Determining if a point is inside a polyhedron

Question

I\'m attempting to determine if a specific point lies inside a polyhedron. In my current implementation, the method I\'m working on take the point we\'re looking for an array of

Answer 1

If your mesh is concave, and not necessarily watertight, that’s rather hard to accomplish.

As a first step, find the point on the surface of the mesh closest to the point. You need to keep track the location, and specific feature: whether the closest point is in the middle of face, on the edge of the mesh, or one of the vertices of the mesh.

If the feature is face, you’re lucky, can use windings to find whether it’s inside or outside. Compute normal to face (don't even need to normalize it, non-unit-length will do), then compute dot( normal, pt - tri[0] ) where pt is your point, tri[0] is any vertex of the face. If the faces have consistent winding, the sign of that dot product will tell you if it’s inside or outside.

If the feature is edge, compute normals to both faces (by normalizing a cross-product), add them together, use that as a normal to the mesh, and compute the same dot product.

The hardest case is when a vertex is the closest feature. To compute mesh normal at that vertex, you need to compute sum of the normals of the faces sharing that vertex, weighted by 2D angles of that face at that vertex. For example, for vertex of cube with 3 neighbor triangles, the weights will be Pi/2. For vertex of a cube with 6 neighbor triangles the weights will be Pi/4. And for real-life meshes the weights will be different for each face, in the range [ 0 .. +Pi ]. This means you gonna need some inverse trigonometry code for this case to compute the angle, probably acos().

If you want to know why that works, see e.g. “Generating Signed Distance Fields From Triangle Meshes” by J. Andreas Bærentzen and Henrik Aanæs.