&gt;: This happens when vertices pass through or very close to z=0. Our &lt;code&gt;eps&lt;/code&gt; value in &lt;code&gt;perspectiveProject&lt;/code&gt; helps prevent this.&lt;/p&gt;

&lt;h2&gt;
  
  
  What You've Learned
&lt;/h2&gt;

&lt;p&gt;In this tutorial, you've built a 3D rendering engine from scratch and learned&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;
&lt;strong&gt;Coordinate Systems&lt;/strong&gt;: How to convert between different coordinate spaces (NDC vs screen space)&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;3D Transformations&lt;/strong&gt;: How rotation matrices transform points in 3D space&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Perspective Projection&lt;/strong&gt;: How dividing by depth (z) creates realistic perspective&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Indexed Geometry&lt;/strong&gt;: Why modern graphics APIs use vertex indices to save memory&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;These are the same fundamental concepts used in engines like Three.js, Unity, and Unreal Engine!&lt;/p&gt;

&lt;h2&gt;
  
  
  Where to Go From Here
&lt;/h2&gt;

&lt;p&gt;Want to take this further? Try these challenges:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Add rotation on multiple axes&lt;/strong&gt;: Combine rotateX, rotateY, and rotateZ &lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Add More Translation&lt;/strong&gt;: Translate X, Translate Y, Translate Z &lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Add camera controls&lt;/strong&gt;: Let users control the camera with mouse/keyboard&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;Learning Resources:&lt;/strong&gt; &lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;a href="https://webglfundamentals.org/" rel="noopener noreferrer"&gt;WebGL Fundamentals&lt;/a&gt; - Deep dive into 3D graphics &lt;/li&gt;
&lt;li&gt;
&lt;a href="https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab" rel="noopener noreferrer"&gt;3Blue1Brown Linear Algebra&lt;/a&gt; - Understanding the math behind transformations &lt;/li&gt;
&lt;li&gt;
&lt;a href="https://www.scratchapixel.com/" rel="noopener noreferrer"&gt;Scratchapixel&lt;/a&gt; - 3D rendering from scratch&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  More Projects Like This
&lt;/h2&gt;

&lt;p&gt;If you enjoyed this tutorial, check out my other 3D graphics projects:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;&lt;a href="https://github.com/Montmont20z/3D-Robot-Model-Renderer" rel="noopener noreferrer"&gt;3D-Robot-Model-Renderer&lt;/a&gt;&lt;/strong&gt; - A hierarchical 3D Gundam-style robot model built from scratch using OpenGL Fixed Pipeline and Win32 API&lt;/li&gt;
&lt;li&gt;
&lt;a href="https://github.com/Montmont20z/Splash-Ground" rel="noopener noreferrer"&gt;Splash Ground&lt;/a&gt; - A fast-paced 3D arena survival / territory-control 3D game&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;See all my work at &lt;a href="https://github.com/Montmont20z" rel="noopener noreferrer"&gt;https://github.com/Montmont20z&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;Questions? Suggestions? Let me know in the comments! 👇&lt;/p&gt;

</description>
      <category>learning</category>
      <category>javascript</category>
      <category>tutorial</category>
      <category>programming</category>
    </item>
    <item>
      <title>I Built a Game Engine from Scratch in C++ (Here's What I Learned)</title>
      <dc:creator>Melvin Cheah</dc:creator>
      <pubDate>Wed, 07 Jan 2026 10:49:57 +0000</pubDate>
      <link>https://forem.com/montmont20z/building-a-game-engine-from-scratch-using-c-my-breakout-clone-journey-20d1</link>
      <guid>https://forem.com/montmont20z/building-a-game-engine-from-scratch-using-c-my-breakout-clone-journey-20d1</guid>
      <description>&lt;h2&gt;
  
  
  I Built a Game Engine from Scratch in C++ (Here's What I Learned)
&lt;/h2&gt;

&lt;p&gt;I crashed my GPU 47 times before I saw my first triangle on screen.&lt;/p&gt;

&lt;p&gt;For 3 months, I built a game engine from scratch in C++ using DirectX 9 and Win32—no Unity, no Unreal, no middleware. Just me, the Windows API, and a lot of segmentation faults.&lt;/p&gt;

&lt;p&gt;This is the story of how building a simple Breakout clone taught me more about game development, graphics programming, and software architecture than years of using Unity ever did.&lt;/p&gt;

&lt;h2&gt;
  
  
  Why Build an Engine?
&lt;/h2&gt;

&lt;p&gt;For years, I built games in Unity. I'd drag and drop GameObjects, attach scripts, hit Play, and watch my game come to life. It was magical—until it wasn't.&lt;/p&gt;

&lt;p&gt;Questions started nagging at me:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;How does Unity actually render my sprites?&lt;/strong&gt; What's happening between &lt;code&gt;GameObject.transform.position = newPos&lt;/code&gt; and pixels on screen?&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Why do people complain about Unreal's performance?&lt;/strong&gt; If it's "optimized," why do developers still struggle?&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Why was Kerbal Space Program's physics so buggy?&lt;/strong&gt; It's Unity—doesn't Unity handle physics automatically?&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;I realized I was using powerful tools without understanding what they were doing under the hood. I was a chef using a microwave, not knowing how heat actually cooks food.&lt;/p&gt;

&lt;p&gt;Then my university professor gave us an assignment: &lt;strong&gt;Build a low-level game engine in C++.&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;No Unity. No libraries. Just C++, DirectX 9, and the Win32 API.&lt;/p&gt;

&lt;p&gt;This was my chance to peek behind the curtain.&lt;/p&gt;

&lt;h2&gt;
  
  
  What I Built: Breakout, But From Scratch
&lt;/h2&gt;

