Spherical Geometry III

A spherical ellipse is similarly defined as an ellipse in the plane. It is the locus of a point X on \({\mathbb S}\) for which the sum of the distances from two fixed points to X is constant. A regular spherical polygon is a cyclic polygon whose edge-lengths are all equal. Using spherical ellipses in this chapter, we show that among cyclic n-gons inscribed in a cap of fixed spherical radius, regular n-gons have the maximum perimeter, and among the n-gons on \({\mathbb S}\) with fixed perimeter, regular n-gons have the maximum area.