Separating Axis Theorem

The Separating Axis Theorem (SAT) provides a way to find the intersection between any n-sided convex 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.

Explanation of Separating Axis Theorem

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:

Given two shapes A and B. Imagine we could isolate a single edge of A and shine a light on it.

Algorithm for Finding the Intersection

Given two polygons A and B:

1. For each edge on A, get the normal n of that edge.
2. Project each vertex v of A onto n. Return the minimum and maximum projection of all vertices.
3. Repeat Step 2 for polygon B.
4. If the min and max projections found in Steps 2 and 3 do NOT overlap, the polygons are not intersecting. Return false.
5. If the projections overlap for each edge of both shapes, the shapes are intersecting. Return true.

SAT Collision Resolution

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:

• Collision Normal: in what direction, point towards object A, did the polygons intersect
• Point of Application: at what point on each object did the objects first intersect
• Relative Velocity: easily found by taking the difference between the two velocities.

Collision Normal

Live Example of Intersection Detection