&lt;p&gt;If you've never played Breakout: you control a paddle at the bottom of the screen, bouncing a ball to destroy bricks at the top. Simple concept, complex implementation.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;My engine features:&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Custom rendering pipeline&lt;/strong&gt; using DirectX 9&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Fixed-timestep game loop&lt;/strong&gt; (60 FPS target)&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;AABB and swept collision detection&lt;/strong&gt; (no tunneling!)&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;State management system&lt;/strong&gt; (Menu → Level 1 → Level 2 → Level 3 → End Game)&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Sound system&lt;/strong&gt; integration&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Sprite animation system&lt;/strong&gt;&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Physics simulation&lt;/strong&gt; (velocity, acceleration, collision response)&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;The result:&lt;/strong&gt; A fully playable Breakout clone running at 60 FPS with ~3,500 lines of C++ code.&lt;/p&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fgithub.com%2Fuser-attachments%2Fassets%2Fe0c36a95-5bd1-4e59-b10c-60ca79667eed" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fgithub.com%2Fuser-attachments%2Fassets%2Fe0c36a95-5bd1-4e59-b10c-60ca79667eed" alt="Breakout" width="720" height="390"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Tech Stack:&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Language: C++17&lt;/li&gt;
&lt;li&gt;Graphics API: DirectX 9 (legacy, but perfect for learning fundamentals)&lt;/li&gt;
&lt;li&gt;Windowing: Win32 API&lt;/li&gt;
&lt;li&gt;Audio: Windows multimedia extensions&lt;/li&gt;
&lt;li&gt;IDE: Visual Studio 2022&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  Architecture Overview: Separation of Concerns
&lt;/h2&gt;

&lt;p&gt;One of my biggest lessons: &lt;strong&gt;good architecture makes or breaks your project.&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;I learned this the hard way (more on that in "Challenges" below), but here's the final structure I landed on:&lt;/p&gt;

&lt;h3&gt;
  
  
  Class Diagram
&lt;/h3&gt;

&lt;p&gt;&lt;a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fzba9oh1huyvqs4xmihvq.png" class="article-body-image-wrapper"&gt;&lt;img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2Fzba9oh1huyvqs4xmihvq.png" alt="Simply Class Diagram" width="800" height="1352"&gt;&lt;/a&gt;&lt;/p&gt;

&lt;h3&gt;
  
  
  Core Components
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;Game Class (The Orchestrator)&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Owns all managers (Renderer, Input, Physics, Sound)&lt;/li&gt;
&lt;li&gt;Manages game state transitions (Menu ↔ Level 1 ↔ Game Over)&lt;/li&gt;
&lt;li&gt;Runs the main game loop&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;MyWindow (Platform Layer)&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Wraps Win32 window creation and message processing&lt;/li&gt;
&lt;li&gt;Handles OS-level events (close, minimize, resize)&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Why separate?&lt;/strong&gt; Platform code should be isolated—makes porting to Linux/Mac easier later&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;Renderer (Graphics Layer)&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Initializes DirectX 9 device&lt;/li&gt;
&lt;li&gt;Manages textures and sprites&lt;/li&gt;
&lt;li&gt;Provides clean API: &lt;code&gt;LoadTexture()&lt;/code&gt;, &lt;code&gt;DrawSprite()&lt;/code&gt;, &lt;code&gt;BeginFrame()&lt;/code&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Key insight:&lt;/strong&gt; The game logic never touches DirectX directly&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;InputManager (User Input)&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Polls keyboard state using DirectInput&lt;/li&gt;
&lt;li&gt;Abstracts raw input into game-meaningful queries: &lt;code&gt;IsKeyDown(DIK_LEFT)&lt;/code&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Why?&lt;/strong&gt; Game code doesn't care about DirectInput—it just wants "left" or "right"&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;PhysicsManager (Collision &amp;amp; Movement)&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;AABB collision detection&lt;/li&gt;
&lt;li&gt;Swept AABB for fast-moving objects (prevents tunneling)&lt;/li&gt;
&lt;li&gt;Collision resolution with restitution&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Lesson learned:&lt;/strong&gt; Separate &lt;em&gt;detection&lt;/em&gt; from &lt;em&gt;resolution&lt;/em&gt; (I didn't know this at first!)&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;SoundManager (Audio)&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Loads and plays sound effects&lt;/li&gt;
&lt;li&gt;Handles background music with looping&lt;/li&gt;
&lt;li&gt;Volume control&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;IGameState (State Pattern)&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Interface for all game states: Menu, Level1, Level2, GameOver, YouWin&lt;/li&gt;
&lt;li&gt;Each state implements: &lt;code&gt;OnEnter()&lt;/code&gt;, &lt;code&gt;Update()&lt;/code&gt;, &lt;code&gt;Render()&lt;/code&gt;, &lt;code&gt;OnExit()&lt;/code&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;This was my "aha!" moment&lt;/strong&gt;—more on this below&lt;/li&gt;
&lt;/ul&gt;

&lt;h3&gt;
  
  
  The Game Loop
&lt;/h3&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="k"&gt;while&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;window&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;ProcessMessages&lt;/span&gt;&lt;span class="p"&gt;())&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="c1"&gt;// 1. Calculate delta time (frame-independent movement)&lt;/span&gt;
    &lt;span class="kt"&gt;float&lt;/span&gt; &lt;span class="n"&gt;dt&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;CalculateDeltaTime&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;

    &lt;span class="c1"&gt;// 2. Update input state&lt;/span&gt;
    &lt;span class="n"&gt;inputManager&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Update&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;

