Exit angle and point of a segment that lies on a triangle in 3D

Question

Knowing the vertices of a 3D triangle, and the x, y coordinates of the projection on the horizontal plane of a point E belonging to the triangle. Also the angle alpha is given, representing the angle respect to the edge AB of a segment that lies on the same plane of the triangle. I would like to find out 3 things:

• for a given alpha, on which side is F
• what is the angle created by the "exit" side with the segment EF (considering always the following vertex in a clockwise way)
• the length of EF
• The length of the segment BF if BC is the exit side (clockwise again)

This is though... and I want to see how it will perform. Thank you.

grid http://www.keplero.com/upps/mesh.jpg

Answer 1

1. Find point E. Draw a line perpendicular to the horizontal plane, passing through E's projection. Point E is the intersection of that line and the plane the triangle lies on. (if the triangle's plane is perpendicular to the horizontal plane, you don't have enough information to find E.)
2. Perform a transformation on points A, B, C, E so that they lie on the horizontal plane. Use only rotations and translations so the angles and distances are preserved. With this step, the problem can be solved in only two dimensions, which simplifies things.
3. Draw a ray extending out from E, that has angle alpha with respect to AB.
4. For each of AB, BC, CA, determine whether the ray extending from E intersects it. Point F is the intersection of the ray and whichever line segment it intersects. (If the ray passes through a vertex of the triangle rather than an edge, then you may not be able to get meaningful answers your questions involving the "exit" side.)
5. Using the position of F, determine the answers of each of your bullet points.
6. Optionally, perform the reverse of the transformation done in step 2, to get the true position of F.