Parameterization and Surface Reconstruction

In this paper we study parameterization as a tool for both constructing and smoothing spatial triangulations. Most of the parameterization methods we study are based on convex combination maps, which have the property that the image of each interior vertex is a convex combination of its neighbours. When mapping an existing triangulation, this latter property ensures that whenever the image of the map is convex, the map is one-to-one, and thus a homeomorphism. At the same time, convex combination maps are very general because every one-to-one piecewise linear map is a convex combination map. Suitable convex combination maps for parameterization arise naturally as discretizations of harmonic maps, but can also be constructed explicitly, in order to satisfy important properties such as linear precision.