    &lt;span class="c1"&gt;// 3. Update current game state&lt;/span&gt;
    &lt;span class="c1"&gt;//    (Menu, Level, GameOver, etc.)&lt;/span&gt;
    &lt;span class="n"&gt;gameState&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;Update&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;dt&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;inputManager&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;physicsManager&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;soundManager&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;

    &lt;span class="c1"&gt;// 4. Render everything&lt;/span&gt;
    &lt;span class="n"&gt;renderer&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;BeginFrame&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;
    &lt;span class="n"&gt;gameState&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;Render&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;renderer&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
    &lt;span class="n"&gt;renderer&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;EndFrame&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;strong&gt;Why this structure?&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Modularity:&lt;/strong&gt; Each system has one job&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Testability:&lt;/strong&gt; Can test physics without rendering&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Maintainability:&lt;/strong&gt; Bug in rendering? Only look in Renderer class&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Scalability:&lt;/strong&gt; Adding a new game state? Just implement IGameState&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  The Rendering Pipeline: From Nothing to Pixels
&lt;/h2&gt;

&lt;p&gt;DirectX 9 has a reputation: it's old (released 2002), verbose, and unforgiving. But that's precisely why it's perfect for learning—you have to understand every step.&lt;/p&gt;

&lt;h3&gt;
  
  
  Initialization: Setting Up DirectX 9
&lt;/h3&gt;

&lt;p&gt;Getting a window to show anything requires &lt;strong&gt;five major steps:&lt;/strong&gt;&lt;/p&gt;

&lt;h4&gt;
  
  
  1. Create the Direct3D9 Interface
&lt;/h4&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="n"&gt;IDirect3D9&lt;/span&gt;&lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="n"&gt;m_direct3D9&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;Direct3DCreate9&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;D3D_SDK_VERSION&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="o"&gt;!&lt;/span&gt;&lt;span class="n"&gt;m_direct3D9&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="c1"&gt;// Failed to create—probably missing DirectX runtime&lt;/span&gt;
    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="nb"&gt;false&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;This creates the main Direct3D object. Think of it as "connecting to the graphics driver."&lt;/p&gt;

&lt;h4&gt;
  
  
  2. Query Display Capabilities
&lt;/h4&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="n"&gt;D3DDISPLAYMODE&lt;/span&gt; &lt;span class="n"&gt;displayMode&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="n"&gt;m_direct3D9&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;GetAdapterDisplayMode&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;D3DADAPTER_DEFAULT&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="o"&gt;&amp;amp;&lt;/span&gt;&lt;span class="n"&gt;displayMode&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;We need to know: What resolution? What color format? This tells us what the monitor supports.&lt;/p&gt;

&lt;h4&gt;
  
  
  3. Configure the Presentation Parameters
&lt;/h4&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="n"&gt;D3DPRESENT_PARAMETERS&lt;/span&gt; &lt;span class="n"&gt;m_d3dPP&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;{};&lt;/span&gt;
&lt;span class="n"&gt;m_d3dPP&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;Windowed&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;TRUE&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;                          &lt;span class="c1"&gt;// Windowed mode (not fullscreen)&lt;/span&gt;
&lt;span class="n"&gt;m_d3dPP&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;BackBufferWidth&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;width&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;                   &lt;span class="c1"&gt;// 800 pixels&lt;/span&gt;
&lt;span class="n"&gt;m_d3dPP&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;BackBufferHeight&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;height&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;                 &lt;span class="c1"&gt;// 600 pixels&lt;/span&gt;
&lt;span class="n"&gt;m_d3dPP&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;BackBufferFormat&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;D3DFMT_UNKNOWN&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;        &lt;span class="c1"&gt;// Match desktop format&lt;/span&gt;
&lt;span class="n"&gt;m_d3dPP&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;BackBufferCount&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;                       &lt;span class="c1"&gt;// Double buffering&lt;/span&gt;
&lt;span class="n"&gt;m_d3dPP&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;SwapEffect&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;D3DSWAPEFFECT_DISCARD&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;       &lt;span class="c1"&gt;// Throw away old frames&lt;/span&gt;
&lt;span class="n"&gt;m_d3dPP&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;EnableAutoDepthStencil&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;TRUE&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;             &lt;span class="c1"&gt;// We need depth testing&lt;/span&gt;
&lt;span class="n"&gt;m_d3dPP&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;AutoDepthStencilFormat&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;D3DFMT_D16&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;      &lt;span class="c1"&gt;// 16-bit depth buffer&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;strong&gt;This is where modern APIs (Vulkan, DX12) get even MORE complex.&lt;/strong&gt; You're essentially telling the GPU: "Here's how I want my window's backbuffer configured."&lt;/p&gt;

&lt;h4&gt;
  
  
  4. Create the Device
&lt;/h4&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="n"&gt;HRESULT&lt;/span&gt; &lt;span class="n"&gt;hr&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;m_direct3D9&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;CreateDevice&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;
    &lt;span class="n"&gt;D3DADAPTER_DEFAULT&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;              &lt;span class="c1"&gt;// Use default GPU&lt;/span&gt;
    &lt;span class="n"&gt;D3DDEVTYPE_HAL&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;                   &lt;span class="c1"&gt;// Hardware acceleration&lt;/span&gt;
    &lt;span class="n"&gt;hWnd&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;                              &lt;span class="c1"&gt;// Window handle&lt;/span&gt;
    &lt;span class="n"&gt;D3DCREATE_HARDWARE_VERTEXPROCESSING&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;  &lt;span class="c1"&gt;// Use GPU for vertex math&lt;/span&gt;
    &lt;span class="o"&gt;&amp;amp;&lt;/span&gt;&lt;span class="n"&gt;m_d3dPP&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
    &lt;span class="o"&gt;&amp;amp;&lt;/span&gt;&lt;span class="n"&gt;m_d3dDevice&lt;/span&gt;
&lt;span class="p"&gt;);&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;strong&gt;This is where I crashed 47 times.&lt;/strong&gt; Wrong parameters? Crash. Unsupported format? Crash. Missing depth buffer? Crash.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Fallback strategy:&lt;/strong&gt; If hardware vertex processing fails (older GPUs), fall back to software:&lt;/p&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;FAILED&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;hr&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="c1"&gt;// Try again with CPU-based vertex processing&lt;/span&gt;
    &lt;span class="n"&gt;hr&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;m_direct3D9&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;CreateDevice&lt;/span&gt;&lt;span class="p"&gt;(...,&lt;/span&gt; &lt;span class="n"&gt;D3DCREATE_SOFTWARE_VERTEXPROCESSING&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="p"&gt;...);&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;h4&gt;
  
