Regular map smoothing

Abstract

A regular map is a family of equivalent polygons, glued together to form a closed surface without boundaries which is vertex, edge and face transitive. The commonly known regular maps are derived from the Platonic solids and some tessellations of the torus. There are also regular maps of genus greater than 1 which are traditionally viewed as finitely generated groups. RMS (Regular Map Smoothing) is a tool for visualizing a geometrical realization of such a group either as a cut-out in the hyperbolic space or as a compact surface in 3−space. It provides also a tool to make the resulting regular map more appealing than before. RMS achieves that by the use of a coloring scheme based on coset enumeration, a Catmull-Clark smoothing scheme and a force-directed algorithm with topology preservation.