Intersecting a minkowski difference from the origin in a direction, how do i find the face im intersecting?

Basically i have a set of vertices on the hull of a minkowski difference of two polyhedra. I want to find the distance from the origin to the hull in some arbitrary predetermined direction. Heres a quick 2D sketch:

So the issue is finding what triangular face/plane the ray is going to intersect. Once i have that plane i simply do a line/plane intersection test. My issue is finding the correct face/plane. Any ideas? Is there some set of dot product/cross product/triple product tests i can do to determine it? Or is it more complicated then that?

If your wondering what this is for im using a GJK algorithm to determine w开发者_如何学JAVAhether two objects are intersecting (which i've got working). If there is a collision i would like to find the penetration depth in a particular direction (which will be the direction of motion of the object).

Project the polyhedron in the direction of the ray, and your problem reduces to 2D, and finding which triangle encloses the origin. To test a single triangle, consider whether a given directed line segment (AB) is going clockwise or counterclockwise with respect to the origin. This is easy to determine with a simple cross-product test: it's counterclockwise iff A x (B-A) > 0.

If all three sides of a triangle have the same sense (clockwise or counterclockwise) then the triangle encloses the origin and that's the face you want.

EDIT:
Since your polyhedron is a hull it is convex, And since it is convex you can search the surface in an efficient way. You can traverse the edges in a very simple "walk uphill/downhill" way to find the two vertices farthest along the the ray in either direction. Then after you project the poyhedron you can start from these two points and do a similar climb toward the origin. This will be O(sqrt(n)).