MASS IN HYPERBOLIC GEOMETRY

Abstract

Archimedes computed the center of mass of several regions and solid bodies [Dijksterhuis], and this fundamental physical notion may very well be due to him. He based his investigations of this concept on the notion of moment as it is used in his Law of the Lever. A hyperbolic version of this law was formulated in the nineteenth century leading to the notion of a hyperbolic center of mass of two point-masses [Andrade, Bonola]. In 1969 Perron extended the notions of mass and center of mass to arbitrary regions of hyperbolic space. In 1987 Gal’perin proposed an axiomatic definition of the center of mass of finite systems of point-masses in Euclidean, hyperbolic and elliptic n-dimensional spaces and proved its uniqueness. Ungar [2004] used the theory of gyrogroups to show that in hyperbolic geometry the center of mass of three point-masses of equal mass coincides with the point of intersection of the medians, a fact that had already been noted by Perron. Some information regarding the centroids of finite point sets in spherical spaces can be found in [Fog, Fabricius-Bjerre].

In this article we begin by offering yet another physical motivation for the hyperbolic Law of the Lever and summarize Perron’s treatment of the subjects of mass and centers of mass (centroids). The masses and centroids of several geometric objects are derived. Surprisingly, the hyperbolic mass formulas are quite similar to the Euclidean ones whereas, as is well known, the formulas for hyperbolic area and volume look nothing like their Euclidean analogs.