An Analogue of Ptolemy's Theorem in Spherical Geometry

Abstract

(pl, P2, . . . X Pn+2) = COS(pip1/P) where i, j=1, 2, * , n+2, and Pi, P2, P * p+2 are n+2 points of the ndimensional spherical Sntp (the surface of a sphere of radius p in euclidean (n+1)-dimensional space, with shorter arc metric, where PPj-=Pipi denotes the distance of the points pi, pj). Haantjes [4] gave a proof of the spherical analogue of the ptolemaic inequality, and [5 ] developed techniques which give a new proof of Ptolemy's Theorem and its converse in the euclidean plane. It is further stated [5] that these techniques give proofs of the analogue of Ptolemy's Theorem and its converse in the spherical and hyperbolic planes. However, his analogue of the converse of Ptolemy's Theorem in the hyperbolic plane is false; e.g., the determinant Isinh2 ppj1/21 =0, where i, j= 1, 2, 3, 4 for four points on a horocycle in the hyperbolic plane (see [6]), and consequently the result for the spherical plane is not immediate. The purpose of this note is to give the geometrical significance of the determinant