ABSTRACT
This chapter reviews some important properties of Laplacians, smooth and discrete. The Laplacian is perhaps the prototypical differential operator for various physical phenomena. The properties mentioned so far play an important role in applications; specifically, in the context of barycentric coordinates, they give rise to mean value coordinates and harmonic coordinates. Discrete Laplacians can be defined on simplicial manifolds or, more generally, on graphs. Positive edge weights are a natural choice if weights resemble transition probabilities of a random walker. Discrete Laplacians with positive weights are always positive semidefinite (Psd) and, just like in the smooth setting, they only have the constant functions in their kernel provided that the graph is connected. The construction of discrete Laplacians based on inner products can be extended from simplicial surfaces to meshes with (not necessarily planar) polygonal faces; for such an extension that is similar to the approach considered for planar polygons.
