# OpenMath Content Dictionary: plangeo1

Canonical URL:
http://www.win.tue.nl/~amc/oz/om/cds/plangeo1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
plangeo1.ocd
CD as XML Encoded OpenMath:
plangeo1.omcd
Defines:
point, are_on_line, assertion, configuration, incident, line, type
Date:
2004-06-01
Version:
0 (Revision 5)
Review Date:
2006-06-01
Status:
experimental


This document is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

The copyright holder grants you permission to redistribute this
document freely as a verbatim copy. Furthermore, the copyright
holder permits you to develop any derived work from this document
provided that the following conditions are met.
a) The derived work acknowledges the fact that it is derived from
this document, and maintains a prominent reference in the
work to the original source.
b) The fact that the derived work is not the original OpenMath
document is stated prominently in the derived work.  Moreover if
both this document and the derived work are Content Dictionaries
then the derived work must include a different CDName element,
chosen so that it cannot be confused with any works adopted by
the OpenMath Society.  In particular, if there is a Content
Dictionary Group whose name is, for example, math' containing
Content Dictionaries named math1', math2' etc., then you should
not name a derived Content Dictionary mathN' where N is an integer.
However you are free to name it private_mathN' or some such.  This
is because the names mathN' may be used by the OpenMath Society
for future extensions.
compilation of derived works, but keep paragraphs a) and b)
intact.  The simplest way to do this is to distribute the derived
work under the OpenMath license, but this is not a requirement.
society at http://www.openmath.org.

  Author: Arjeh Cohen


This CD defines symbols for planar Euclidean geometry.

## point

Description:

The symbol is used to indicate a point of planar Euclidean geometry by a variable. The point may (but need not) be subject to constraints. The symbol takes the variable as the first argument and the constraints as further arguments.

Example:
Given two lines l and m, a point A on l and m is defined by:
$\mathrm{point}\left(A,\mathrm{incident}\left(A,l\right),\mathrm{incident}\left(A,m\right)\right)$
Signatures:
sts

 [Next: line] [Last: are_on_line] [Top]

## line

Description:

The symbol is used to indicate a line of planar Euclidean geometry by a variable. The line may (but need not) be subject to constraints. The symbol takes the variable as the first argument and the constraints as further arguments.

Example:
Given points A and B, a line l through A and B is defined by:
$\mathrm{line}\left(l,\mathrm{incident}\left(A,l\right),\mathrm{incident}\left(B,l\right)\right)$
Signatures:
sts

 [Next: incident] [Previous: point] [Top]

## incident

Description:

The symbol represents the logical incidence function which is a binary function taking arguments representing geometric objects like points and lines and returning a boolean value. It is true if and only if the first argument is incident to the second.

Example:
That a point A is incident to a line l is given by:
$\mathrm{incident}\left(A,l\right)$
Signatures:
sts

 [Next: configuration] [Previous: line] [Top]

## configuration

Description:

The symbol represents a configuration in Euclidean planar geometry consisting of a sequence of geometric objects like points, lines, etc, but also of other configurations.

Example:
The configuration of a point A and a line l incident to A is defined by:
$\mathrm{configuration}\left(\mathrm{point}\left(A\right),\mathrm{line}\left(l,\mathrm{incident}\left(A,l\right)\right)\right)$
Example:
The prevous configuration of a point A and a line l incident with A can be extended by adding a second point B incident with l:
$\mathrm{configuration}\left(\mathrm{configuration}\left(\mathrm{point}\left(A\right),\mathrm{line}\left(l,\mathrm{incident}\left(A,l\right)\right)\right),\mathrm{point}\left(B,\mathrm{incident}\left(B,l\right)\right)\right)$
Example:
We describe a triangle on the distinct points A, B, C and lines a, b, c:
$\mathrm{configuration}\left(\mathrm{point}\left(A\right),\mathrm{point}\left(B,¬\left(A=B\right)\right),\mathrm{line}\left(c,\mathrm{incident}\left(c,A\right),\mathrm{incident}\left(c,B\right)\right),\mathrm{point}\left(C,¬\mathrm{incident}\left(C,c\right)\right),\mathrm{line}\left(a,\mathrm{incident}\left(a,B\right),\mathrm{incident}\left(a,C\right)\right),\mathrm{line}\left(b,\mathrm{incident}\left(b,A\right),\mathrm{incident}\left(b,C\right)\right)\right)$
Signatures:
sts

 [Next: type] [Previous: incident] [Top]

## type

Description:

The symbol represents the type of the basic geometric objects: points, lines, configuration.

Commented Mathematical property (CMP):
If A and B are objects of the same type, then they are not incident.
Formal Mathematical property (FMP):
$\mathrm{type}\left(A\right)=\mathrm{type}\left(B\right)⇒¬\mathrm{incident}\left(A,B\right)$
Signatures:
sts

 [Next: assertion] [Previous: configuration] [Top]

## assertion

Description:

The symbol is a constructor with two arguments. Its first argument should be a configuration, its second argument a statement about the configuration, called thesis. When applied to a configuration C and a thesis T, the OpenMath object assertion(C,T) expresses the assertion that T holds in C.

Example:
The assertion that two distinct lines meet in only one point can be expressed as follows using the assertion symbol.
$\mathrm{assertion}\left(\mathrm{configuration}\left(\mathrm{point}\left(A\right),\mathrm{point}\left(B\right),\mathrm{line}\left(l,\mathrm{incident}\left(A,l\right),\mathrm{incident}\left(B,l\right)\right),\mathrm{line}\left(m,\mathrm{incident}\left(A,m\right),\mathrm{incident}\left(B,m\right),¬\left(l=m\right)\right),A=B\right)\right)$
Signatures:
sts

 [Next: are_on_line] [Previous: type] [Top]

## are_on_line

Description:

The statement that a set of points is collinear.

Example:
This example states that A, B, C, and D are collinear.
$\mathrm{are_on_line}\left(A,B,C,D\right)$
Signatures:
sts

 [First: point] [Previous: assertion] [Top]