Suppose I have a set of points in 3D which are all co-planar, and which describe the vertices of a convex polygon. I know the coordinates of all of these vertices, I know the unit normal to the plane of the polygon, and I know the order (in a right-hand sense with respect to the normal) in which the vertices go around the polygon.
I also have a polyhedron, which is in general non-convex, and the surface of which is made of triangles. It is a triangle mesh like the one shown here. It describes a completely enclosed volume (no holes in the surface). I know the coordinates of all the vertices of all the triangles that make up the polyhedron. For a given vertex, I know the list of which triangles it is a vertex of.
One other thing I know, which may or may not be useful, is that the polygon lies "in the vicinity of" a particular pair of vertices of the polyhedron, say the ones with indices $i$ and $j$. So if there were to be some kind of a search of the surface of the polyhedron, that would be the place to start.
My task is to find the area of the intersection between the polygon and the polyhedron. I believe this is a well-defined quantity. I need an algorithm to calculate this, hopefully efficiently but at this point I'll take anything.
I can classify each polygon vertex as either inside or outside the polyhedron. I've written a ray-casting algorithm that computes this fairly accurately and efficiently. That means that each edge of the polygon can be classified as either "both vertices inside", "both vertices outside", or "one in, one out." I can calculate the point of intersection between a given polygon edge and the polyhedron by checking a bunch of nearby triangles.
But that's not enough. For instance, just because both vertices of an edge are inside doesn't mean the whole edge is inside. A "finger" of the polyhedron might intersect an edge without touching either of its endpoints. Furthermore, the polyhedron might intersect the polygon in such a way that it doesn't touch any of the edges at all. Clearly, one must also calculate the intersections between the triangles' edges and the polygon as well.
It gets a bit too complicated for me at that point, and I'm wondering if there is some established algorithm for this?
Perhaps something like: Project the polyhedron onto the plane containing the polygon and then... I don't know.