ordered geometry (original) (raw)

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1 Definition

For 0≤i<n, let

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

and

Bi⁢(a)={(p,q)∣(p,a,q)∈Bi}.

For any a∈Pi, the set is symmetricMathworldPlanetmathPlanetmathPlanetmath and anti-reflexive.

We say that the hyperplaneMathworldPlanetmath a∈Pi 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,ℓ,q)∈B1 whereℓ is a line, then p,q and ℓ necessarily lie on a common plane π.

The above diagram seems to suggest that ℓ “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

In fact, in axiom S1, it can be shown that exactly one of (q,r) and (r,p) is in B1⁢(ℓ). 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 ℓ lie on plane π and that ℓ is between p and q. Then we say that p and q are on the opposite sides of line ℓ. Two points are on the same side of line ℓ if they are not on the opposite sides of ℓ. If r is a third point (distinct from p,q) that lies on π and not on ℓ, 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.

2 Remarks

References

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