Schultz - A system of Axioms for Hyperbolic Geometry.pdf - Topologia i Geometria - renvox

2008 Birkhauser Verlag Basel/Switzerland

0047-2468/010185-8, published online 21 November 2008

DOI 10.1007/s00022-008-1903-9

Journal of Geometry

A System of Axioms for Hyperbolic Geometry

John W. Schutz

This paper is dedicated to my parents,

Edith Schutz ( nee Edit Kertesz, 9 / 09 / 1917 Budapest – 4 / 07 / 2005) and

Steven Joseph Schutz ( Istvan Jozsef Schutz, b. 12 / 05 / 1912 , Budapest )

Abstract. Three–dimensional hyperbolic geometry is characterized using ax-

ioms of order, incidence, dimension, continuity and, instead of an axiom of

parallels, there is an axiom of “rigidity” and, rather than several axioms of

congruence, there is one axiom of symmetry. It is claimed that this system of

axioms is simpler than the system of independent axioms of Moore [15].

Mathematics Subject Classiﬁcation (2000). 50A10, 51M10.

Keywords. Axioms, hyperbolic geometry.

1. Introduction

Hyperbolic geometry will be developed from an axiomatic system which can be

shown to specify the interior of an ellipsoid with the projective cross–ratio as an

invariant metric.

Projective geometry was axiomatised by von Staudt [25,26] and described in terms

of homogeneous coordinates by Plucker [17,18]. Cayley [6] showed that the projec-

tive cross–ratio could be used as a projectively invariant measure of distance and

then Klein [13] developed the ﬁrst model of hyperbolic geometry which consisted

of the interior of an ellipse in which line segments are equipped with the pro-

jective cross-ratio as an invariant measure of distance. The development of these

ideas is described in considerable detail by Torretti [27] and these and subsequent

developments in non–Euclidean geometry are described by Karzel and Kroll [12]

and reviewed by Karzel [11]. A complete description of hyperbolic geometry and

projective metrics in terms of homogeneous coordinates is given by Busemann and

Kelly [5].

J. geom. 90 (2008), 185–192

186

J. W. Schutz

J. Geom.

Klein [14] ﬁrst expounded the idea that projective geometry could be developed

as an embedding space for a a geometry of “points” and “lines” with a relation of

“intermediacy”. Veblen [28] used this approach to specify Euclidean and hyperbolic

geometry with an axiomatic system whose twelve axioms allow the given geometry

to be embedded in a three–dimensional projective space over the reals with a metric

deﬁned by the choice of a polarity on the “plane at inﬁnity”.

In the present axiomatic system we consider undeﬁned sets of “points” and “lines”

with a relation of “intermediacy” which satisfy the ﬁrst eleven (of twelve) axioms

of Veblen, together with one additional relation of “congruence” and two further

axioms of “isotropy” and “rigidity”. The ﬁrst eleven axioms correspond to those

of Veblen and imply that the geometry may be considered as a convex region of

projective space. The axiom of “isotropy” then restricts the region to be either

an ane subspace or the interior of an ellipsoid, by a theorem of Aleksandrov [1],

Busemann [4] and Schutz [24]. Finally the axiom of “rigidity” excludes the ane

subspace and the only remaining possibility is the Klein model of hyperbolic ge-

ometry in which the metric is deﬁned by the “motions” or automorphisms of the

space.

Even though the the cited results of Aleksandrov and Busemann use deep results

from transformation group theory, the result of Schutz uses standard results of

Euclidean, ane and projective geometry. Consequently hyperbolic geometry can

be developed axiomatically using elementary techniques.

2. The system of axioms

The name “ordered geometry” has been used by Coxeter [7] to describe the “ge-

ometry of serial order” which can be developed in terms of a single undeﬁned

ternary relation of order called “betweenness” or “intermediacy”. This geometry

can be developed without any mention of congruence and without any axiom of

uniqueness of parallelism. The discussion of Coxeter is based on that of Veblen [28]

but is more detailed and provides detailed proofs with references to acknowledge

the many sources [2, 3, 8, 9, 16, 21–23, 29]. Veblen actually states a total of twelve

axioms but the twelfth axiom is not required for “ordered geometry”.

