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A one-to-one correspondence between the elements of a [[Projective space|projective space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p0751901.png" /> (projective subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p0751902.png" />) and the equivalence classes of ordered finite subsets of elements of a [[Skew-field|skew-field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p0751903.png" />. Projective coordinates of subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p0751904.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p0751905.png" /> (also called Grassmann coordinates) are defined in terms of coordinates of the points (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p0751906.png" />-dimensional subspaces) lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p0751907.png" />. Therefore it suffices to define the projective coordinates of the points of a projective space.
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$#C+1 = 122 : ~/encyclopedia/old_files/data/P075/P.0705190 Projective coordinates
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Suppose that in the collection of rows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p0751908.png" /> of elements of a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p0751909.png" /> that are not simultaneously zero (they are also called homogeneous point coordinates) a left (right) equivalence relation is introduced: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519010.png" /> if there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519013.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519014.png" />. Then the collection of equivalence classes is in one-to-one correspondence with the collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519015.png" /> of points of the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519016.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519017.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519018.png" /> is interpreted as the set of straight lines of the left (right) vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519019.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519020.png" />), then the homogeneous coordinates of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519021.png" /> have the meaning of the coordinates of the vectors belonging to the straight line that represents this point, and the projective coordinates have the meaning of the collection of all such coordinates.
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In the general case, projective coordinates of points of a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519022.png" /> relative to some basis are introduced by purely projective means (under the necessary condition that the [[Desargues assumption|Desargues assumption]] holds in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519023.png" />) as follows:
+
A one-to-one correspondence between the elements of a [[Projective space|projective space]]  $  \Pi _ {n} ( K) $(
 +
projective subspaces  $  S _ {q} $)
 +
and the equivalence classes of ordered finite subsets of elements of a [[Skew-field|skew-field]] $  K $.
 +
Projective coordinates of subspaces  $  S _ {q} $
 +
for  $  q > 0 $(
 +
also called Grassmann coordinates) are defined in terms of coordinates of the points ( $  0 $-
 +
dimensional subspaces) lying in  $  S _ {q} $.  
 +
Therefore it suffices to define the projective coordinates of the points of a projective space.
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519025.png" /> independent points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519026.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519027.png" /> is called a simplex. In this case the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519029.png" /> are also independent and determine a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519030.png" />, which is called a face of this simplex. There exists a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519031.png" /> that lies on none of the faces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519032.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519033.png" /> be any permutation of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519034.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519037.png" /> turn out to be independent and determine some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519038.png" />. Next, the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519039.png" /> also determine some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519040.png" />, and since the sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519042.png" /> is the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519045.png" /> have exactly one common point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519046.png" /> that lies in none of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519047.png" />-dimensional subspaces determined by the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519049.png" />; in this case the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519050.png" /> are also independent. Thus one obtains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519051.png" /> points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519052.png" />, including the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519054.png" />, which constitute a frame of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519055.png" />; the simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519056.png" /> is its skeleton.
+
Suppose that in the collection of rows  $  ( x  ^ {0} \dots x  ^ {n} ) = x $
 +
of elements of a skew-field  $  K $
 +
that are not simultaneously zero (they are also called homogeneous point coordinates) a left (right) equivalence relation is introduced: $  x \sim y $
 +
if there is a $  \lambda \in K $
 +
such that $  x  ^ {i} = \lambda y  ^ {i} $(
 +
$  x  ^ {i} = y  ^ {i} \lambda $),
 +
$  i = 0 \dots n $.  
 +
Then the collection of equivalence classes is in one-to-one correspondence with the collection  $  {\mathcal P} $
 +
of points of the projective space  $  P _ {n}  ^ {l} ( K) $(
 +
respectively, $  P _ {n}  ^ {r} ( K) $).  
 +
If  $  {\mathcal P} $
 +
is interpreted as the set of straight lines of the left (right) vector space  $  A _ {n+} 1  ^ {l} ( K) $(
 +
respectively, $  A _ {n+} 1  ^ {r} ( K) $),  
 +
then the homogeneous coordinates of a point  $  M $
 +
have the meaning of the coordinates of the vectors belonging to the straight line that represents this point, and the projective coordinates have the meaning of the collection of all such coordinates.
  
