The Third Motivation for Spherical Geometry

Abstract

Historically, spherical geometry has developed mainly on the terrestrial globe and the celestial globe. In this paper, we will introduce the third globe “visual globe” centered at our eyes. When we see the external world, we are under an illusion that our view screen is a plane. Locally it is true, however, globally it is more natural to think that our view screen is a sphere centered at our eyes. In this concept, we will study “visual angles” of a rectangle in the space. An angle in the space changes its visual angle according to our viewpoint. This visual angle is easily measured on the visual globe and there is a very simple relation among the visual angles of a rectangle. We will also realize the visual angle on a two dimensional Euclidean plane along with the dynamic geometry software Cabri II Plus. Furthermore, we will find out that the simple relation above is also true in hyperbolic geometry.