OpenMath Content Dictionary: plangeo3

Canonical URL:

plangeo3.ocd

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 2)

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.

xml prefix mathml

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

distance ( A, B)

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

Signatures:

sts

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.

xml prefix mathml

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

perpendicular ( L, M)

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

Signatures:

sts

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.

xml prefix mathml

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

parallel ( L, M)

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

Signatures:

sts

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

xml prefix mathml

<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>

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

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

Signatures:

sts

radius

Description:

The radius of a circle.

Example:

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

xml prefix mathml

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

radius ( C, 1)

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

Signatures:

sts

radius_of

Description:

Gives the radius of a circle.

Example:

The radius of the circle C is given by

xml prefix mathml

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

radius_of ( C)

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

Signatures:

sts

center

Description:

Defines the center of a circle.

Example:

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

xml prefix mathml

<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>

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

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

Signatures:

sts

center_of

Description:

Gives the center of the circle

Example:

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

xml prefix mathml

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

center_of ( C)

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

Signatures:

sts

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.

xml prefix mathml

<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>

are_on_circle ( A, B, C, D)

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

Signatures:

sts

angle

Description:

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

Example:

Notice that

xml prefix mathml

<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>

angle (corner ( A, B, C) )

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

xml prefix mathml

<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>

angle (corner ( C, B, A) )

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

are not the same.

Signatures:

sts

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.

xml prefix mathml

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

midpoint ( A, B)

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

equals

xml prefix mathml

<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>

midpoint (segment ( A, B) )

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

Signatures:

sts

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.

xml prefix mathml

<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>

is_midpoint ( C, A, B)

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

Signatures:

sts

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.

xml prefix mathml

<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>

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

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

Formal Mathematical property (FMP):

xml prefix mathml

<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>

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

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

Signatures:

sts

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:

xml prefix mathml

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

perpbisector ( A, B)

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

Signatures:

sts

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:

xml prefix mathml

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

altitude ( p, L)

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

Signatures:

sts

perpline

Description:

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

Example:

xml prefix mathml

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

perpline ( p, L)

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

Signatures:

sts

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:

xml prefix mathml

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

polarline ( p, C)

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

Signatures:

sts

tangent

Description:

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

Example:

xml prefix mathml

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

tangent ( L, C)

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

Signatures:

sts

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:

xml prefix mathml

<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>

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

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

Signatures:

sts