On each straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519057.png" /> there are three points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519058.png" />; suppose they play the role of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519059.png" /> in the definition of the skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519060.png" /> of the projective geometry under consideration (see [[Projective algebra|Projective algebra]]). The skew-fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519062.png" /> are isomorphic to one another, and the isomorphism is established by a projective correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519063.png" /> between the points of the two lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519065.png" /> such that the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519066.png" /> correspond to the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519067.png" />. The element of the skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519068.png" /> corresponding to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519069.png" /> of the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519070.png" /> is called the projective coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519071.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519072.png" /> in the scale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519073.png" />. In particular, the projective coordinate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519074.png" /> is always 1, while the projective coordinate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519075.png" /> in the scale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519076.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519077.png" />.
+
In the general case, projective coordinates of points of a projective space  $  \Pi _ {n} $
 +
relative to some basis are introduced by purely projective means (under the necessary condition that the [[Desargues assumption|Desargues assumption]] holds in  $  \Pi _ {n} $)  
 +
as follows:
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519078.png" /> be a point of the space that lies on none of the faces of the simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519079.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519080.png" /> that together with some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519081.png" /> forms a frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519082.png" />. If one uses the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519083.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519084.png" /> in the above construction of the frame, then one obtains a sequence of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519085.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519086.png" /> lies in the subspace determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519087.png" /> (but lies in none of the faces of the simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519088.png" /> formed by these points). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519089.png" /> be the coordinate of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519090.png" /> (lying on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519091.png" />) in the scale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519092.png" />. If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519093.png" /> are distinct, then
+
A set  $  \sigma _ {n} $
 +
of  $  n+ 1 $
 +
independent points  $  A _ {0} \dots A _ {n} $
 +
of the space  $  \Pi _ {n} $
 +
is called a simplex. In this case the points  $  A _ {0} \dots A _ {i-} 1 $,
 +
$  A _ {i+} 1 \dots A _ {n} $
 +
are also independent and determine a subspace  $  S _ {n-} 1 = \Sigma  ^ {i} $,
 +
which is called a face of this simplex. There exists a point $  E $
 +
that lies on none of the faces $  \Sigma  ^ {i} $.
 +
Let  $  i _ {0} \dots i _ {n} $
 +
be any permutation of the numbers  $  0 \dots n $.  
 +
The points  $  A _ {i _ {k+} 1 }  \dots A _ {i _ {n}  } $,
 +
$  k \geq  0 $,
 +
and  $  E $
 +
turn out to be independent and determine some $  S _ {n-} k $.  
 +
Next, the points  $  A _ {i _ {0}  } \dots A _ {i _ {k}  } $
 +
also determine some  $  S _ {k} $,
 +
and since the sum of $  S _ {k} $
 +
and  $  S _ {n-} k $
 +
is the entire space  $  \Pi _ {n} $,  
 +
$  S _ {k} $
 +
and  $  S _ {n-} k $
 +
have exactly one common point  $  E _ {i _ {0}  \dots i _ {k} } $
 +
that lies in none of the $  ( k- 1) $-
 +
dimensional subspaces determined by the points  $  A _ {i _ {0}  } \dots A _ {i _ {j-} 1 }  $,
 +
$  A _ {i _ {j+} 1 }  \dots A _ {i _ {k}  } $;
 +
in this case the points  $  E _ {i _ {0}  \dots i _ {k} } , A _ {i _ {k+} 1 }  \dots A _ {i _ {n}  } $
 +
are also independent. Thus one obtains  $  2  ^ {n+} 1 - 1 $
 +
points $  E _ {i _ {0}  \dots i _ {k} } $,
 +
including the points  $  E _ {i} = A _ {i} $,
 +
$  E _ {0 \dots n }  = E $,
 +
which constitute a frame of the space  $  S _ {n} = \Pi _ {n} $;
 +
the simplex  $  \sigma _ {n} $
 +
is its skeleton.
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519094.png" />;
+
On each straight line  $  A _ {i} A _ {j} $
 +
there are three points  $  A _ {i} , A _ {j} , E _ {ij} $;
 +
suppose they play the role of the points  $  O , U , E $
 +
in the definition of the skew-field  $  K $
 +
of the projective geometry under consideration (see [[Projective algebra|Projective algebra]]). The skew-fields  $  K ( A _ {i} , E _ {ij} , A _ {j} ) $
 +
and  $  K ( A _ {k} , E _ {kl} , A _ {l} ) $
 +
are isomorphic to one another, and the isomorphism is established by a projective correspondence  $  T _ {ij}  ^ {kl} $
 +
between the points of the two lines  $  A _ {i} A _ {j} $
 +
and  $  A _ {k} A _ {l} $
 +
such that the points  $  A _ {k} , E _ {kl} , A _ {l} $
 +
correspond to the points  $  A _ {i} , E _ {ij} , A _ {j} $.
 +
The element of the skew-field  $  K $
 +
corresponding to a point  $  P $
 +
of the straight line  $  A _ {i} A _ {j} $
 +
is called the projective coordinate  $  p $
 +
of the point  $  P $
 +
in the scale  $  ( A _ {i} , E _ {ij} , A _ {j} ) $.
 +
In particular, the projective coordinate of  $  E _ {ij} $
 +
is always 1, while the projective coordinate of  $  P $
 +
in the scale  $  ( A _ {j} , E _ {ij} , A _ {i} ) $
 +
is  $  p  ^ {*} = p ^ {-} 1 $.
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519095.png" />.
+
Let  $  P $
 +
be a point of the space that lies on none of the faces of the simplex  $  \sigma _ {n} $:
 +
$  A _ {0} \dots A _ {n} $
 +
that together with some point  $  E $
 +
forms a frame  $  R $.  
 +
If one uses the point  $  P $
 +
instead of  $  E $
 +
in the above construction of the frame, then one obtains a sequence of points  $  P _ {i} , P _ {ij} , P _ {ijk} \dots $
 +
where  $  P _ {i _ {0}  \dots i _ {k} } $
 +
lies in the subspace determined by  $  A _ {i _ {0}  } \dots A _ {i _ {k}  } $(
 +
but lies in none of the faces of the simplex  $  \sigma _ {k} $
 +
formed by these points). Let  $  p _ {ij} $
 +
be the coordinate of a point  $  P _ {ij} $(
 +
lying on  $  A _ {i} A _ {j} $)
 +
in the scale  $  ( A _ {i} , E _ {ij} , A _ {j} ) $.  
 +
If the  $  i , j , k $
 +
are distinct, then
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519096.png" /> be an arbitrary element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519097.png" /> different from zero, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p07519099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190100.png" /> (in this case it turns out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190101.png" />). Then the collection of equivalent rows determined by various elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190102.png" /> gives the projective coordinates of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190103.png" /> with respect to the frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190104.png" />.
+
1) p _ {ij} p _ {ji} = 1 $;
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190105.png" /> lies in the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190106.png" /> determined by the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190107.png" /> but in none of the faces of the simplex determined by these points. Let the collection of equivalent rows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190108.png" /> be the projective coordinates of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190109.png" /> with respect to a frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190110.png" /> of the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190111.png" /> determined by a simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190112.png" /> and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190113.png" />. Then the projective coordinates of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190114.png" /> with respect to the frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190115.png" /> are given as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190117.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190118.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190119.png" />.
+
2)  $  p _ {ik} p _ {kj} p _ {ji} = 1 $.
  