  
  5. Create the Sprite Renderer
&lt;/h4&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="n"&gt;ID3DXSprite&lt;/span&gt;&lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="n"&gt;m_spriteBrush&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="n"&gt;D3DXCreateSprite&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;m_d3dDevice&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="o"&gt;&amp;amp;&lt;/span&gt;&lt;span class="n"&gt;m_spriteBrush&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;DirectX 9's &lt;code&gt;ID3DXSprite&lt;/code&gt; is a helper for 2D games. It batches sprite draws and handles transformations.&lt;/p&gt;




&lt;h3&gt;
  
  
  Rendering Each Frame
&lt;/h3&gt;

&lt;p&gt;Once initialized, every frame follows this pattern:&lt;/p&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="kt"&gt;void&lt;/span&gt; &lt;span class="n"&gt;Renderer&lt;/span&gt;&lt;span class="o"&gt;::&lt;/span&gt;&lt;span class="n"&gt;BeginFrame&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="c1"&gt;// Clear the screen to black&lt;/span&gt;
    &lt;span class="n"&gt;m_d3dDevice&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;Clear&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nb"&gt;NULL&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;D3DCLEAR_TARGET&lt;/span&gt; &lt;span class="o"&gt;|&lt;/span&gt; &lt;span class="n"&gt;D3DCLEAR_ZBUFFER&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; 
                        &lt;span class="n"&gt;D3DCOLOR_XRGB&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt; &lt;span class="mf"&gt;1.0&lt;/span&gt;&lt;span class="n"&gt;f&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;

    &lt;span class="n"&gt;m_d3dDevice&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;BeginScene&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;           &lt;span class="c1"&gt;// Start recording draw calls&lt;/span&gt;
    &lt;span class="n"&gt;m_spriteBrush&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;Begin&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;D3DXSPRITE_ALPHABLEND&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;  &lt;span class="c1"&gt;// Enable alpha blending for sprites&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;

&lt;span class="kt"&gt;void&lt;/span&gt; &lt;span class="n"&gt;Renderer&lt;/span&gt;&lt;span class="o"&gt;::&lt;/span&gt;&lt;span class="n"&gt;DrawSprite&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="k"&gt;const&lt;/span&gt; &lt;span class="n"&gt;SpriteInstance&lt;/span&gt;&lt;span class="o"&gt;&amp;amp;&lt;/span&gt; &lt;span class="n"&gt;sprite&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="c1"&gt;// Apply transformations (position, rotation, scale)&lt;/span&gt;
    &lt;span class="n"&gt;D3DXMATRIX&lt;/span&gt; &lt;span class="n"&gt;transform&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;CalculateTransform&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;sprite&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
    &lt;span class="n"&gt;m_spriteBrush&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;SetTransform&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="o"&gt;&amp;amp;&lt;/span&gt;&lt;span class="n"&gt;transform&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;

    &lt;span class="c1"&gt;// Draw the texture&lt;/span&gt;
    &lt;span class="n"&gt;m_spriteBrush&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;Draw&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;sprite&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;texture&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="o"&gt;&amp;amp;&lt;/span&gt;&lt;span class="n"&gt;sourceRect&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nb"&gt;nullptr&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="nb"&gt;nullptr&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;sprite&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;color&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;

&lt;span class="kt"&gt;void&lt;/span&gt; &lt;span class="n"&gt;Renderer&lt;/span&gt;&lt;span class="o"&gt;::&lt;/span&gt;&lt;span class="n"&gt;EndFrame&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="n"&gt;m_spriteBrush&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;End&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;          &lt;span class="c1"&gt;// Finish sprite batch&lt;/span&gt;
    &lt;span class="n"&gt;m_d3dDevice&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;EndScene&lt;/span&gt;&lt;span class="p"&gt;();&lt;/span&gt;        &lt;span class="c1"&gt;// Stop recording&lt;/span&gt;
    &lt;span class="n"&gt;m_d3dDevice&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;Present&lt;/span&gt;&lt;span class="p"&gt;(...);&lt;/span&gt;      &lt;span class="c1"&gt;// Flip backbuffer to screen (VSYNC happens here)&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;strong&gt;Key Concept: Double Buffering&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;We draw to a "backbuffer" (off-screen), then &lt;code&gt;Present()&lt;/code&gt; swaps it with the screen's front buffer. This prevents tearing (seeing half-drawn frames).&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Performance Note:&lt;/strong&gt; Each &lt;code&gt;DrawSprite()&lt;/code&gt; call is relatively expensive. In a real engine, you'd batch hundreds of sprites into fewer draw calls. For Breakout (~50 bricks max), it doesn't matter.&lt;/p&gt;

&lt;h2&gt;
  
  
  Challenges &amp;amp; Solutions: Where I Failed (And What I Learned)
&lt;/h2&gt;

