Plane Geometry - Bruce E. Shapiro

56 SECTION 13. INCIDENCE GEOMETRYExample 13.4 The Cartesian Plane. This is the traditional example weuse in geometry and it obeys all the rules of incidence geometry. Define apoint as an ordered pair of real numbers {x, y}. Then a line is the collectionof all points ax + by + c = 0 for some choice of real numbers a, b, c. Theusual notation for this set is R 2 .Example 13.5 Surface of a Sphere. Consider a unit sphere centered atthe origin in normal 3-space. The surface of this sphere is given by the setof all points {x, y, z} such thatx 2 + y 2 + z 2 = 1Define a point as any point on the surface of the sphere, and define aline as any great circle on the plane (a great circle is the intersection ofthe unit sphere with any plane that goes through the center of sphere; orequivalently, it is any circle on the sphere whose radius is equal to the radiusof the sphere, which is 1). This geometry does not obey incidence geometrybecause any two antipodal points (points at opposite poles of the sphere)are on an infinite number of common lines. This violates Incidence Axiom1. Furthermore, there are no parallel lines in this geometry because anytwo great circles meet.Example 13.6 The Klein Disk. Consider the interior of the unit diskcentered at the origin of R 2 . This is the set of all points such thatx 2 + y 2 < 1Define a point as any ordered pair of numbers (x, y) such that x 2 + y 2 < 1,and define a line as any the part of any Euclidean line that lies inside thiscircle. See figure 13.4. The Klein Disk obeys incidence geometry but doesnot obey Euclid’s fifth postulate.Definition 13.5 (Parallel Lines) Two lines l and m are said to be parallelif there is no point P such that P lies on both l and m. We denote this byl ‖ m.Euclid’s fifth axiom is equivalent to the following statement:Axiom 13.6 (Euclidian Parallel Postulate) For every line l and forevery point P ∉ l there is exactly one line m such that P lies on m andm ‖ l.Four point geometry and geometry of the Cartesian plane each satisfy theEuclidean Parallel Postulate.There are two other possible parallel postulates that are incompatible withthe Euclidean Parallel Postulate but which lead to consistent geometries.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012