Any collection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190120.png" /> left (right) equivalent rows constructed by the above method corresponds to one and only one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190121.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075190/p075190122.png" /> and therefore defines projective coordinates in it.
+
Let  $  x _ {0} $
 +
be an arbitrary element of  $  K $
 +
different from zero, and let  $  x _ {i} = x _ {0} p _ {i0} $,
 +
$  x \neq 0 $,
 +
$  i = 1 \dots n $(
 +
in this case it turns out that  $  p _ {ij} = x _ {j}  ^ {-} 1 x _ {i} $).
 +
Then the collection of equivalent rows determined by various elements  $  x _ {0} $
 +
gives the projective coordinates of the point  $  P $
 +
with respect to the frame  $  R $.
 +
 
 +
Suppose that  $  P $
 +
lies in the subspace  $  S _ {k} $
 +
determined by the points  $  A _ {i _ {0}  } \dots A _ {i _ {k}  } $
 +
but in none of the faces of the simplex determined by these points. Let the collection of equivalent rows  $  ( x _ {i _ {0}  } \dots x _ {i _ {k}  } ) $
 +
be the projective coordinates of a point  $  P $
 +
with respect to a frame  $  R $
 +
of the subspace  $  S _ {k} $
 +
determined by a simplex  $  \sigma _ {k} $
 +
and a point  $  E _ {i _ {0}  \dots i _ {k} } $.
 +
Then the projective coordinates of the point  $  P $
 +
with respect to the frame  $  R $
 +
are given as follows:  $  y _ {i} = x _ {i} $,
 +
$  i = i _ {0} \dots i _ {k} $;
 +
$  y _ {i} = 0 $,
 +
$  i \neq i _ {0} \dots i _ {k} $.
 +
 
