J. geom. 90 (2008), 185 192
© 2008 Birkhäuser Verlag Basel/Switzerland
0047-2468/010185-8, published online 21 November 2008
Journal of Geometry
DOI 10.1007/s00022-008-1903-9
A System of Axioms for Hyperbolic Geometry
John W. Schutz
This paper is dedicated to my parents,
Edith Schutz (nee Edit Kertész, 9/09/1917 Budapest  4/07/2005) and
Steven Joseph Schutz (István József Schütz, 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 Plücker [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].
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 affine 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 affine
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, affine 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

H = P, L, [ . . . ]
where P is a set whose elements are called points, L is a set of subsets of P called
lines and [. . . ] is a ternary relation on the set of points of P 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 |AB| where A and B are
the end-points of the segment. The ray |AB :={A, B}*"|AB|*"{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 |AB|, |BC|, |CA|, 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 [AFB] 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.
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 |AB|, |BC|, |CD|, |DA|, |AC|, |BD|. 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 A0, A1, A2, . . . " L with the order property
[Ai Ai+1 Ai+2] (for all i) is called a linearly ordered sequence and will be denoted
as [A0 A1 A2 . . . ]. The set B = {Bb : "Ai = A0, [A0 Ai Bb]; Bb " L} is called

the set of bounds of the sequence: if B is non empty we say that the sequence is
bounded and, if there is a bound Bc "Bsuch that [A0 Bc Bb] (for all Bb "B\{Bc})
we say that Bc 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 affine subspace) of three dimensional projective space
P3 as we will now show.
Lemma 1 (Convex subset). The set of points P is a convex open subset of some
affine subset of P3.
Proof. We first consider the special case of a a plane subspace  of H. A line
through a point A "P meets the boundary "P in a point B (where the prime in-
dicates that the point does not belong to P). The set of points P 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 P)"  into two components or  sides . Each other line (of ) 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.
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  separates the
remaining set of points of P into two disconnected subsets or  sides : if we exclude
the plane , the remaining subset of planes which contain the line l and which
intersect P, form two disconnected components. Through A take any line which is
not contained in : 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 P.
2.3. The axiom of symmetry
Axiom 12 (Isotropy). There is a point A such that, for each pair of distinct rays
|AB and |AC , there is an automorphism of H = P, L, [ . . . ] which maps |AB
onto |AC .
This axiom ensures that the convex domain is either a three dimensional affine
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 V be a convex open subset of some
affine subset of P3. If V is isotropic with respect to some point A " V, then V is
either an ellipsoid or an affine subspace of P3.
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
of H.
If there is an automorphism of H = P, L, [ . . . ] which maps a segment |AB| onto
another segment |CD|, we say that the segments |AB| and |CD| are congruent and
<"
we write |AB| |CD|. 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
|AB| is not congruent to a subsegment of itself.
The concept of congruence is based upon the existence of automorphisms of
H = P, L, [ . . . ] which are projectivities of the embedding three dimensional
projective space. An affine subspace of a projective space has automorphisms of
any segment onto any other segment: the axiom of rigidity therefore excludes the
possibility of an affine space, which leaves the interior of an ellipsoid as the only
possible subset of points corresponding to P. The definition of congruence im-
plies that the concept of length of segments must be defined by the projectively
invariant cross ratio.
190 J. W. Schutz J. Geom.
Thus the present system of thirteen axioms specifies the Klein [13, 14] model of
hyperbolic geometry. A complete exposition of projective and hyperbolic geometry
in terms of homogeneous coordinates may be found in Busemann and Kelly [5].
3. Independence models
Each of the two additional axioms is independent of all other axioms as is demon-
strated by the following independence models.
3.1. Independence model for Axiom 12
An independence model can be specified with a set of points P interior to any
open three dimensional non ellipsoidal convex subset of R3 such as, for example,
the interior of the quartic
x4 + x4 + x4 d" 1
1 2 3
where the lines L are the intersections of lines of R3 with P.
A second type of independence model can be specified with P = R2 × (-1, 1)
and with a similar definition of the set of lines. In this model there are some lines
which satisfy the property of  rigidity so Axiom 13 is satisfied and clearly all
other axioms apart from the Axiom of Isotropy (Axiom 12) are also satisfied.
3.2. Independence model for Axiom 13
Clearly A3 is not rigid and yet it satisfies all the other axioms.
3.3. Independence of Axioms 1 11
In the present system of axioms, the first ten correspond to Veblen s [28] system
of axioms for affine geometry, while the eleventh axiom (of weak completeness)
is equivalent to Veblen s Axiom XI (of continuity). Veblen s independence proofs
therefore apply to the mutual independence of the first eleven axioms, each from
the others, but it still remains to be shown that Axioms 1 11 are independent
of the entire present system of axioms which includes the two additional axioms
(Axioms 12 and 13).
4. Conclusion
We have demonstrated that the first eleven axioms of Veblen together with the
axiom of isotropy and the axiom of rigidity are sufficient to characterise hyperbolic
geometry using elementary techniques of affine and projective geometry. It could be
claimed that the present axiomatic system with its thirteen axioms is simpler than
previous axiomatic systems: the axiomatic system of Coxeter [7] has similar axioms
of order together with additional axioms of congruence and parallels, Redéi s [19,
Vol. 90 (2008) A System of Axioms for Hyperbolic Geometry 191
20] system has similar axioms of order and several axioms of motion, while the
system of independent axioms of Moore [15] has a total of fifteen axioms which
includes four independent axioms of congruence.
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John W. Schutz
Department of Mathematics and Statistics
La Trobe University
Bendigo, Victoria 3552
Australia
e-mail: [e-mail]