ThreeDgeo3.ocd

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

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          this document, and maintains a prominent reference in the 
          work to the original source.
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          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
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       c) The derived work is distributed under terms that allow the
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</CDComment>

<CDName>ThreeDgeo3</CDName>
<CDURL>http://nash.sip.ucm.es/LAD-3D/3DgeoCDs/ThreeDgeo3.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>set_affine_coordinates</Name>
<Description> 
Defines the affine coordinates of an a point in 3-dimensional Euclidean space. 
Takes the point as first argument and the vector with the coordinates as second argument.
</Description>


<Example>
The description of the point A with affine coordinates (4.8,0.6,10.2) is given by:

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="ThreeDgeo1" name="point"/>
    <OMV name="A"/>
    <OMA>
      <OMS cd="ThreeDgeo3" name="set_affine_coordinates"/>
      <OMV name="A"/>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
        <OMF dec =" 4.8 "/>
        <OMF dec =" 0.6 "/>
        <OMF dec =" 10.2 "/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ> 

</Example>

</CDDefinition>

<CDDefinition>
<Name>distance</Name>
<Description> 
The distance between two affine points in 3-dimensional Euclidean space is the Euclidean distance.
The distance between two geometric objects O and O' in 3-dimensional Euclidean space is the infimum of the
distances between two affine points, one on O and one on O'.  
</Description>

<Example> The distance between two points A and B is given by

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="ThreeDgeo3" name="distance"/>
    <OMV name="A"/>
    <OMV name="B"/>
  </OMA>
</OMOBJ>
 
</Example>

</CDDefinition>

<CDDefinition>
<Name>configuration</Name>
<Description> 
The symbol represents a configuration in Euclidean 3-dimensional geometry consisting of a sequence of geometric objects like points, lines,  etc, but also of other configurations.
</Description>

<Example> The configuration of a point A and a line l incident to A 
is defined by:

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="ThreeDgeo3" name="configuration"/>
    <OMA>
      <OMS cd="ThreeDgeo1" name="point"/> 
      <OMV name="A"/>
    </OMA>
    <OMA>
      <OMS cd="ThreeDgeo1" name="line"/>
      <OMV name="l"/>
      <OMA>
        <OMS cd="ThreeDgeo2" name="incident"/>
        <OMV name="A"/>
        <OMV name="l"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

</Example>

<Example>
The following is the description of the configuration consisting on two different points A and B and the line l determined by them:

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="ThreeDgeo3" name="configuration"/>
    <OMA>
      <OMS cd="ThreeDgeo1" name="point"/>
      <OMV name="A"/>
    </OMA>
    <OMA>
      <OMS cd="ThreeDgeo1" name="point"/>
      <OMV name="B"/>
      <OMA>
        <OMS cd="logic1" name="not"/>
        <OMA>
          <OMS cd="relation1" name="eq"/>
          <OMV name="A"/>
          <OMV name="B"/>
        </OMA>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="ThreeDgeo1" name="line"/>
      <OMV name="l"/>
      <OMA>
        <OMS cd="ThreeDgeo2" name="incident"/>
        <OMV name="A"/>
        <OMV name="l"/>
      </OMA>
      <OMA>
        <OMS cd="ThreeDgeo2" name="incident"/>
        <OMV name="B"/>
        <OMV name="l"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

</Example>

</CDDefinition>

<CDDefinition>
<Name>assertion</Name>
<Description> 
The symbol is a constructor with two arguments.
Its first argument is a 3-dimensional Euclidean geometry 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.
</Description>

<Example> The assertion that two distinct intersecting lines meet in only one point
can be expressed as follows using the assertion symbol.

