OpenMath Content Dictionary: plangeo3

Canonical URL:

http://www.openmath.org/cd/plangeo3.ocd

CD Base:

http://www.openmath.org/cd

CD File:

plangeo3.ocd

CD as XML Encoded OpenMath:

plangeo3.omcd

Defines:

altitude, angle, arc, are_on_circle, center, center_of, center_of_gravity, circle, distance, is_midpoint, midpoint, parallel, perpbisector, perpendicular, perpline, polarline, radius, radius_of, tangent

Date:

2004-06-01

Version:

0 (Revision 3)

Review Date:

2006-06-01

Status:

experimental

---

This CD defines symbols for planar Euclidean geometry related to distance.

---

distance

Description:

The distance between two affine points is the Euclidean distance. The distance between two geometric objects O and O' is the infimum of the distances between two affine points, one on O and one on O'.

Example:

The distance between two points A and B.

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
  <OMS cd="plangeo3" name="distance"/>
   <OMV name="A"/>
   <OMV name="B"/>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol cd="plangeo3">distance</csymbol><ci>A</ci><ci>B</ci></apply></math>

Prefix

distance ( A, B)

Popcorn

plangeo3.distance($A, $B)

Rendered Presentation MathML

$$\mathrm{distance}(A,B)$$

Signatures:

sts

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[Next: perpendicular] [Last: arc] [Top]

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perpendicular

Description:

perpendicular is a binary boolean function with input two lines, halflines or segments. Its value is true whenever the two inputs are perpendicular to each other.

Example:

Testing perpendicularity of two lines L and M.

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
  <OMS cd="plangeo3" name="perpendicular"/>
   <OMV name="L"/>
   <OMV name="M"/>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol cd="plangeo3">perpendicular</csymbol><ci>L</ci><ci>M</ci></apply></math>

Prefix

perpendicular ( L, M)

Popcorn

plangeo3.perpendicular($L, $M)

Rendered Presentation MathML

$$\mathrm{perpendicular}(L,M)$$

Signatures:

sts

---

[Next: parallel] [Previous: distance] [Top]

---

parallel

Description:

parallel is a binary boolean function with input two lines, halflines or segments. Its value is true whenever the two inputs are parallel.

Example:

Testing parallelism of two lines L and M.

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
  <OMS cd="plangeo3" name="parallel"/>
   <OMV name="L"/>
   <OMV name="M"/>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol cd="plangeo3">parallel</csymbol><ci>L</ci><ci>M</ci></apply></math>

Prefix

parallel ( L, M)

Popcorn

plangeo3.parallel($L, $M)

Rendered Presentation MathML

$$\mathrm{parallel}(L,M)$$

Signatures:

sts

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[Next: circle] [Previous: perpendicular] [Top]

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circle

Description:

The symbol represents a circle. The circle may be subject to constraints.

Example:

The circle C with center A and radius 1 is given by

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="plangeo3" name="circle"/>
    <OMV name="C"/>
    <OMA>
      <OMS cd="plangeo3" name="center"/>
      <OMV name="C"/>
      <OMV name="A"/>
    </OMA>
    <OMA>
      <OMS cd="plangeo3" name="radius"/>
      <OMV name="C"/>
      <OMI>1</OMI>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="plangeo3">circle</csymbol>
  <ci>C</ci>
  <apply><csymbol cd="plangeo3">center</csymbol><ci>C</ci><ci>A</ci></apply>
  <apply><csymbol cd="plangeo3">radius</csymbol><ci>C</ci><cn type="integer">1</cn></apply>
 </apply>
</math>

Prefix

circle ( C, center ( C, A) , radius ( C, 1) )

Popcorn

plangeo3.circle($C, plangeo3.center($C, $A), plangeo3.radius($C, 1))

Rendered Presentation MathML

$$\mathrm{circle}(C,\mathrm{center}(C,A),\mathrm{radius}(C,1))$$

Signatures:

sts

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[Next: radius] [Previous: parallel] [Top]

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radius

Description:

The radius of a circle.

Example:

The assertion that the radius of the circle C is 1:

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
    <OMS cd="plangeo3" name="radius"/>
    <OMV name="C"/>
    <OMI>1</OMI>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="plangeo3">radius</csymbol><ci>C</ci><cn type="integer">1</cn></apply>
</math>

Prefix

radius ( C, 1)

Popcorn

plangeo3.radius($C, 1)

Rendered Presentation MathML

$$\mathrm{radius}(C,1)$$

Signatures:

sts

---

[Next: radius_of] [Previous: circle] [Top]

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radius_of

Description:

Gives the radius of a circle.

