Parametrization of Meshes

Many algorithms rely on mesh parametrization. In particular, the mapping from a mesh to a 2D domain (and vice versa) is essential to rendering 3D models and an essential component of e.g. remeshing. In this chapter, we study several algorithms for such flattening of a patch of disk topology. The basic algorithm—quite similar to mesh smoothing—is introduced, and various vertex weights are covered, ranging from simple uniform weights to mean value and harmonic weights which result in less distortion. Finally, we discuss the so called natural boundary conditions which allow us to flatten the mesh with minimum angle distortion in the least squares sense.