Local Geodesic Parametrization: An Ant's Perspective
Abstract
Two dimensional parameterizations of meshes is a dynamic field of
research. Most works focus on parameterizing complete surfaces,
attempting to satisfy various constraints on distances and angles
and produce a 2D map with minimal errors. Except for developable
surfaces no single map can be devoid of errors, and a
parametrization produced for one purpose usually doesn't suit
others.
This work presents a different viewpoint. We try and acquire the
perspective of an ant living on the surface. The point on which it
stands is the center of its world, and importance diminishes from
there onward. Distances and angles measured relative to its
position have higher importance than those measured elsewhere.
Hence, the local parametrization of the geodesic neighborhood
should convey this perspective by mostly preserving geodesic
distances from the center. We present a method for producing such
overlapping local-parametrization for each vertex on the mesh. Our
method provides an accurate rendition of the local area of each
vertex and can be used for several purposes, including clustering
algorithms which focus on local areas of the surface within a
certain window such as Mean Shift.