Example:

The radius of the circle C is given by

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
    <OMS cd="plangeo3" name="radius_of"/>
    <OMV name="C"/>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol cd="plangeo3">radius_of</csymbol><ci>C</ci></apply></math>

Prefix

radius_of ( C)

Popcorn

plangeo3.radius_of($C)

Rendered Presentation MathML

$$\mathrm{radius\_of}\left(C\right)$$

Signatures:

sts

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[Next: center] [Previous: radius] [Top]

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center

Description:

Defines the center of a circle.

Example:

The circle C with center A and radius 1 is given by

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA>
    <OMS cd="plangeo3" name="circle"/>
    <OMV name="C"/>
    <OMA>
      <OMS cd="plangeo3" name="center"/>
      <OMV name="C"/>
      <OMV name="A"/>
    </OMA>
    <OMA>
      <OMS cd="plangeo3" name="radius"/>
      <OMV name="C"/>
      <OMI>1</OMI>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="plangeo3">circle</csymbol>
  <ci>C</ci>
  <apply><csymbol cd="plangeo3">center</csymbol><ci>C</ci><ci>A</ci></apply>
  <apply><csymbol cd="plangeo3">radius</csymbol><ci>C</ci><cn type="integer">1</cn></apply>
 </apply>
</math>

Prefix

circle ( C, center ( C, A) , radius ( C, 1) )

Popcorn

plangeo3.circle($C, plangeo3.center($C, $A), plangeo3.radius($C, 1))

Rendered Presentation MathML

$$\mathrm{circle}(C,\mathrm{center}(C,A),\mathrm{radius}(C,1))$$

Signatures:

sts

---

[Next: center_of] [Previous: radius_of] [Top]

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center_of

Description:

Gives the center of the circle

Example:

If C is the circle of the previous example then the following gives A.

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
      <OMS cd="plangeo3" name="center_of"/>
      <OMV name="C"/>
    </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol cd="plangeo3">center_of</csymbol><ci>C</ci></apply></math>

Prefix

center_of ( C)

Popcorn

plangeo3.center_of($C)

Rendered Presentation MathML

$$\mathrm{center\_of}\left(C\right)$$

Signatures:

sts

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[Next: are_on_circle] [Previous: center] [Top]

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are_on_circle

Description:

The statement that a set of points is on one circle.

Example:

This example states that A, B, C, and D lie on one circle.

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
  <OMS cd="plangeo3" name="are_on_circle"/>
  <OMV name="A"/>
  <OMV name="B"/>
  <OMV name="C"/>
  <OMV name="D"/>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="plangeo3">are_on_circle</csymbol>
  <ci>A</ci>
  <ci>B</ci>
  <ci>C</ci>
  <ci>D</ci>
 </apply>
</math>

Prefix

are_on_circle ( A, B, C, D)

Popcorn

plangeo3.are_on_circle($A, $B, $C, $D)

Rendered Presentation MathML

$$\mathrm{are\_on\_circle}(A,B,C,D)$$

Signatures:

sts

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[Next: angle] [Previous: center_of] [Top]

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angle

Description:

Angle of a corner, always measured in positive (anti-clockwise) direction.

Example:

Notice that

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
  <OMS cd="plangeo3" name="angle"/>
    <OMA>
    <OMS cd="plangeo2" name="corner"/>
      <OMV name="A"/>
      <OMV name="B"/>
      <OMV name="C"/>
    </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="plangeo3">angle</csymbol>
  <apply><csymbol cd="plangeo2">corner</csymbol><ci>A</ci><ci>B</ci><ci>C</ci></apply>
 </apply>
</math>

Prefix

angle (corner ( A, B, C) )

Popcorn

plangeo3.angle(plangeo2.corner($A, $B, $C))

Rendered Presentation MathML

$$\mathrm{angle}\left(\mathrm{corner}(A,B,C)\right)$$

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
  <OMS cd="plangeo3" name="angle"/>
    <OMA>
    <OMS cd="plangeo2" name="corner"/>
      <OMV name="C"/>
      <OMV name="B"/>
      <OMV name="A"/>
    </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="plangeo3">angle</csymbol>
  <apply><csymbol cd="plangeo2">corner</csymbol><ci>C</ci><ci>B</ci><ci>A</ci></apply>
 </apply>
</math>

Prefix

angle (corner ( C, B, A) )

Popcorn

plangeo3.angle(plangeo2.corner($C, $B, $A))

Rendered Presentation MathML

$$\mathrm{angle}\left(\mathrm{corner}(C,B,A)\right)$$

are not the same.