 +
Any collection of  $  n+ 1 $
 +
left (right) equivalent rows constructed by the above method corresponds to one and only one point $  P $
 +
of the space $  \Pi _ {n} $
 +
and therefore defines projective coordinates in it.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''1''' , Cambridge Univ. Press (1947) {{MR|0028055}} {{ZBL|0796.14002}} {{ZBL|0796.14003}} {{ZBL|0796.14001}} {{ZBL|0157.27502}} {{ZBL|0157.27501}} {{ZBL|0055.38705}} {{ZBL|0048.14502}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''1''' , Cambridge Univ. Press (1947) {{MR|0028055}} {{ZBL|0796.14002}} {{ZBL|0796.14003}} {{ZBL|0796.14001}} {{ZBL|0157.27502}} {{ZBL|0157.27501}} {{ZBL|0055.38705}} {{ZBL|0048.14502}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
For transformations of projective coordinates see e.g. [[Collineation|Collineation]]; [[Projective transformation|Projective transformation]].
 
For transformations of projective coordinates see e.g. [[Collineation|Collineation]]; [[Projective transformation|Projective transformation]].

Revision as of 08:08, 6 June 2020


A one-to-one correspondence between the elements of a projective space $ \Pi _ {n} ( K) $( projective subspaces $ S _ {q} $) and the equivalence classes of ordered finite subsets of elements of a skew-field $ K $. Projective coordinates of subspaces $ S _ {q} $ for $ q > 0 $( also called Grassmann coordinates) are defined in terms of coordinates of the points ( $ 0 $- dimensional subspaces) lying in $ S _ {q} $. Therefore it suffices to define the projective coordinates of the points of a projective space.

Suppose that in the collection of rows $ ( x ^ {0} \dots x ^ {n} ) = x $ of elements of a skew-field $ K $ that are not simultaneously zero (they are also called homogeneous point coordinates) a left (right) equivalence relation is introduced: $ x \sim y $ if there is a $ \lambda \in K $ such that $ x ^ {i} = \lambda y ^ {i} $( $ x ^ {i} = y ^ {i} \lambda $), $ i = 0 \dots n $. Then the collection of equivalence classes is in one-to-one correspondence with the collection $ {\mathcal P} $ of points of the projective space $ P _ {n} ^ {l} ( K) $( respectively, $ P _ {n} ^ {r} ( K) $). If $ {\mathcal P} $ is interpreted as the set of straight lines of the left (right) vector space $ A _ {n+} 1 ^ {l} ( K) $( respectively, $ A _ {n+} 1 ^ {r} ( K) $), then the homogeneous coordinates of a point $ M $ have the meaning of the coordinates of the vectors belonging to the straight line that represents this point, and the projective coordinates have the meaning of the collection of all such coordinates.

In the general case, projective coordinates of points of a projective space $ \Pi _ {n} $ relative to some basis are introduced by purely projective means (under the necessary condition that the Desargues assumption holds in $ \Pi _ {n} $) as follows:

A set $ \sigma _ {n} $ of $ n+ 1 $ independent points $ A _ {0} \dots A _ {n} $ of the space $ \Pi _ {n} $ is called a simplex. In this case the points $ A _ {0} \dots A _ {i-} 1 $, $ A _ {i+} 1 \dots A _ {n} $ are also independent and determine a subspace $ S _ {n-} 1 = \Sigma ^ {i} $, which is called a face of this simplex. There exists a point $ E $ that lies on none of the faces $ \Sigma ^ {i} $. Let $ i _ {0} \dots i _ {n} $ be any permutation of the numbers $ 0 \dots n $. The points $ A _ {i _ {k+} 1 } \dots A _ {i _ {n} } $, $ k \geq 0 $, and $ E $ turn out to be independent and determine some $ S _ {n-} k $. Next, the points $ A _ {i _ {0} } \dots A _ {i _ {k} } $ also determine some $ S _ {k} $, and since the sum of $ S _ {k} $ and $ S _ {n-} k $ is the entire space $ \Pi _ {n} $, $ S _ {k} $ and $ S _ {n-} k $ have exactly one common point $ E _ {i _ {0} \dots i _ {k} } $ that lies in none of the $ ( k- 1) $- dimensional subspaces determined by the points $ A _ {i _ {0} } \dots A _ {i _ {j-} 1 } $, $ A _ {i _ {j+} 1 } \dots A _ {i _ {k} } $; in this case the points $ E _ {i _ {0} \dots i _ {k} } , A _ {i _ {k+} 1 } \dots A _ {i _ {n} } $ are also independent. Thus one obtains $ 2 ^ {n+} 1 - 1 $ points $ E _ {i _ {0} \dots i _ {k} } $, including the points $ E _ {i} = A _ {i} $, $ E _ {0 \dots n } = E $, which constitute a frame of the space $ S _ {n} = \Pi _ {n} $; the simplex $ \sigma _ {n} $ is its skeleton.

