A new method for researching and constructing spherical bicentric polygons based on geometric mapping

Abstract

A bicentric polygon is a special planar polygon that has both a circumcircle and an incircle. Many interesting properties of a planar bicentric polygon have been discovered. However, searching the geometric properties of a spherical bicentric polygon with elementary geometric method is very difficult. In this paper, we propose a special mapping for a pair of circles between a plane and a sphere. This mapping provides us a new method to research the spherical bicentric polygons. With this mapping, we can translate the Fuss relation, which describes the relation of the circumcircle and incircle of a bicentric polygon, from a planar bicentric polygon to a spherical polygon. Based on the mapping, we can derive the geometric properties of spherical bicentric polygons from the planar bicentric polygons, and construct a series of spherical bicentric n-polygons from a planar bicentric n-polygon.