Plane Geometry - Bruce E. Shapiro

SECTION 53. SPHERICAL GEOMETRY 315Figure 53.2: Betweenness fails on a circle.The angle between two spherical lines is defined as the angle betweentangent lines to the great circles in R 3 .In neutral geometry (including Euclidean and Hyperbolic geometry) anytwo distinct lines can only intersect at a single point; consequently, thepolygon with the least number of sides is a triangle.In spherical geometry it is possible to define a two-sided polygon, becauseany two distinct lines intersect at two antipodal points. This object iscalled a lune. A lune is sometimes called a sector; it resembles a sector ofan orange.Theorem 53.8 The two interior angles of a lune are congruent.Figure 53.3: A spherical lune resembles an orange segment.PolesαRevised: 18 Nov 2012 « CC BY-NC-ND 3.0.