On each straight line $ A _ {i} A _ {j} $ there are three points $ A _ {i} , A _ {j} , E _ {ij} $; suppose they play the role of the points $ O , U , E $ in the definition of the skew-field $ K $ of the projective geometry under consideration (see Projective algebra). The skew-fields $ K ( A _ {i} , E _ {ij} , A _ {j} ) $ and $ K ( A _ {k} , E _ {kl} , A _ {l} ) $ are isomorphic to one another, and the isomorphism is established by a projective correspondence $ T _ {ij} ^ {kl} $ between the points of the two lines $ A _ {i} A _ {j} $ and $ A _ {k} A _ {l} $ such that the points $ A _ {k} , E _ {kl} , A _ {l} $ correspond to the points $ A _ {i} , E _ {ij} , A _ {j} $. The element of the skew-field $ K $ corresponding to a point $ P $ of the straight line $ A _ {i} A _ {j} $ is called the projective coordinate $ p $ of the point $ P $ in the scale $ ( A _ {i} , E _ {ij} , A _ {j} ) $. In particular, the projective coordinate of $ E _ {ij} $ is always 1, while the projective coordinate of $ P $ in the scale $ ( A _ {j} , E _ {ij} , A _ {i} ) $ is $ p ^ {*} = p ^ {-} 1 $.

Let $ P $ be a point of the space that lies on none of the faces of the simplex $ \sigma _ {n} $: $ A _ {0} \dots A _ {n} $ that together with some point $ E $ forms a frame $ R $. If one uses the point $ P $ instead of $ E $ in the above construction of the frame, then one obtains a sequence of points $ P _ {i} , P _ {ij} , P _ {ijk} \dots $ where $ P _ {i _ {0} \dots i _ {k} } $ lies in the subspace determined by $ A _ {i _ {0} } \dots A _ {i _ {k} } $( but lies in none of the faces of the simplex $ \sigma _ {k} $ formed by these points). Let $ p _ {ij} $ be the coordinate of a point $ P _ {ij} $( lying on $ A _ {i} A _ {j} $) in the scale $ ( A _ {i} , E _ {ij} , A _ {j} ) $. If the $ i , j , k $ are distinct, then

1) $ p _ {ij} p _ {ji} = 1 $;

2) $ p _ {ik} p _ {kj} p _ {ji} = 1 $.

Let $ x _ {0} $ be an arbitrary element of $ K $ different from zero, and let $ x _ {i} = x _ {0} p _ {i0} $, $ x \neq 0 $, $ i = 1 \dots n $( in this case it turns out that $ p _ {ij} = x _ {j} ^ {-} 1 x _ {i} $). Then the collection of equivalent rows determined by various elements $ x _ {0} $ gives the projective coordinates of the point $ P $ with respect to the frame $ R $.

Suppose that $ P $ lies in the subspace $ S _ {k} $ determined by the points $ A _ {i _ {0} } \dots A _ {i _ {k} } $ but in none of the faces of the simplex determined by these points. Let the collection of equivalent rows $ ( x _ {i _ {0} } \dots x _ {i _ {k} } ) $ be the projective coordinates of a point $ P $ with respect to a frame $ R $ of the subspace $ S _ {k} $ determined by a simplex $ \sigma _ {k} $ and a point $ E _ {i _ {0} \dots i _ {k} } $. Then the projective coordinates of the point $ P $ with respect to the frame $ R $ are given as follows: $ y _ {i} = x _ {i} $, $ i = i _ {0} \dots i _ {k} $; $ y _ {i} = 0 $, $ i \neq i _ {0} \dots i _ {k} $.

Any collection of $ n+ 1 $ left (right) equivalent rows constructed by the above method corresponds to one and only one point $ P $ of the space $ \Pi _ {n} $ and therefore defines projective coordinates in it.

References

[1] W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , 1 , Cambridge Univ. Press (1947) MR0028055 Zbl 0796.14002 Zbl 0796.14003 Zbl 0796.14001 Zbl 0157.27502 Zbl 0157.27501 Zbl 0055.38705 Zbl 0048.14502

Comments

For transformations of projective coordinates see e.g. Collineation; Projective transformation.

How to Cite This Entry:
Projective coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_coordinates&oldid=48317
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article