**5.3 Hilbert's Axioms
Modified for
Plane Elliptic Geometry
**

—

** Introductory Notes.**
Consider
three distinct collinear points

In order to use the Riemann Sphere model with the following axiom set, we identify each pair of antipodal points as a single point since two distinct lines are incident with a unique point, i.e., the Modified Riemann Sphere model.

**Undefined Terms** *point,
line, lie,
separate, *and
*congruence.*

I.1. For every two distinct points

AandB,there exists a linemthat contains each of the pointsAandB.

I.2. For every two distinct pointsAandB,there is no more than one linemthat contain each of the pointsAandB.

I.3. There exist at least two distinct points on a line. There exist at least three points that do not lie on a line.

I.4.(Elliptic Parallel Postulate)For any two distinct linesmandn, there exists a unique pointAcontained by linesmandn.

*Group II. Axioms of Separation (or Axioms of Order)
*

II.1. For any three distinct points

A, BandCon a linem,there is a pointDonmsuch thatAandBseparatesCandD(denoted (A, B|C, D).II.2. If (

A, B|C, D), then the pointsA, B, C,andDare distinct and collinear.II.3. If (

A, B|C, D), then (B, A|C, D) and (C, D|A, B).

II.4. If (A, B|C, D), thenAandCdoes not separateBandD.

II.5. For any four distinct collinear pointsA, B, C,andD,(A, B|C, D) or (A, C|B, D) or (A, D|B,C).

II.6. For any five distinct collinear pointsA, B, C, D,andE,if (A, B|D, E), then either (A, B|C, D) or (A, B|C, E).

II.7. Separation of points is invariant under a perspectivity, i.e., if (A, B|C, D)andthere is a perspectivity mappingA, B, C,andDon lineptothe corresponding pointsA', B', C',andD'on linep', then (A', B'|C', D').

Since in neutral geometry the definitions of segments, rays, angles, and triangles all depended on betweenness of points, the definitions all need to be revised based on the separation of points.

** Exercise 5.1.1. ** Use the concepts from the separation axioms to
write definitions for each of the following.

(a) segment

(b) ray (Note that in the spherical model a ray is a line.)

(c) angle (Note that at the vertex of an angle there are four
angles. Consider how to distinguish which is the particular angle desired.)

(d) triangle (Note that three noncollinear points determine more
than one triangle.)

*Group III. Axioms of Congruence*

III.1. If

A, Bare two points on a linem, andA'is a point on the same or on another linem'then it is always possible to find a pointB'on a given side of the linem'throughA'such that the segmentABis congruent to the segmentA'B'(denotedAB≅A'B').

III.2. Congruence of segments is an equivalence relation.

III.3. If two segments are congruent, then their mutually complementary segments are congruent.

III.4. On the linemletABandBCbe two segments which except forBhave no point in common. Furthermore, on the same or on another linem'letA'B'andB'C'be two segments which except forB'also have no point in common. In that case, ifAB≅A'B'andBC≅B'C',thenAC≅A'C'.III.5. Let ∠(

h,k)be an angle andm'a line andlet a definite side ofm'. Leth'be a ray on the linem'that emanates from the pointO'. Then there exists one and only one rayk'such that the angle ∠(h,k)is congruent to the angle ∠(h',k')and at the same time all interior points of the angle ∠(h',k')lie on the given side ofm'.Symbolically ∠(h, k)≅∠(h',k').Every angle is congruent to itself.

III.6. If for two trianglesABCandA'B'C'the congruencesAB≅A'B', AC≅A'C',∠BAC≅ ∠B'A'C'hold, then the congruence ∠ABC≅ ∠A'B'C'is also satisfied.

** Group IV. Axiom of Parallels** - Elliptic
geometry has no parallel lines and hence the elliptic parallel postulate
should probably be called the elliptic

See Group I. Axioms of Incidence - I.4.

(Elliptic Parallel Postulate)For any two distinct linesmandn, there exists a unique pointAcontained by linesmandn.

*Group V. Axiom of Continuity*

V. (

Dedekind's Axiom) Assume the set of all points on a segmentABis the union,S_{1}US_{2}, of two nonempty subsets of segmentABsuch that no point ofS_{1}separates two points ofS_{2}. Then there is a unique pointCin the interior of segmentABsuch that (A, C|P_{1},P_{2}) if and only ifP_{1}is inS_{1}andP_{2}is inS_{2}andCis neitherP_{1}norP_{2}.

**Defined Terms**

- Consider two distinct points
*A*and*B*on a line*m,*then there are points*C*and*D*on line*m*where (*A, B*|*C, D*). The union of the set of the two points*A*and*B*and the set of all points*X*where (*A, B*|*C, X*) is called*segment**AB/C*. The points*A*and*B*are called the*end**points*of the segment*AB/C*. The set of all points*X*where (*A, B*|*C, X*) is called*interior of segment**AB/C*. The set of all points*Y*where (*A, B*|*Y, D*) is called*exterior**of segment**AB/C*. The segment*AB/C*and segment*AB/D*are called*mutually complementary segments.*Note that three points are often needed to name a segment since points*A*and*B*define two mutually complementary segments, segment*AB/C*and segment*AB/D.*Further note that the interior of segment*AB/C*is the exterior of segment*AB/D.* - Let
*A, A', O, B*be four points of a line*m*such that*O*and*B*do not separate*A*and*A'*. The points*A, A'*are then said to lie*on the line m**on one and the same side of the point O*and the points*A, B*are said to lie*on the line m on different sides of the point O*. The totality of the points of the line*m*that lie on the same side of*O*is called a*ray*emanating from*O.*The point*O**is the**vertex**of the ray emanating from**O**and**A*is a*relative point*of the ray emanating from*O.* - Let
*h*and*k*be any two distinct rays emanating from*O*and lying on distinct lines. Further, let*AB*be a segment with*A*and*B*relative points of rays*h*and*k,*respectively, distinct from*O.*The pair of rays*h, k*is called an*angle*and is denoted by ∠(*h,k)*or ∠(*k,h)*or*∠**AOB*. Let*C*be an exterior point of segment*AB*. All points that lie in the interior of segments*XY/Z*where*X*and*Y*are relative points of*h*and*k*, respectively*,*and*Z*is a point on line*OC*are said to lie in the*interior*of the angle ∠(*h,k) or*.*∠**AOB* - Let
*AB, BC,*and*CA*be three segments where*A, B,*and*C*are three points which do not lie on the same line. The union of the three segments*AB, BC,*and*CA*is a*triangle*called*triangle ABC. (Note that the noncollinear points A, B, and C define more than one triangle.)*

** Exercise 5.1.2. **(a) Prove a
segment has at least three points.

(b) Prove a segment has infinitely many points.

* Exercise 5.1.3.*
Show that the Riemann Sphere where each pair of antipodal points is a point
satisfies each of the axioms.

_____________

Coxeter, H. S. M., *Non-Euclidean Geometry,* Fourth
Edition. University of Toronto Press, 1961.

Greenberg, Marvin J., *Euclidean and Non-Euclidean
Geometries: Development and History*. San Francisco: W. H. Freeman and
Company, 1974.

Hilbert, David, *Foundations of Geometry (Grundlagen der Geometrie)*,
Second English Edition trans. by Unger,L. LaSalle: Open Court Publishing
Company, 1971 (1899).

Riemann geometry. L.A. Sidorov (originator), *Encyclopedia of Mathematics.*
URL:
http://www.encyclopediaofmath.org/index.php?title=Riemann_geometry&oldid=15725

© Copyright 2013 - Timothy Peil |