What's the most efficient way to detect triangle-triangle intersections?

Question

How can I tell whether two triangles intersect in 2D Euclidean space? (i.e. classic 2D geometry) given the (X,Y) coordinates of each vertex in each triangle.

Answer 1

As stated, you'll need to check that a point is inside a triangle. The simplest way to check if a point is inside a closed polygon is to draw a straight line in any direction from the point and count how many times the line crosses a vertex. If the answer is odd then the point is in the polygon, even, then it's outside.

The simplest straight line to check is the one going horizontally to the right of the point (or some other perpendicular direction). This makes the check for vertex crossing nearly trivial. The following checks should suffice:

• Is the point's y-coordinate between the y-coordinates of the two end points of the vertex? No, then doesn't cross.

• Is the point's x-coordinate greater than the furthest right end point of the vertex? Yes, then doesn't cross.

• Is the point's x-coordinate less than the furthest left end point of the vertex? Yes, then does cross.

• If the cases above fail, then you can use the cross product of the vector representing the vertex and a vector formed from the end of the vertex to the point. A negative answer will indicate the point lies on one side of the vertex, a positive answer on the other side of the vertex, and a zero answer on the vertex. This works because a cross product involves taking the sine of two vectors.