Signatures:

sts

---

[Next: midpoint] [Previous: are_on_circle] [Top]

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midpoint

Description:

The midpoint between two points or two endpoints of a segment.

Example:

The midpoint of two points A and B is the same as the midpoint of the segment S on A and B.

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
  <OMS cd="plangeo3" name="midpoint"/>
      <OMV name="A"/>    
      <OMV name="B"/>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol cd="plangeo3">midpoint</csymbol><ci>A</ci><ci>B</ci></apply></math>

Prefix

midpoint ( A, B)

Popcorn

plangeo3.midpoint($A, $B)

Rendered Presentation MathML

$$\mathrm{midpoint}(A,B)$$

equals

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
  <OMS cd="plangeo3" name="midpoint"/>
  <OMA>
    <OMS cd="plangeo2" name="segment"/>
    <OMV name="A"/>
    <OMV name="B"/>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="plangeo3">midpoint</csymbol>
  <apply><csymbol cd="plangeo2">segment</csymbol><ci>A</ci><ci>B</ci></apply>
 </apply>
</math>

Prefix

midpoint (segment ( A, B) )

Popcorn

plangeo3.midpoint(plangeo2.segment($A, $B))

Rendered Presentation MathML

$$\mathrm{midpoint}\left(\mathrm{segment}(A,B)\right)$$

Signatures:

sts

---

[Next: is_midpoint] [Previous: angle] [Top]

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is_midpoint

Description:

The statement that one point is the midpoint of two others.

Example:

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

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
  <OMS cd="plangeo3" name="is_midpoint"/>
  <OMV name="C"/>    
  <OMV name="A"/>    
  <OMV name="B"/>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="plangeo3">is_midpoint</csymbol><ci>C</ci><ci>A</ci><ci>B</ci></apply>
</math>

Prefix

is_midpoint ( C, A, B)

Popcorn

plangeo3.is_midpoint($C, $A, $B)

Rendered Presentation MathML

$$\mathrm{is\_midpoint}(C,A,B)$$

Signatures:

sts

---

[Next: center_of_gravity] [Previous: midpoint] [Top]

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center_of_gravity

Description:

Center of gravity of a number of points.

Example:

The center of gravity G of three points A, B and C is defined as follows.

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
<OMS cd="plangeo1" name="point"/>
  <OMV name="G"/>
  <OMA>
    <OMS cd="plangeo3" name="center_of_gravity"/>
      <OMV name="A"/>    
      <OMV name="B"/>
      <OMV name="C"/>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="plangeo1">point</csymbol>
  <ci>G</ci>
  <apply><csymbol cd="plangeo3">center_of_gravity</csymbol><ci>A</ci><ci>B</ci><ci>C</ci></apply>
 </apply>
</math>

Prefix

point ( G, center_of_gravity ( A, B, C) )

Popcorn

plangeo1.point($G, plangeo3.center_of_gravity($A, $B, $C))

Rendered Presentation MathML

$$\mathrm{point}(G,\mathrm{center\_of\_gravity}(A,B,C))$$

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
<OMS cd="relation1" name="eq"/>
  <OMA>
    <OMS cd="plangeo3" name="center_of_gravity"/>
    <OMV name="A"/>    
    <OMV name="B"/>
  </OMA>
  <OMA>
    <OMS cd="plangeo3" name="midpoint"/>
    <OMV name="A"/>    
    <OMV name="B"/>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="plangeo3">center_of_gravity</csymbol><ci>A</ci><ci>B</ci></apply>
  <apply><csymbol cd="plangeo3">midpoint</csymbol><ci>A</ci><ci>B</ci></apply>
 </apply>
</math>

Prefix

eq (center_of_gravity ( A, B) , midpoint ( A, B) )

Popcorn

plangeo3.center_of_gravity($A, $B) = plangeo3.midpoint($A, $B)

Rendered Presentation MathML

$$\mathrm{center\_of\_gravity}(A,B)=\mathrm{midpoint}(A,B)$$

Signatures:

sts

---

[Next: perpbisector] [Previous: is_midpoint] [Top]

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perpbisector

Description:

Given two distinct points A and B, this is the line of all points at equal distance to both A and B.

