# Compute mesh of the projection of a 3D surface triangulation

Given a triangulated surface in $\Bbb{R}^3$ we can simply project it on a plane. This will result in a family of triangles which do not form a mesh of the projection for the following reasons:

• each point of the projection is covered by at least two triangles (see the picture).
• the orientation of the triangles is not right
• things may get more complicated for more complex surfaces

Even if the projection is not a mesh, every point of the projected surface is inside one of the triangles, so this could give a good idea of the projection.

Is there a way to extract/construct a mesh of the projection, starting from the projection of the surface?

• I guess you could try to compute the convex hull of the projected point set and mesh it, but you'd still have to figure out how to deal with multiply-connected surfaces like the torus in your example. – sssssssssssss Mar 25 '18 at 22:30
• I think that you can take your projected mesh and then some cleanup. Using, for example, MeshLab: meshlab.net – nicoguaro Mar 25 '18 at 23:46

• Thank you for the ideas. Using Delaunay directly on the projected points fills the holes in the projection. I'll try looking more closely, since Matlab has the ability to pass some constraints into delaunayTriangulation. – Beni Bogosel Mar 26 '18 at 13:45