2D Rotated Rectangle Collision
by Eric Meythaler

The Method

The problem with checking for collision between two rotated rectangles is really a matter of being able to decide when they’re not colliding. The simple intersection test used by the Microsoft InteresectRect() function will check if the minimum and maximum x and y values of rectangle B are within the minimum and maximum x and y values of rectangle A. This method works fine for axis-aligned rectangles, but when dealing with rotated rectangles we need something a little more complex.

Figure 2. Standard Bounds-based Collision Check

As you can see, the minimum x value of B lies within the space defined by the minimum and maximum x values of A. Additionally, the minimum y value of B lies within the space defined by the minimum and maximum y values of A. With simple bounds based collision detection this would register as a collision, when it clearly is not.

Step 1

The first step in this method is to determine the axes that we will be projecting our vertices onto. The separating axis theorem states that we must have an axis that is perpendicular to each of the edges of our two polygons.

Figure 3. The Eight Perpendicular Axes

As you can see, we end up with eight axes. You should also immediately see the benefits of using rectangles. Firstly, each edge has an opposite edge which shares an identical axis, we can take advantage of this to lower the number of axes that need checked to four. Secondly, the angle that exists between any two adjacent edges on a rectangle is 90 degrees. As such, for any edge of a rectangle, both of the adjacent edges are perpendicular to it. This means that we can calculate our four axes to be as such:

Axis1.x = A.UR.x - A.UL.x
Axis1.y = A.UR.y - A.UL.y
Axis2.x = A.UR.x - A.LR.x
Axis2.y = A.UR.y - A.LR.y
Axis3.x = B.UL.x - B.LL.x
Axis3.y = B.UL.y - B.LL.y
Axis4.x = B.UL.x - B.UR.x
Axis4.y = B.UL.y - B.UR.y

Meaning that Axis 1 is the resultant vector of the upper-right corner of A minus the upper-left corner of A and so on. This gives us four axes, each of which is perpendicular to two opposite edges of one of the rectangles, meaning that for each edge we have an axis that is perpendicular to it.

Figure 4. Our Four Axes

Step 2

The next step is to project the vectors representing the four corners of each rectangle onto each of the axes. If you know how to do matrix projections then you should have no problem doing this. If you understand vectors, but have forgotten how to do projections, then here is the equation for the projection of rectangle A’s upper-right corner onto Axis 1:

Here is the equation expanded out into scalar math and simplified:

It is important to note that the only difference between these two equations is that we’re multiplying by Axis 1’s x coordinate at the end of the first equation and we’re multiplying by Axis 1’s y coordinate at the end of the second equation. That will give you the x and y coordinates of A.UR projected onto Axis 1. As an example, let’s pretend that A.UR is at location (2, 6) and Axis 1 is represented by the vector (3, 4):

Therefore, in this example, the x coordinate of A.UR projected onto Axis 1 is 3.6 and the y coordinate is 4.8.

Figure 5. Vectors Projected Onto Axis 1

The Method Steps 3 & 4

Contents

	Introduction
	The Method Steps 1 & 2
	The Method Steps 3 & 4
	Optimizations

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