Axioms and basic theorems for spherical geometry
How do our definitions, axioms, and basic theorems for Euclidean geometry have to be changed for spherical geometry?
Our model for spherical geometry is a sphere of radius 1. Thus the points of this geometry are points on the sphere, and the lines of this geometry are great circles on the sphere. A great circle (or geodesic) is the intersection of the sphere with a plane through the center of the sphere.
The distance between two points on the sphere is the length of the shortest great circle arc between the two points. The maximum distance between two points is π, half the circumference of the sphere. The distance between two points A and B is π if and only if A and B are antipodal, i.e. A and B are endpoints of a diameter of the sphere.
Angles are measured as in 3D Euclidean geometry. In other words, given a point O on the sphere and two points A and B, with A not equal to O and A not antipodal to O, and B not equal to O and B not antipodal to O, the angle AOB is the angle between the shorter great circle arcs OA and OB, which is the angle between their tangent vectors at O.
Area is also measured as in 3D Euclidean geometry. (The area of a region can be computed as a surface integral.) The area of the whole sphere is 4π.
Let's see how our axioms hold up in spherical geometry.
(1) Distance. This axiom holds in spherical geometry. But one has to realize that the distance between two points A and B is the length of the shorter of the two great circle arcs joining A and B. Thus the "triangle inequality" is not really about triangles -- it is about distance. The definition of a circle is unchanged, but the radius must be less than π. Note that the circle of radius r with center A is the same set as the circle of radius (π - r) with center the antipodal point A'.
(2) Line. Instead of saying that a line is a set parametrized by the real number line, we say that a line is a set parametrized by the unit circle in the (x,y) plane. (In other words, we use a circular ruler, as in the Lenart sphere kit.) And the distance between A and B corresponds to the length of the shorter circular arc on the unit circle. The line axiom holds in spherical geometry with this definition.
It is convenient to define a ray with endpoint A to be half of a great circle between A and its antipode A', where the ray contains the point A but not the point A'.
A triangle is a set consisting of three great circle arcs AB, BC, CA. Note that a triangle is not determined by its set of vertices, and a triangle, with ABC not on the same great circle, determines two regions, which we call complementary triangle regions. (Similarly, an n-gon is consists of n great circle arcs A1A2, A2A3, . . . , AnA1.)
(3) Angle. This axiom holds in spherical geometry.
(4) Between. If A and B are not antipodal, we say that C is between A and B if C lies on the shorter geodesic arc joining A and B. If A and B are antipodal, we say that every point is between A and B! Betweenness for rays is defined just as in Euclidean geometry, but with our new definition of ray. With these modifications, the betweenness axiom holds in spherical geometry.
(5) Side-Angle-Side Congruence. This is true in spherical geometry.
(6) Parallel. There are no parallel lines in spherical geometry! Two distinct lines always intersect in two points, and these points are antipodal.
(7) Area. This axiom holds in spherical geometry, except for the fact that there are no rectangles in spherical geometry. Instead of a formula for the area of a rectangle, we have the angle excess formula: The area of an n-gon is the sum of the angles of the n-gon minus (n-2)π.
Basic theorems in spherical geometry:
Set 1: Isosceles triangles - All these theorems are true in spherical geometry.
Set 2: Parallels - None of these theorems are true in spherical geometry.
Set 3: Congruence - Both ASA congruence and SSS congruence are true in spherical geometry. (There is a proof of SSS congruence that does not use parallels or the triangle angle sum theorem.)
Set 4: Area - There are no parallelograms in spherical geometry. The Euclidean formula for the area of a triangle does not hold in spherical geometry.
Set 5: None of the similarity theorems are true in spherical geometry, except when the similarity ratio is 1, i.e. the only similar triangles are congruent triangles! The Euclidean Pythagorean Theorem does not hold in spherical geometry. (But there is a generalization of the Pythagorean Theorem using spherical trigonometry.)
11/22/08