&lt;h3&gt;
  
  
  Challenge 1: Architecture Disaster (Week 3)
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;The Problem:&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;I made the classic beginner mistake: &lt;strong&gt;I started coding without designing.&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;My first attempt looked like this:&lt;/p&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="k"&gt;class&lt;/span&gt; &lt;span class="nc"&gt;Game&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="n"&gt;Renderer&lt;/span&gt; &lt;span class="n"&gt;renderer&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
    &lt;span class="n"&gt;InputManager&lt;/span&gt; &lt;span class="n"&gt;input&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;

    &lt;span class="c1"&gt;// OH NO—game logic mixed into Game class!&lt;/span&gt;
    &lt;span class="n"&gt;Paddle&lt;/span&gt; &lt;span class="n"&gt;paddle&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
    &lt;span class="n"&gt;Ball&lt;/span&gt; &lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
    &lt;span class="n"&gt;Brick&lt;/span&gt; &lt;span class="n"&gt;bricks&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;50&lt;/span&gt;&lt;span class="p"&gt;];&lt;/span&gt;

    &lt;span class="kt"&gt;void&lt;/span&gt; &lt;span class="n"&gt;Update&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
        &lt;span class="c1"&gt;// Handle input&lt;/span&gt;
        &lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;input&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;IsKeyDown&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;LEFT&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt; &lt;span class="n"&gt;paddle&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;x&lt;/span&gt; &lt;span class="o"&gt;-=&lt;/span&gt; &lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;

        &lt;span class="c1"&gt;// Update physics&lt;/span&gt;
        &lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;x&lt;/span&gt; &lt;span class="o"&gt;+=&lt;/span&gt; &lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;velocityX&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;

        &lt;span class="c1"&gt;// Check collisions&lt;/span&gt;
        &lt;span class="k"&gt;for&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="k"&gt;auto&lt;/span&gt;&lt;span class="o"&gt;&amp;amp;&lt;/span&gt; &lt;span class="n"&gt;brick&lt;/span&gt; &lt;span class="o"&gt;:&lt;/span&gt; &lt;span class="n"&gt;bricks&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
            &lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;CollidesWith&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;brick&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
                &lt;span class="n"&gt;brick&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;alive&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nb"&gt;false&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
            &lt;span class="p"&gt;}&lt;/span&gt;
        &lt;span class="p"&gt;}&lt;/span&gt;

        &lt;span class="c1"&gt;// ...300 more lines of spaghetti code&lt;/span&gt;
    &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="p"&gt;};&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;strong&gt;This worked fine&lt;/strong&gt;—until I needed to add a menu screen.&lt;/p&gt;

&lt;p&gt;Suddenly I realized: &lt;strong&gt;How do I switch between Menu and Level1?&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;My code had no concept of "states." Everything was hardcoded into one giant &lt;code&gt;Update()&lt;/code&gt; function. Adding a menu meant:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Wrapping everything in &lt;code&gt;if (currentState == PLAYING)&lt;/code&gt;
&lt;/li&gt;
&lt;li&gt;Duplicating input handling for menu vs. gameplay&lt;/li&gt;
&lt;li&gt;Managing which objects exist when&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;It was a mess. I was 2 weeks in and facing a complete rewrite.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;The Solution: State Pattern&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;I asked my lecturer (and ChatGPT) for advice. The answer: &lt;strong&gt;State Pattern.&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="c1"&gt;// Interface that all game states implement&lt;/span&gt;
&lt;span class="k"&gt;class&lt;/span&gt; &lt;span class="nc"&gt;IGameState&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
&lt;span class="nl"&gt;public:&lt;/span&gt;
    &lt;span class="k"&gt;virtual&lt;/span&gt; &lt;span class="kt"&gt;void&lt;/span&gt; &lt;span class="n"&gt;OnEnter&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;GameServices&lt;/span&gt;&lt;span class="o"&gt;&amp;amp;&lt;/span&gt; &lt;span class="n"&gt;services&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
    &lt;span class="k"&gt;virtual&lt;/span&gt; &lt;span class="kt"&gt;void&lt;/span&gt; &lt;span class="n"&gt;Update&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="kt"&gt;float&lt;/span&gt; &lt;span class="n"&gt;dt&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="p"&gt;...)&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
    &lt;span class="k"&gt;virtual&lt;/span&gt; &lt;span class="kt"&gt;void&lt;/span&gt; &lt;span class="n"&gt;Render&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;Renderer&lt;/span&gt;&lt;span class="o"&gt;&amp;amp;&lt;/span&gt; &lt;span class="n"&gt;renderer&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
    &lt;span class="k"&gt;virtual&lt;/span&gt; &lt;span class="kt"&gt;void&lt;/span&gt; &lt;span class="n"&gt;OnExit&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;GameServices&lt;/span&gt;&lt;span class="o"&gt;&amp;amp;&lt;/span&gt; &lt;span class="n"&gt;services&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="p"&gt;};&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Now each screen is its own class:&lt;/p&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="k"&gt;class&lt;/span&gt; &lt;span class="nc"&gt;MenuState&lt;/span&gt; &lt;span class="o"&gt;:&lt;/span&gt; &lt;span class="k"&gt;public&lt;/span&gt; &lt;span class="n"&gt;IGameState&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt; &lt;span class="cm"&gt;/* menu logic */&lt;/span&gt; &lt;span class="p"&gt;};&lt;/span&gt;
&lt;span class="k"&gt;class&lt;/span&gt; &lt;span class="nc"&gt;Level1&lt;/span&gt; &lt;span class="o"&gt;:&lt;/span&gt; &lt;span class="k"&gt;public&lt;/span&gt; &lt;span class="n"&gt;IGameState&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt; &lt;span class="cm"&gt;/* level 1 logic */&lt;/span&gt; &lt;span class="p"&gt;};&lt;/span&gt;
&lt;span class="k"&gt;class&lt;/span&gt; &lt;span class="nc"&gt;GameOverState&lt;/span&gt; &lt;span class="o"&gt;:&lt;/span&gt; &lt;span class="k"&gt;public&lt;/span&gt; &lt;span class="n"&gt;IGameState&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt; &lt;span class="cm"&gt;/* game over logic */&lt;/span&gt; &lt;span class="p"&gt;};&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;The &lt;code&gt;Game&lt;/code&gt; class just delegates to the current state:&lt;/p&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="k"&gt;class&lt;/span&gt; &lt;span class="nc"&gt;Game&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="n"&gt;std&lt;/span&gt;&lt;span class="o"&gt;::&lt;/span&gt;&lt;span class="n"&gt;unique_ptr&lt;/span&gt;&lt;span class="o"&gt;&amp;lt;&lt;/span&gt;&lt;span class="n"&gt;IGameState&lt;/span&gt;&lt;span class="o"&gt;&amp;gt;&lt;/span&gt; &lt;span class="n"&gt;currentState&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;

