A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing

Topology: How Are Objects Connected? 173

Figure 11.12: The Euler characteristic of a polygonal mesh P of genus g

with V vertices, E edges, and F faces is defined as χ(P ) = V − E + F ,

and always equals 2(1−g). Comparing the left and the middle figures, we

observe that for a triangle mesh, each step of a face-subdivision adds one

new vertex, three new edges, and two additional faces, thus leaving χ(P)

unchanged. By induction, we argue that any number of such subdivision

steps keeps the Euler characteristic unchanged. It changes only due to

topological surgeries that modify the genus of the surface, for example,

due to a crack, as seen in the right figure. ...