Plane Geometry - Bruce E. Shapiro

Section 53Spherical GeometryNow we are ready to example geometry confined to the surface of the unitsphere S. We will assume that the sphere is embedded in a Euclidean R 3space and centered at the origin.Definition 53.1 A great circle on S is the locus of points formed by theintersection of S with any plane in R 3 passing through the center of S.Definition 53.2 A spherical line on S is a great circle.Then we can – almost – say that two points define a line.restrict ourselves to omit points at poles:We have toDefinition 53.3 To points A and B ∈ S are said to be antipodal if theylie on a diameter of S.The problem with antipodal points is that there are infinite number of greatcircles connecting them - simply because they are collinear with the centerof S in R 3 there are an infinite number of planes containing the diameter.Then the axioms of incidence on a sphere become:Axiom 53.4 For any two points A, B ∈ S that are not antipodal, there isa unique spherical line connecting them.Axiom 53.5 Two distinct spherical lines intersect at precisely two antipodalpoints.As an immediate consequence we have the following result:Corollary 53.6 Parallel spherical lines do not exist.Thus spherical geometry is a model for the elliptic parallel postulate.313