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="ThreeDgeo3" name="assertion"/>
    <OMA>
      <OMS cd="ThreeDgeo3" name="configuration"/>
      <OMA>
        <OMS cd="ThreeDgeo1" name="point"/> 
        <OMV name="A"/>
      </OMA>
      <OMA>
        <OMS cd="ThreeDgeo1" name="point"/> 
        <OMV name="B"/>
      </OMA>
      <OMA>
        <OMS cd="ThreeDgeo1" name="line"/>
        <OMV name="l"/>
        <OMA>
          <OMS cd="ThreeDgeo2" name="incident"/>
          <OMV name="A"/>
          <OMV name="l"/>
        </OMA>
        <OMA>
          <OMS cd="ThreeDgeo2" name="incident"/>
          <OMV name="B"/>
          <OMV name="l"/>
        </OMA>
      </OMA>
      <OMA>
        <OMS cd="ThreeDgeo1" name="line"/>
        <OMV name="m"/>
        <OMA>
          <OMS cd="ThreeDgeo2" name="incident"/>
          <OMV name="A"/>
          <OMV name="m"/>
        </OMA>
        <OMA>
          <OMS cd="ThreeDgeo2" name="incident"/>
          <OMV name="B"/>
          <OMV name="m"/>
        </OMA>
        <OMA>
          <OMS cd="logic1" name="not"/>
          <OMA>
            <OMS cd="relation1" name="eq"/>
            <OMV name="l"/>
            <OMV name="m"/>
          </OMA>
        </OMA>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMV name="A"/>
      <OMV name="B"/>
    </OMA>
  </OMA>
</OMOBJ>

</Example>

</CDDefinition>

<CDDefinition>
<Name>locus</Name>
<Description> 
The symbol is used to indicate by a variable the locus set traced by a point T in a 3-dimensional Euclidean geometry configuration C. That is, the set of all points satisfying the conditions imposed on T in the configuration C. 
The locus may (but need not) be defined by constraints on the point T additional to those in the configuration.
The symbol takes the variable as the first argument, the tracer point T as second argument and the additional constraints as further arguments.
</Description>


<Example> 
The following example describes a configuration with the locus set L of all points C equidistant to two given points A and B.

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0"> 
  <OMA>
    <OMS cd="ThreeDgeo3" name="configuration"/>
    <OMA>
      <OMS cd="ThreeDgeo1" name="point"/>
      <OMV name="A"/>
    </OMA>
    <OMA>
      <OMS cd="ThreeDgeo1" name="point"/>
      <OMV name="B"/>
    </OMA>
    <OMA>
      <OMS cd="ThreeDgeo1" name="point"/>
      <OMV name="C"/>
    </OMA>
    <OMA>
      <OMS cd="ThreeDgeo3" name="locus"/>
      <OMV name="L"/>
      <OMV name="C"/>
      <OMA>
        <OMS cd="relation1" name="eq"/>
        <OMA>
          <OMS cd="ThreeDgeo3" name="distance"/>
          <OMV name="A"/>
          <OMV name="C"/>
        </OMA>
        <OMA>
          <OMS cd="ThreeDgeo3" name="distance"/>
          <OMV name="B"/>
          <OMV name="C"/>
        </OMA>
      </OMA>
    </OMA>
  </OMA> 
</OMOBJ> 

</Example>

</CDDefinition>

<CDDefinition>
<Name>discovery</Name>
<Description> 
The symbol is used to describe the task of finding necessary conditions for some properties to hold in a geometric configuration in 3-dimensional Euclidean geometry.
The symbol takes a configuration as the first argument and the properties for which necessary conditions are to be sought as further arguments.
</Description>

<Example> 
The following example encodes the task of finding necessary conditions for a point C to be equidistant to the points A and B.

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0"> 
  <OMA>
    <OMS cd="ThreeDgeo3" name="discovery"/>
    <OMA>
      <OMS cd="ThreeDgeo3" name="configuration"/>
      <OMA>
        <OMS cd="ThreeDgeo1" name="point"/>
        <OMV name="A"/>
      </OMA>
      <OMA>
        <OMS cd="ThreeDgeo1" name="point"/>
        <OMV name="B"/>
      </OMA>
      <OMA>
        <OMS cd="ThreeDgeo1" name="point"/>
        <OMV name="C"/>
      </OMA>
    </OMA> 
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="ThreeDgeo3" name="distance"/>
        <OMV name="A"/>
        <OMV name="C"/>
      </OMA>
      <OMA>
        <OMS cd="ThreeDgeo3" name="distance"/>
        <OMV name="B"/>
        <OMV name="C"/>
      </OMA>
   </OMA>
  </OMA>
</OMOBJ> 

</Example>
</CDDefinition>

</CD>