Circular Reasoning

Abstract

This chapter examines the unique characteristics of the sphere. We live on a small part of a large sphere. Our senses tell us it is flat and that geometric relationships are Euclidean with straight lines and distances measured in feet of meters. In this world, there are parallel lines such as roads and railroad tracks and objects come in various sizes and can enlarge or reduce any image, reserving all the angles as we see them. But on a sphere, none of these impressions are true. For example, we can make an infinite number of similar triangles simply by changing the ratio of the sides. But not on a sphere, all triangles are unique—there are no similar triangles. They are either the same or they are congruent. This chapter also discusses how great circles define point intersections, spherical polygons, such as lunes, gores, and quadrilaterals. It also shows how their surface and dihedral angles are measured and areas calculated. There are several coordinate systems for representing points on a sphere. Some spherical problems are easier to solve in one system than another and often, one system is converted to another. These conventions and techniques are explained.