ordered geometry


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1 Definition

Let (A,B) be a linear ordered geometry, where A=(P,n,I) is an incidence geometry, and B is a strict betweenness relation. Recall that P is partitioned into disjoint sets P0,,Pn, where n is a positive integer.

For 0i<n, let

Bi = {(p,a,q)P0×Pi×P0p,q do not lie on a,
and there exists a point r lying on a such that (p,r,q)B},



For any aPi, the set is symmetricMathworldPlanetmathPlanetmathPlanetmath and anti-reflexive.

We say that the hyperplaneMathworldPlanetmath aPi is between p and q if (p,q)Bi(a).
We see that B0=B.
Let’s look at the case when i=1. If (p,,q)B1 where is a line, then p,q and necessarily lie on a common plane π.

The above diagram seems to suggest that “separates π into two regions”. However, this is not true in general without the next axiom.
An ordered geometry (A,B) is a linear ordered geometry such that

  • S1

    for any three non-collinear points p,q,r, and any line lying on the same plane π generated by p,q,r, if (p,q)B1() and if r does not lie on , then at least one of (q,r),(r,p)B1().

In fact, in axiom S1, it can be shown that exactly one of (q,r) and (r,p) is in B1(). This axiom says that “a line lying on a plane separates the plane into two mutually exclusive subsets”.
Each subset is called an (open) half plane of the line.
A closed half plane is just the union of one of its open half planes and the line itself.
Suppose points p,q and line lie on plane π and that is between p and q. Then we say that p and q are on the opposite sides of line . Two points are on the same side of line if they are not on the opposite sides of . If r is a third point (distinct from p,q) that lies on π and not on , then according to axiom S1 above, r must be on the same side of either p or q (but not both!). Same sidedness is an equivalence relationMathworldPlanetmath on points of A.

3 An equivalentMathworldPlanetmathPlanetmathPlanetmath characterization of axiom S1 is in the form of Pasch’s theorem.

The ten conditions or axioms (seven betweenness, two collinearity, and one “separationMathworldPlanetmathPlanetmath” axioms) are sometimes called the “order axioms” of A.

It is customary, in an ordered geometry, to identify each element of P by its shadow (http://planetmath.org/IncidenceGeometry) (a subset of P0), and we shall do so in this discussion. A line, for example, will then consist of points that are incidentMathworldPlanetmathPlanetmath with it, as opposed to an abstract element of P1. Hence, we shall also confuse the notation pq with I0(pq).

2 Remarks

  • Law of Trichotomy on a strict betweenness relation: Let B be a strict betweenness relation. If p,q,r are collinear, then exactly one of (p,q,r), (q,r,p), or (r,p,q)B.

  • In an ordered geometry, one can define familiar concepts, such as a line segment, a ray, even an angle, using the order axioms above. For example, Bp*q is called the open line segment between p and q, and is more commonly denoted by pq¯, or (p,q). A closed line segment between p and q is just {p}pq¯{q}, denoted by [p,q]. From the third remark under betweenness relation, pq¯=Bp*q=Bq*p=qp¯. The points p and q are called the end points of pq¯.

  • A ray is defined to be Bpq. For a more detailed discussion, see the entry on ray (http://planetmath.org/Ray).

  • pq=B(p,q).

  • pq¯pq. The inclusion is strict, since there exists a point r such that (p,q,r)B by order axiom S1. r lies on the pq and is clearly distinct from both p and q.

  • Any line segment pq¯ in an ordered geometry, in to being orderable, is linearly orderable, thanks to the Law of Trichotomy.

  • It fact, , defined on a line segment, can be extended to a linear order defined on the line that includes the segment (see the last remark above on betweenness relation). This shows that every line in an ordered geometry can be linearly ordered.


  • 1 D. Hilbert, Foundations of Geometry, Open Court Publishing Co. (1971)
  • 2 K. Borsuk and W. Szmielew, Foundations of Geometry, North-Holland Publishing Co. Amsterdam (1960)
  • 3 R. Hartshorne, GeometryMathworldPlanetmath: Euclid and Beyond, Springer (2000)
Title ordered geometry
Canonical name OrderedGeometry
Date of creation 2013-03-22 15:28:21
Last modified on 2013-03-22 15:28:21
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 43
Author CWoo (3771)
Entry type Definition
Classification msc 51G05
Synonym open interval
Synonym closed interval
Synonym interval
Related topic PaschsTheorem
Defines half plane
Defines side of line
Defines open line segment
Defines closed line segment
Defines opposite sides
Defines open half plane
Defines closed half plane
Defines end points
Defines open line segment
Defines closed line segment