A Geometric Approach to the Reflections of Regular Maps

A reflection of a regular map on a Riemarm surface fixes some simple closed curves, which are called mirrors. Each mirror passes through some of the geometric points (vertices, face-centers and edge-centers) of the map such that these points form a periodic sequence which we call the pattern of the mirror. For every mirror there exist two particular conformal automorphisms of the map that fix the mirror setwise and rotate it in opposite directions. We call these automorphisms the rotary automorphisms of the mirror. In this paper we first introduce the notion of pattern and then describe the patterns of mirrors on surfaces. We also determine the rotary automorphisms of mirrors. Finally, we give some necessary conditions under which all reflections of a regular map are conjugate.