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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<link rel="stylesheet" href="/index.css">
<title>Physics for Games</title>
<link rel="shortcut icon" href="favicon/favicon.ico" type="image/x-icon">
<meta name="description" content="A place to learn all about real-time physics simulations through descriptions, code snippets, and example programs all written in C++ and OpenGL.">
<meta name="og:description" content="A place to learn all about real-time physics simulations through descriptions, code snippets, and example programs all written in C++ and OpenGL.">
</head>
<body>
<header>
<h1><a title="physicsforgames.com" href="/">Physics for Games</a></h1>
</header>
<main>
<nav>
<ul class="outer-tree">
<li><a href="/">Introduction</a></li>
<li>
<span>🏀<span>2D</span></span>
<ul class="inner-tree">
<li><label>Rigidbody</label></li>
<li><a title="/2d/rigidbody/rigidbody_1.html" href="/2d/rigidbody/rigidbody_1.html">Linear Forces</a></li>
<li><a title="/2d/rigidbody/rigidbody_2.html" href="/2d/rigidbody/rigidbody_2.html">Rotational Forces</a></li>
<li><a title="/2d/rigidbody/rigidbody_3.html" href="/2d/rigidbody/rigidbody_3.html">Collisions</a></li>
<li><label>Collisions</label></li>
<li><a title="/2d/_collisions/rectangle_line.html" href="/2d/_collisions/rectangle_line.html">Rectangle-Line</a></li>
<li><a title="/2d/_collisions/rectangle_rectangle.html" href="/2d/_collisions/rectangle_rectangle.html">Rectangle-Rectangle</a></li>
<li><a title="/2d/_collisions/polygon_polygon.html" href="/2d/_collisions/polygon_polygon.html">Separating Axis Theorem</a></li>
</ul>
</li>
<li>
<span>🌠<span>3D</span></span>
<ul class="inner-tree">
<li><label>Rigidbody</label></li>
<li><a title="/3d/rigidbody.html" href="/3d/rigidbody.html">Rigidbody in 3D</a></li>
</ul>
</li>
<li>
<span>🔧<span>WebAssembly</span></span>
<ul class="inner-tree">
<li><a title="/intro/intro.html" href="/intro/intro.html">Introduction</a></li>
</ul>
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<li>
<span>🛈<span>About</span></span>
<ul class="inner-tree">
<li><a title="/roadmap.html" href="/roadmap.html">Roadmap</a></li>
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<script src="./polygon_polygon/dist/output.js"></script>
<script>
window.onload = function() {
var lPlayElement = document.getElementById('gl_canvas_play'),
lStopElement = document.getElementById('gl_canvas_stop');
lPlayElement.addEventListener('click', function() {
lPlayElement.style.display = 'none';
lStopElement.style.display = 'block';
});
lStopElement.addEventListener('click', function() {
lStopElement.style.display = 'none';
lPlayElement.style.display = 'block';
});
}
</script>
<article>
<h1>Separating Axis Theorem</h1>
<section>
<p>
The Separating Axis Theorem (SAT) provides a way to find the intersection between any <i>n</i>-sided <a href='https://ianqvist.blogspot.com/2009/09/convex-polygon-based-collision.html'>convex</a> polygon or circle. In this tutorial, I will explain how this theorem works, and how you can use it to both detect and resolve collisions in your simulation.
</p>
</section>
<section>
<h2>Explanation of Separating Axis Theorem</h2>
<p>
SAT makes use of vector projection to figure out whether or not two concave polygons are intersecting. The way to think about it is this:
<br/>
<br/>
Given two shapes <b>A</b> and <b>B</b>.
Imagine we could isolate a single edge of A and shine a light on it.
</p>
</section>
<section>
<h2>Algorithm for Finding the Intersection</h2>
<p>
Given two polygons <b>A</b> and <b>B</b>:
<ol>
<li>For each edge on <b>A</b>, get the normal <i>n</i> of that edge.</li>
<li>Project each vertex <i>v</i> of <b>A</b> onto <i>n</i>. Return the minimum and maximum projection of all vertices.</li>
<li>Repeat Step 2 for polygon <b>B</b>.</li>
<li>If the min and max projections found in Steps 2 and 3 do <b>NOT</b> overlap, the polygons are not intersecting. Return false.</li>
<li>If the projections overlap for each edge of both shapes, the shapes are intersecting. Return true.</li>
</ol>
And that is all there is to <i>finding</i> the intersection between two convex polygons.
</p>
</section>
<section>
<h2>SAT Collision Resolution</h2>
<p>
Now that we know our objects have intersecting, we want to be able to send them tumbling away from each other to simulate a collision. To do this, we will need to find the following things:
<ul>
<li><b>Collision Normal</b>: in what direction, point towards object <b>A</b>, did the polygons intersect</li>
<li><b>Point of Application</b>: at what point on each object did the objects first intersect</li>
<li><b>Relative Velocity</b>: easily found by taking the difference between the two velocities.
</ul>
<h3>Collision Normal</h3>
<p>
</p>
</p>
</section>
<section>
<h2>
Live Example of Intersection Detection
</h2>
<div class="opengl_canvas_container">
<canvas id="gl_canvas" width="800" height="600"></canvas>
<button id="gl_canvas_play" class="play_button">
Play
</button>
<button id="gl_canvas_stop" class="stop_button">
Stop
</button>
</div>
</section>
<footer id="references">
<h2>References</h2>
<ul>
<li><a href="https://en.wikipedia.org/wiki/Vector_projection">Vector Projection Wikapedia</a></li>
</ul>
</footer>
</article>
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