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<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>
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</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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