    &lt;span class="kt"&gt;void&lt;/span&gt; &lt;span class="n"&gt;Update&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="kt"&gt;float&lt;/span&gt; &lt;span class="n"&gt;dt&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
        &lt;span class="n"&gt;currentState&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;Update&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;dt&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="p"&gt;...);&lt;/span&gt;  &lt;span class="c1"&gt;// Let the state handle it&lt;/span&gt;
    &lt;span class="p"&gt;}&lt;/span&gt;

    &lt;span class="kt"&gt;void&lt;/span&gt; &lt;span class="nf"&gt;ChangeState&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;std&lt;/span&gt;&lt;span class="o"&gt;::&lt;/span&gt;&lt;span class="n"&gt;unique_ptr&lt;/span&gt;&lt;span class="o"&gt;&amp;lt;&lt;/span&gt;&lt;span class="n"&gt;IGameState&lt;/span&gt;&lt;span class="o"&gt;&amp;gt;&lt;/span&gt; &lt;span class="n"&gt;newState&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
        &lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;currentState&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="n"&gt;currentState&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;OnExit&lt;/span&gt;&lt;span class="p"&gt;(...);&lt;/span&gt;
        &lt;span class="n"&gt;currentState&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;std&lt;/span&gt;&lt;span class="o"&gt;::&lt;/span&gt;&lt;span class="n"&gt;move&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;newState&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
        &lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;currentState&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="n"&gt;currentState&lt;/span&gt;&lt;span class="o"&gt;-&amp;gt;&lt;/span&gt;&lt;span class="n"&gt;OnEnter&lt;/span&gt;&lt;span class="p"&gt;(...);&lt;/span&gt;
    &lt;span class="p"&gt;}&lt;/span&gt;
&lt;span class="p"&gt;};&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;strong&gt;What I Learned:&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Design before code&lt;/strong&gt; (ESPECIALLY for 1,000+ line projects)&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Separation of concerns&lt;/strong&gt; makes code flexible&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Refactoring hurts, but teaches more than getting it right the first time&lt;/strong&gt;&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;The State Pattern is everywhere—React components, game engines, even operating systems use it. This lesson alone was worth the 3 months.&lt;/p&gt;




&lt;h3&gt;
  
  
  Challenge 2: The Ball Goes Through Bricks (Tunneling)
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;The Problem:&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;My first collision detection looked like this:&lt;/p&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;OverlapsAABB&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;brick&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="n"&gt;brick&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;alive&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nb"&gt;false&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
    &lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;velocityY&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;velocityY&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;  &lt;span class="c1"&gt;// Bounce&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;strong&gt;This worked at 60 FPS... until the ball moved too fast.&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;At high speeds, the ball would &lt;strong&gt;tunnel&lt;/strong&gt;—pass completely through a brick between frames:&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight plaintext"&gt;&lt;code&gt;Frame 1: Ball is here     →  [    ]
                                ↓
Frame 2: Ball is here         [    ]  ← Ball skipped the brick!
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;The ball moved 50 pixels, but the brick was only 32 pixels wide. By the next frame, the ball was &lt;em&gt;already past&lt;/em&gt; the brick, so the overlap check returned false.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;First Failed Solution: Smaller Time Steps&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;I tried updating physics 120 times per second instead of 60. This helped but didn't solve it—at very high velocities, tunneling still occurred.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;The Real Solution: Swept AABB&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;I needed &lt;strong&gt;continuous collision detection&lt;/strong&gt;—checking not just "are they overlapping now?" but "&lt;strong&gt;will they overlap at any point during this frame's movement?&lt;/strong&gt;"&lt;/p&gt;