The geometry resulting from Veblen’s [28] Axioms I to XI is the geometry of

incidence and order of a three–dimensional convex open domain. It will be shown

in Lemma 1 that this geometry can be embedded as a convex open domain in a

three–dimensional projective space over the reals. Any other equivalent system of

axioms, such as the axiomatic system of Coxeter [7], could also be used. Veblen

claims and establishes the mutual independence of the ﬁrst twelve axioms and, in a

subsequent article, Moore [15] also uses a modiﬁed system of mutually independent

axioms.

Vol. 90 (2008)

A System of Axioms for Hyperbolic Geometry

187

2.1. Undeﬁned basis

Hyperbolic geometry is

= P

, [ ... ]

where

is a set whose elements are called points ,

is a set of subsets of

called

lines and [ ... ] is a ternary relation on the set of points of

called a betweenness

relation .

2.2. Axioms of incidence and order

Axiom 1. There exist at least two distinct points.

Axiom 2. If points A, B, C are in the order [ ABC ] , they are in the order [ CBA ].

Axiom 3. If points A, B, C are in the order [ ABC ] , they are not in the order

[ BCA ].

Axiom 4. If points A, B, C are in the order [ ABC ], then A is distinct from C .

Axiom 5. If A and B are any two distinct points, there exists a point C such that

A, B, C are in the order [ ABC ] .

For distinct points A, B ,the line

AB :=

{

A, B

}∪ X :[ ABX ] , [ AXB ] , [ XAB ] ,X

∈P .

The points X in the order [ AXB ] constitute the segment

where A and B are

the end-points of the segment. The ray

{

A, B

}∪|

|∪{

X :[ ABX ]: X

∈P}

is the set of all points on the half–line from A in the direction of B .

Axiom 6. If points C and D ( C

= D ) lie on the line AB , then A lies on the line

CD .

Axiom 7. If there exist three distinct points, there exist three points A , B , C not

in any of the orders [ ABC ] , [ BCA ], or [ CAB ].

Three distinct points not lying on the same line are the vertices of a triangle ABC,

whose sides are the segments

, and whose boundary consists of its

vertices and the points of its sides.

Axiom 8. If three distinct points A, B ,and C do not lie on the same line, and D

and E are two points in the orders [ BCD ] and [ CEA ], then a point F exists in

the order [ AF B ] and such that D, E, F lie on the same line.

A point O is in the interior of a triangle if it lies on a segment, the end-points of

which are points of diﬀerent sides of the triangle. The set of such points O is the

interior of the triangle.

If A, B, C form a triangle, the plane ABC consists of all points collinear with any

two points of the sides of the triangle.

188

J. W. Schutz

J. Geom.

Axiom 9. If there exist three points not lying in the same line, there exists a plane

ABC such that there is a point D not lying in the plane ABC .

. The points of

faces, edges, and vertices constitute the surface of the tetrahedron.

If A, B, C, D are the vertices of a tetrahedron, the space ABCD consists of all

points collinear with any two points of the faces of the tetrahedron.

Axiom 10. If there exist four points neither lying in the same line nor lying in the

same plane, there exists a space ABCD such that there is no point E not collinear

with two points of the space, ABCD .

The next axiom resembles the second–order geometric axiom of the same name in

the axiom systems of Hilbert [10], Veblen [28] and Moore [15].

For a given line L , a sequence of points A 0 ,A 1 ,A 2 ,...

∈

{

B b :

∀

A i

= A 0 , [ A 0 A i B b ]; B b ∈

}

is called

the set of bounds of the sequence :if

is non–empty we say that the sequence is

bounded and, if there is a bound B c ∈B

such that [ A 0 B c B b ] (for all B b ∈B\{

B c }

)

we say that B c is a closest bound.

