ThreeDgeo2.ocd

<CD xmlns="http://www.openmath.org/OpenMathCD">
<CDComment>

     This document is distributed in the hope that it will be useful, 
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     provided that the following conditions are met.
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          work to the original source.
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          both this document and the derived work are Content Dictionaries
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</CDComment>

<CDName>ThreeDgeo2</CDName>
<CDURL>http://nash.sip.ucm.es/LAD-3D/3DgeoCDs/ThreeDgeo2.ocd</CDURL>
<CDReviewDate>2017-12-31</CDReviewDate>
<CDStatus>experimental</CDStatus>
<CDDate>2008-01-21</CDDate>
<CDVersion>0</CDVersion>
<CDRevision>3</CDRevision>
<CDComment>
  Author: Jesús Escribano
</CDComment>

<Description> 
This CD defines symbols for 3-dimensional Euclidean geometry
</Description> 

<CDDefinition>
<Name>incident</Name>
<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.
</Description>

<Example> 
That a point A is incident to a line l is given by: 

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="ThreeDgeo2" name="incident"/>
    <OMV name="A"/>
    <OMV name="l"/>
  </OMA>
</OMOBJ>

</Example>

</CDDefinition>

<CDDefinition>
<Name>is_midpoint</Name>
<Description> The statement that one point is the midpoint of two others.
</Description>

<Example> 
This example states that C is the midpoint of A and B.

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="ThreeDgeo2" name="is_midpoint"/>
    <OMV name="C"/>    
    <OMV name="A"/>    
    <OMV name="B"/>
  </OMA>
</OMOBJ>

</Example>

</CDDefinition>

<CDDefinition>
<Name>circle_center</Name>
<Description> 
The statement that a circle in 3-dimensional Euclidean space has a given point as center. 
Takes the circle as first argument and the point as second argument.
</Description>

<Example> 
The circle c with center at A and passing through the point B is given by:

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="ThreeDgeo1" name="circle"/>
    <OMV name="c"/>
    <OMA>
      <OMS cd="ThreeDgeo2" name="circle_center"/>
      <OMV name="c"/>
      <OMV name="A"/>
    </OMA>
    <OMA>
      <OMS cd="ThreeDgeo2" name="incident"/>
      <OMV name="B"/>
      <OMV name="c"/>
    </OMA>
  </OMA>
</OMOBJ>

</Example>

</CDDefinition>

<CDDefinition>
<Name>sphere_center</Name>
<Description> 
The statement that a sphere in 3-dimensional Euclidean space has a given point as center. 
Takes the sphere as first argument and the point as second argument.
</Description>

<Example> 
The sphere s with center at A and passing through the point B is given by:

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="ThreeDgeo1" name="sphere"/>
    <OMV name="s"/>
    <OMA>
      <OMS cd="ThreeDgeo2" name="sphere_center"/>
      <OMV name="s"/>
      <OMV name="A"/>
    </OMA>
    <OMA>
      <OMS cd="ThreeDgeo2" name="incident"/>
      <OMV name="B"/>
      <OMV name="s"/>
    </OMA>
  </OMA>
</OMOBJ>

</Example>

</CDDefinition>

<CDDefinition>
<Name>perpendicular</Name>
<Description> 
The symbol represents a binary boolean function with input two lines or segments.
Its value is true whenever the first argument is perpendicular to the second.
</Description>

<Example> 
This example states that the lines l and m are perpendicular.

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="ThreeDgeo2" name="perpendicular"/>
    <OMV name="l"/>
    <OMV name="m"/>
  </OMA>
</OMOBJ> 

</Example>

</CDDefinition>

<CDDefinition>
<Name>line_parallel</Name>
<Description> 
The symbol represents a binary boolean function with input two lines or segments.
Its value is true whenever the first argument is parallel to the second.
</Description>

<Example> 
This example states that the lines l and m are parallel.

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="ThreeDgeo2" name="line_parallel"/>
    <OMV name="l"/>
    <OMV name="m"/>
  </OMA>
</OMOBJ> 

</Example>

</CDDefinition>

<CDDefinition>
<Name>plane_parallel</Name>
<Description> 
The symbol represents a binary boolean function with input two planes.
Its value is true whenever the first argument is parallel to the second.
</Description>

<Example> 
This example states that the planes m and n are parallel.

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="ThreeDgeo2" name="plane_parallel"/>
    <OMV name="m"/>
    <OMV name="n"/>
  </OMA>
</OMOBJ> 

</Example>

</CDDefinition>

<CDDefinition>
<Name>normal</Name>
<Description> 
The symbol represents a binary boolean function with a line as first argument and a plane as second argument.
Its value is true whenever the first argument is normal to the second.
</Description>

<Example> 
This example states that the line l is normal to the plane p.

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="ThreeDgeo2" name="normal"/>
    <OMV name="l"/>
    <OMV name="p"/>
  </OMA>
</OMOBJ> 

</Example>

</CDDefinition>

<CDDefinition>
<Name>are_on_line</Name>
<Description> 
The symbol is a boolean n-ary function. Its arguments should be points. 
When applied to a sequence of points in 3-dimensional Euclidean space, its evaluated to true if and only if there is a line on which all arguments lie.
</Description>

<Example> 
This example states that the points A, B, C, and D are collinear.
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="ThreeDgeo2" name="are_on_line"/>
    <OMV name="A"/>
    <OMV name="B"/>
    <OMV name="C"/>
    <OMV name="D"/>
  </OMA>
</OMOBJ>

</Example>

</CDDefinition>

<CDDefinition>
<Name>are_on_plane</Name>
<Description> 
The symbol is a boolean n-ary function. Its arguments should be points. 
When applied to a sequence of points in 3-dimensional Euclidean space, its evaluated to true if and only if there is a plane on which all arguments lie.
</Description>

<Example> 
This example states that the points A, B, C, and D are coplanar.
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="ThreeDgeo2" name="are_on_plane"/>
    <OMV name="A"/>
    <OMV name="B"/>
    <OMV name="C"/>
    <OMV name="D"/>
  </OMA>
</OMOBJ>

</Example>

</CDDefinition>

</CD>