&lt;p&gt;This is called &lt;strong&gt;swept AABB&lt;/strong&gt; (or &lt;strong&gt;ray-swept box&lt;/strong&gt;). Instead of checking the ball's current position, I treat the ball's movement as a ray:&lt;/p&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="kt"&gt;bool&lt;/span&gt; &lt;span class="nf"&gt;SweepAABB&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;
    &lt;span class="n"&gt;Vector3&lt;/span&gt; &lt;span class="n"&gt;ballPos&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;Vector2&lt;/span&gt; &lt;span class="n"&gt;ballSize&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
    &lt;span class="n"&gt;Vector3&lt;/span&gt; &lt;span class="n"&gt;displacement&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;  &lt;span class="c1"&gt;// Where the ball will move this frame&lt;/span&gt;
    &lt;span class="n"&gt;Vector3&lt;/span&gt; &lt;span class="n"&gt;brickPos&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;Vector2&lt;/span&gt; &lt;span class="n"&gt;brickSize&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;
    &lt;span class="kt"&gt;float&lt;/span&gt;&lt;span class="o"&gt;&amp;amp;&lt;/span&gt; &lt;span class="n"&gt;timeOfImpact&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;   &lt;span class="c1"&gt;// When in [0,1] does collision happen?&lt;/span&gt;
    &lt;span class="n"&gt;Vector3&lt;/span&gt;&lt;span class="o"&gt;&amp;amp;&lt;/span&gt; &lt;span class="n"&gt;hitNormal&lt;/span&gt;     &lt;span class="c1"&gt;// Which side did we hit?&lt;/span&gt;
&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="c1"&gt;// Calculate when the ball's edges cross the brick's edges&lt;/span&gt;
    &lt;span class="kt"&gt;float&lt;/span&gt; &lt;span class="n"&gt;xEntryTime&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;...;&lt;/span&gt; &lt;span class="c1"&gt;// Math for X-axis entry&lt;/span&gt;
    &lt;span class="kt"&gt;float&lt;/span&gt; &lt;span class="n"&gt;yEntryTime&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;...;&lt;/span&gt; &lt;span class="c1"&gt;// Math for Y-axis entry&lt;/span&gt;

    &lt;span class="kt"&gt;float&lt;/span&gt; &lt;span class="n"&gt;overallEntry&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;max&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;xEntryTime&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;yEntryTime&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;

    &lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;overallEntry&lt;/span&gt; &lt;span class="o"&gt;&amp;lt;&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt; &lt;span class="o"&gt;||&lt;/span&gt; &lt;span class="n"&gt;overallEntry&lt;/span&gt; &lt;span class="o"&gt;&amp;gt;&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
        &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="nb"&gt;false&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;  &lt;span class="c1"&gt;// No collision this frame&lt;/span&gt;
    &lt;span class="p"&gt;}&lt;/span&gt;

    &lt;span class="n"&gt;timeOfImpact&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;overallEntry&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="nb"&gt;true&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;Now my collision loop looks like:&lt;/p&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="n"&gt;Vector3&lt;/span&gt; &lt;span class="n"&gt;displacement&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;velocity&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="n"&gt;dt&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="kt"&gt;float&lt;/span&gt; &lt;span class="n"&gt;toi&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="n"&gt;Vector3&lt;/span&gt; &lt;span class="n"&gt;normal&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;

&lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;SweepAABB&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;displacement&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;brick&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;toi&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;normal&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="c1"&gt;// Move ball to exactly the collision point&lt;/span&gt;
    &lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;position&lt;/span&gt; &lt;span class="o"&gt;+=&lt;/span&gt; &lt;span class="n"&gt;displacement&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="n"&gt;toi&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;

    &lt;span class="c1"&gt;// Bounce&lt;/span&gt;
    &lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;normal&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;x&lt;/span&gt; &lt;span class="o"&gt;!=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;velocity&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;x&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;velocity&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;x&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
    &lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;normal&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;y&lt;/span&gt; &lt;span class="o"&gt;!=&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;velocity&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;y&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;velocity&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;y&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;

    &lt;span class="n"&gt;brick&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;alive&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nb"&gt;false&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;strong&gt;Result:&lt;/strong&gt; No more tunneling, even at 1000 pixels/second.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;What I Learned:&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Discrete collision detection (overlap checks) fails at high speeds&lt;/strong&gt;&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;Continuous collision detection (swept/ray-based) is essential for fast-moving objects&lt;/strong&gt;&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;This is why bullets in games use raycasts, not overlap checks&lt;/strong&gt;&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;The math was painful (lots of min/max comparisons), but understanding this concept changed how I think about physics in games.&lt;/p&gt;




&lt;h3&gt;
  
  
  Challenge 3: Collision Detection ≠ Collision Resolution
&lt;/h3&gt;

&lt;p&gt;&lt;strong&gt;The Confusion:&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;When I started, I thought "collision detection" and "collision resolution" were the same thing. They're not.&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;
&lt;strong&gt;Detection&lt;/strong&gt; = "Did these two objects hit?"&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Resolution&lt;/strong&gt; = "Okay, now what do we DO about it?"&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;My first attempt mixed them together:&lt;/p&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;OverlapsAABB&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;paddle&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;velocityY&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;velocityY&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;  &lt;span class="c1"&gt;// This is resolution!&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;This caused bugs:&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Ball would "stick" to the paddle&lt;/li&gt;
&lt;li&gt;Multiple collisions in one frame would cancel out&lt;/li&gt;
&lt;li&gt;Overlapping objects would vibrate&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;The Fix: Separate Phases&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;cpp&lt;br&gt;
&lt;/p&gt;

&lt;div class="highlight js-code-highlight"&gt;
&lt;pre class="highlight cpp"&gt;&lt;code&gt;&lt;span class="c1"&gt;// Phase 1: Detection (PhysicsManager)&lt;/span&gt;
&lt;span class="kt"&gt;bool&lt;/span&gt; &lt;span class="n"&gt;hit&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;SweepAABB&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;paddle&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;timeOfImpact&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;normal&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;

&lt;span class="c1"&gt;// Phase 2: Resolution (also PhysicsManager, but separate function)&lt;/span&gt;
&lt;span class="k"&gt;if&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;hit&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="p"&gt;{&lt;/span&gt;
    &lt;span class="c1"&gt;// Move ball to contact point&lt;/span&gt;
    &lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;position&lt;/span&gt; &lt;span class="o"&gt;+=&lt;/span&gt; &lt;span class="n"&gt;displacement&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="n"&gt;timeOfImpact&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;