Axiom 11 (Continuity). Any bounded linearly–ordered sequence of points has a

closest bound.

These ﬁrst eleven axioms imply that the set of points is an open convex three–

dimensional) subset (of an ane subspace) of three–dimensional projective space

Lemma 1 (Convex subset). The set of points

is a convex open subset of some

a ne subset of

3 .

Proof. We ﬁrst consider the special case of a a plane subspace Π of

. A line

through a point A

∈P

meets the boundary ∂

in a point B (where the prime in-

satisﬁes the order

properties implied by Axioms 1–11 (which are used by Coxeter [7] to establish an

“ordered” or “descriptive” geometry) so the line AB separates the remaining set of

points of

). The set of points

Π into two components or “sides”. Each other line (of Π) through B

intersects at most one of these “sides” and the two “sides” specify disconnected

subsets of lines (through B ). These subsets meet the projective extension of any

other line through A in open (disconnected) segments. If the point A is excluded

from this projective line, the continuity property applies to the remaining linearly

ordered subset, so there is at least one point between the two open segments.

Therefore, through the point B , there is some line l which does not meet

P∩

If A, B, C ,and D are four points not lying in the same plane, they form a tetrahe-

dron ABCD whose faces are the interiors of the triangles ABC, BCD, CDA, DAB

(if the triangles exist) whose vertices are the four points, A, B, C, and D ,and

whose edges are the segments

L with the order property

[ A i A i +1 A i +2 ] (for all i ) is called a linearly–ordered sequence and will be denoted

as [ A 0 A 1 A 2 ... ]. The set

3 as we will now show.

dicates that the point does not belong to

Vol. 90 (2008)

A System of Axioms for Hyperbolic Geometry

189

To extend the proof to three–dimensions, we consider the set of planes which

contain the line l . As in the two–dimensional case, the plane Π separates the

remaining set of points of

and which

, form two disconnected components. Through A take any line which is

not contained in Π: the two disconnected components meet the projective extension

of this line in open (disconnected) segments and, as in the previous case, there is

now some plane (containing l ) which belongs to neither component and therefore

contains no points of

2.3. The axiom of symmetry

Axiom 12 (Isotropy). There is a point A such that, for each pair of distinct rays

and

, there is an automorphism of

, [ ... ]

which maps

onto

This axiom ensures that the convex domain is either a three–dimensional ane

space or the interior of an ellipsoid, as follows from characterisations of ellipsoids

by Aleksandrov [1], Busemann [4] and Schutz [24] which may be stated as:

Theorem 2 (Projective isotropy theorem). Let

be a convex open subset of some

a ne subset of

3 .If

is isotropic with respect to some point A

∈V

, then

either an ellipsoid or an a ne subspace of

3 .

2.4. The axiom of rigidity

The next axiom is related to a concept of congruence which is based on the idea

that equality of segments may be demonstrated by the motion of a “rigid ruler”.

This concept is either imposed upon the geometry or, if a ruler is regarded as

existing within a space, the “rigidity” is related to the possible motions of the

space and, accordingly, the concept of congruence is deﬁned by the automorphisms

, [ ... ]

which maps a segment

onto

another segment

, we say that the segments

and

are congruent and

| =

. The axiom of rigidity will be stated for a single segment.

Axiom 13 (Rigidity). There are two distinct points A, B such that the segment

is not congruent to a subsegment of itself.

which are projectivities of the embedding three–dimensional

projective space. An ane subspace of a projective space has automorphisms of

any segment onto any other segment: the axiom of rigidity therefore excludes the

possibility of an ane space, which leaves the interior of an ellipsoid as the only

possible subset of points corresponding to

, [ ... ]

. The deﬁnition of congruence im-

plies that the concept of length of segments must be deﬁned by the projectively

invariant cross–ratio.

into two disconnected subsets or “sides”: if we exclude

the plane Π, the remaining subset of planes which contain the line l

intersect

If there is an automorphism of

we write

The concept of congruence is based upon the existence of automorphisms of

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