  Mapping a shape to some parametric domain is a fundamental tool in graphics
and scientific computing. In practice, a map between two shapes is commonly
represented by two meshes with same connectivity and different embedding. The
standard approach is to input a meshing of one of the two domains plus a
function that projects its boundary to the other domain, and then solve for the
position of the interior vertices. Inspired by basic principles in mesh
generation, in this paper we present the reader a novel point of view on mesh
parameterization: we consider connectivity as an additional unknown, and assume
that our inputs are just two boundaries that enclose the domains we want to
connect. We compute the map by simultaneously growing the same mesh inside both
shapes.This change in perspective allows us to recast the parameterization
problem as a mesh generation problem, granting access to a wide set of mature
tools that are typically not used in this setting. Our practical outcome is a
provably robust yet trivial to implement algorithm that maps any planar shape
with simple topology to a homotopic domain that is weakly visible from an inner
convex kernel. Furthermore, we speculate on a possible extension of the
proposed ideas to volumetric maps, listing the major challenges that arise.
Differently from prior methods, for which we show that a volumetric extension
is not possible, our analysis leaves us reasoneable hopes that the robust
generation of volumetric maps via compatible mesh generation could be obtained
in the future.
