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

114 Laplace and Poisson

recall here that in the simplest form, a discrete mesh Laplacian of

a vertex f ∈ R

3

, denoted by ∆(f), is

∆(f) = f −

1

d

d

X

k=1

f

k

,

where d is the degree (number of neighbors) of the node f, and f

k

are neighbors of vertex f.

The above equation enables generalizing the principle expressed

in Equation (7.7) to irregular meshes. Setting the Laplacian ∆(f)

of a given mesh to zero while imposing some boundary conditions

leads to the construction of a minimal surface that passes through

some preset locations. The surface will look like a membrane and

have spikes around the fixed locations. On the other hand, min-

imizing the norm of ∆(f) of