Am I understanding correctly? I don't have the necessary background for the paper, but it sounds like in 3D space they capture the x and y coordinate of each point (so like an orthographic projection, not the screen coordinates) but for the z coordinate instead of storing actual values for each point they store the difference between the z coordinates of the endpoints of each edge.

That sounds like the object would still only be viewable from one direction, but when doing things like the heat diffusion you can consider each triangle to be at whatever depth makes sense for it to be touching the triangle you're reaching it from. So the same bit of geometry could be at different depths depending how you get there when movimg along the surface (a bit like a Riemann surface, I suppose). Is that right?