    &lt;span class="c1"&gt;// Apply restitution (bounciness)&lt;/span&gt;
    &lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;velocity&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;Reflect&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;ball&lt;/span&gt;&lt;span class="p"&gt;.&lt;/span&gt;&lt;span class="n"&gt;velocity&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;normal&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="n"&gt;restitution&lt;/span&gt;&lt;span class="p"&gt;;&lt;/span&gt;
&lt;span class="p"&gt;}&lt;/span&gt;
&lt;/code&gt;&lt;/pre&gt;

&lt;/div&gt;



&lt;p&gt;&lt;strong&gt;What I Learned:&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Physics engines have &lt;strong&gt;detection&lt;/strong&gt; and &lt;strong&gt;response&lt;/strong&gt; as separate systems&lt;/li&gt;
&lt;li&gt;This separation allows for complex scenarios (multiple overlaps, conveyor belts, one-way platforms)&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;YouTube tutorials often skip this distinction&lt;/strong&gt;—they work for simple cases but fail for complex games&lt;/li&gt;
&lt;/ul&gt;

&lt;h2&gt;
  
  
  What I'd Do Differently
&lt;/h2&gt;

&lt;p&gt;If I built this again from scratch:&lt;/p&gt;

&lt;ol&gt;
&lt;li&gt;
&lt;strong&gt;Design first, code second&lt;/strong&gt; (spend 2-3 days on architecture diagrams)&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Use a modern graphics API&lt;/strong&gt; (DirectX 11 or OpenGL 4.5) instead of DX9

&lt;ul&gt;
&lt;li&gt;DX9 is fine for learning, but dated (no compute shaders, limited pipeline control)&lt;/li&gt;
&lt;/ul&gt;
&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Write unit tests for physics&lt;/strong&gt; (I caught so many bugs by hand-testing that could've been automated)&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Implement an entity-component system (ECS)&lt;/strong&gt; instead of inheritance-based game objects&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Add a debug overlay earlier&lt;/strong&gt; (FPS counter, collision visualization saved me hours of debugging)&lt;/li&gt;
&lt;li&gt;
&lt;strong&gt;Profile from day one&lt;/strong&gt; (I didn't measure performance until week 8—wasted time optimizing the wrong things)&lt;/li&gt;
&lt;/ol&gt;




&lt;h2&gt;
  
  
  Conclusion &amp;amp; Next Steps
&lt;/h2&gt;

&lt;p&gt;Building a game engine from scratch was &lt;strong&gt;hard&lt;/strong&gt;—way harder than Unity tutorials made game development seem.&lt;/p&gt;

&lt;p&gt;But that's the point.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;What I Gained:&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Deep understanding of rendering pipelines&lt;/li&gt;
&lt;li&gt;Practical knowledge of physics simulation&lt;/li&gt;
&lt;li&gt;Appreciation for what Unity/Unreal abstract away&lt;/li&gt;
&lt;li&gt;Confidence to debug low-level issues&lt;/li&gt;
&lt;li&gt;A portfolio piece that stands out&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;What's Next:&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;Port this engine to &lt;strong&gt;DirectX 11&lt;/strong&gt; (modern pipeline, compute shaders)&lt;/li&gt;
&lt;li&gt;Build a &lt;strong&gt;voxel engine&lt;/strong&gt; (Minecraft-style, using Three.js for web)&lt;/li&gt;
&lt;li&gt;Experiment with &lt;strong&gt;Vulkan&lt;/strong&gt; (the final boss of graphics APIs)&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;&lt;strong&gt;Links:&lt;/strong&gt;&lt;/p&gt;

&lt;ul&gt;
&lt;li&gt;📦 &lt;a href="https://github.com/Montmont20z/Breakout-GameEngine" rel="noopener noreferrer"&gt;Source Code (GitHub)&lt;/a&gt;
&lt;/li&gt;
&lt;li&gt;🎮 &lt;a href="https://github.com/Montmont20z/Breakout-GameEngine/releases/tag/Release" rel="noopener noreferrer"&gt;Download &amp;amp; Play&lt;/a&gt;
&lt;/li&gt;
&lt;li&gt;🌐 &lt;a href="https://montmont20z.netlify.app/" rel="noopener noreferrer"&gt;My Portfolio&lt;/a&gt;
&lt;/li&gt;
&lt;li&gt;📧 &lt;a href="https://montmont20z.netlify.app/contact" rel="noopener noreferrer"&gt;Let's Connect&lt;/a&gt;
&lt;/li&gt;
&lt;li&gt;Originally published on &lt;a href="https://montmont20z.hashnode.dev/i-built-a-game-engine-from-scratch-in-c-heres-what-i-learned" rel="noopener noreferrer"&gt;Hashnode&lt;/a&gt;
&lt;/li&gt;
&lt;/ul&gt;




&lt;p&gt;&lt;strong&gt;If you're thinking about building your own engine:&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Do it. Not to replace Unity, but to &lt;strong&gt;understand&lt;/strong&gt; Unity.&lt;/p&gt;

&lt;p&gt;You'll struggle. You'll debug cryptic errors at 2 AM. You'll question why you didn't just use Godot.&lt;/p&gt;

&lt;p&gt;But when you finally see that ball bounce for the first time—compiled from YOUR code, rendered by YOUR pipeline, colliding with YOUR physics—you'll understand why people say &lt;strong&gt;"reinventing the wheel is the best way to learn how wheels work."&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Questions? Suggestions? Let me know in the comments! 👇&lt;/p&gt;

</description>
      <category>programming</category>
      <category>gamedev</category>
      <category>learning</category>
      <category>cpp</category>
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