Example:

OpenMath XML (source)

 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
  <OMS cd="plangeo3" name="perpbisector"/>
      <OMV name="A"/>    
      <OMV name="B"/>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol cd="plangeo3">perpbisector</csymbol><ci>A</ci><ci>B</ci></apply></math>

Prefix

perpbisector ( A, B)

Popcorn

plangeo3.perpbisector($A, $B)

Rendered Presentation MathML

$$\mathrm{perpbisector}(A,B)$$

Signatures:

sts

---

[Next: altitude] [Previous: center_of_gravity] [Top]

---

altitude

Description:

Given a point p and a line L, this defines the segment starting at p and ending in the unique point of L closest to p.

Example:

OpenMath XML (source)

 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
  <OMS cd="plangeo3" name="altitude"/>
      <OMV name="p"/>    
      <OMV name="L"/>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol cd="plangeo3">altitude</csymbol><ci>p</ci><ci>L</ci></apply></math>

Prefix

altitude ( p, L)

Popcorn

plangeo3.altitude($p, $L)

Rendered Presentation MathML

$$\mathrm{altitude}(p,L)$$

Signatures:

sts

---

[Next: perpline] [Previous: perpbisector] [Top]

---

perpline

Description:

Given a point p and a line L, this defines the line through p perpendicular to L.

Example:

OpenMath XML (source)

 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
  <OMS cd="plangeo3" name="perpline"/>
      <OMV name="p"/>    
      <OMV name="L"/>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol cd="plangeo3">perpline</csymbol><ci>p</ci><ci>L</ci></apply></math>

Prefix

perpline ( p, L)

Popcorn

plangeo3.perpline($p, $L)

Rendered Presentation MathML

$$\mathrm{perpline}(p,L)$$

Signatures:

sts

---

[Next: polarline] [Previous: altitude] [Top]

---

polarline

Description:

Given a point p and a circle C this defines the polar line of p with respect to C.

Commented Mathematical property (CMP):

If p is incident with the circle C, the polar line of p with respect to C is the tangent line at p with respect to C.

Example:

OpenMath XML (source)

 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
  <OMS cd="plangeo3" name="polarline"/>
      <OMV name="p"/>    
      <OMV name="C"/>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol cd="plangeo3">polarline</csymbol><ci>p</ci><ci>C</ci></apply></math>

Prefix

polarline ( p, C)

Popcorn

plangeo3.polarline($p, $C)

Rendered Presentation MathML

$$\mathrm{polarline}(p,C)$$

Signatures:

sts

---

[Next: tangent] [Previous: perpline] [Top]

---

tangent

Description:

Given a line L and a circle C this boolean checks whether L is a tangent line to C.

Example:

OpenMath XML (source)

 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
  <OMS cd="plangeo3" name="tangent"/>
      <OMV name="L"/>    
      <OMV name="C"/>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol cd="plangeo3">tangent</csymbol><ci>L</ci><ci>C</ci></apply></math>

Prefix

tangent ( L, C)

Popcorn

plangeo3.tangent($L, $C)

Rendered Presentation MathML

$$\mathrm{tangent}(L,C)$$

Signatures:

sts

---

[Next: arc] [Previous: polarline] [Top]

---

arc

Description:

an arc of a circle M from A to B is the set of points of M that are encountered when traversing the circle clockwise from A to B.

Example:

OpenMath XML (source)

 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA>
  <OMS cd="plangeo3" name="arc"/>
  <OMA>
    <OMS cd="plangeo3" name="circle"/>
    <OMV name="M"/>
  </OMA>
  <OMA>
    <OMS cd="plangeo1" name="point"/>
    <OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="plangeo1" name="point"/>
    <OMV name="B"/>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="plangeo3">arc</csymbol>
  <apply><csymbol cd="plangeo3">circle</csymbol><ci>M</ci></apply>
  <apply><csymbol cd="plangeo1">point</csymbol><ci>A</ci></apply>
  <apply><csymbol cd="plangeo1">point</csymbol><ci>B</ci></apply>
 </apply>
</math>

Prefix

arc (circle ( M) , point ( A) , point ( B) )

Popcorn

plangeo3.arc(plangeo3.circle($M), plangeo1.point($A), plangeo1.point($B))

Rendered Presentation MathML

$$\mathrm{arc}(\mathrm{circle}\left(M\right),\mathrm{point}\left(A\right),\mathrm{point}\left(B\right))$$

Signatures:

sts

---

[First: distance] [Previous: tangent] [Top]