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The coordinates of a straight line in three-dimensional space, the six numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p0728901.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p0728902.png" />, of which the first three are the coordinates of the direction vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p0728903.png" /> for the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p0728904.png" /> and the second three are the moments of this vector about the origin. Let the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p0728905.png" /> pass through the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p0728906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p0728907.png" /> with projective coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p0728908.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p0728909.png" />, respectively; the Plücker coordinates for this line are the numbers
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289010.png" /></td> </tr></table>
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The Plücker coordinates are used in line geometry. They were first considered by J. Plücker (1869). Sometimes, instead of the Plücker coordinates one uses the Klein coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289011.png" />, which are related to the Plücker ones as follows:
+
The coordinates of a straight line in three-dimensional space, the six numbers  $  p _ {01} , p _ {02} , p _ {03} , p _ {23} , p _ {31} $,
 +
and  $  p _ {12} $,  
 +
of which the first three are the coordinates of the direction vector  $  l $
 +
for the straight line  $  L $
 +
and the second three are the moments of this vector about the origin. Let the line  $  L $
 +
pass through the points  $  X $
 +
and  $  Y $
 +
with projective coordinates $  ( x _ {0} : \dots :  x _ {3} ) $
 +
and  $  ( y _ {0} : \dots : y _ {3} ) $,  
 +
respectively; the Plücker coordinates for this line are the numbers
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289012.png" /></td> </tr></table>
+
$$
 +
p _ {ik}  = x _ {i} y _ {k} - x _ {k} y _ {i} .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289013.png" /></td> </tr></table>
+
The Plücker coordinates are used in line geometry. They were first considered by J. Plücker (1869). Sometimes, instead of the Plücker coordinates one uses the Klein coordinates  $  ( x _ {0} : \dots : x _ {5} ) $,
 +
which are related to the Plücker ones as follows:
  
More generally, one naturally considers the Plücker coordinates as coordinates of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289014.png" />-dimensional vector subspace of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289015.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289016.png" />. Then they are understood as the set of numbers equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289017.png" />-subdeterminants of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289018.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289019.png" /> with as columns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289021.png" />, the coordinate columns (in some basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289022.png" />) of the basis vectors of a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289024.png" /> are the components of a column <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289026.png" />, then the Plücker coordinates (or Grassmann coordinates) are the numbers
+
$$
 +
p _ {01}  = x _ {0} + x _ {1} ,\ \
 +
p _ {02}  = x _ {2} + x _ {3} ,\ \
 +
p _ {03}  = x _ {4} + x _ {5} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289027.png" /></td> </tr></table>
+
$$
 +
p _ {23}  = x _ {0} - x _ {1} ,\  p _ {31}  = x _ {2} - x _ {3} ,\  p _ {12}  = x _ {4} - x _ {5} .
 +
$$
  
The Plücker coordinates are anti-symmetric in all indices. The number of significant Plücker coordinates is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289028.png" />.
+
More generally, one naturally considers the Plücker coordinates as coordinates of a  $  p $-dimensional vector subspace of an  $  n $-dimensional vector space  $  V $.
 +
Then they are understood as the set of numbers equal to  $  ( p \times p) $-subdeterminants of the  $  ( n \times p) $-matrix  $  A( a _ {1} \dots a _ {p} ) $
 +
with as columns  $  a _ {i} $,
 +
$  1 \leq  i \leq  p $,
 +
the coordinate columns (in some basis for  $  V $)
 +
of the basis vectors of a subspace  $  W $.  
 +
If  $  a _ {i}  ^ {j} $
 +
are the components of a column  $  a _ {i} $,
 +
$  1 \leq  i \leq  p $,
 +
then the Plücker coordinates (or Grassmann coordinates) are the numbers
  
When the basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289029.png" /> is changed and the basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289030.png" /> is fixed, the Plücker coordinates are all multiplied by the same non-zero number. When the basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289031.png" /> is changed and the basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289032.png" /> is fixed, the Plücker coordinates transform as the components of a contravariant tensor of valency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289033.png" /> (see [[Poly-vector|Poly-vector]]). Two subspaces coincide if and only if their Plücker coordinates, calculated in the same basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289034.png" />, differ only by a non-zero factor.
+
$$
 +
u ^ {i _ {1} \dots i _ {p} }  = \left |
  
A vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289035.png" /> belongs to a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289036.png" /> if the linear equations
+
\begin{array}{lll}
 +
a _ {1} ^ {i _ {i} }  &\cdots  &a _ {p} ^ {i _ {1} }  \\
 +
\vdots  &\ddots  &\vdots  \\
 +
a _ {1} ^ {i _ {p} }  &\cdots  &a _ {p} ^ {i _ {p} }  \\
 +
\end{array}
 +
\right |  = \
 +
p! a _ {1} ^ {[ i _ {1} } \dots a _ {p} ^ { {} i _ {p} ] } ,\ \
 +
1 \leq  i _  \nu  \leq  n.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289037.png" /></td> </tr></table>
+
The Plücker coordinates are anti-symmetric in all indices. The number of significant Plücker coordinates is  $  ( {} _ {p}  ^ {n} ) $.
  
with coefficients that are the Plücker coordinates for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289038.png" />, are fulfilled. In these equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289039.png" /> are all possible sets of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289040.png" />.
+
When the basis of  $  W $
 +
is changed and the basis for  $  V $
 +
is fixed, the Plücker coordinates are all multiplied by the same non-zero number. When the basis of  $  V $
 +
is changed and the basis for $  W $
 +
is fixed, the Plücker coordinates transform as the components of a contravariant tensor of valency  $  p $
 +
(see [[Poly-vector|Poly-vector]]). Two subspaces coincide if and only if their Plücker coordinates, calculated in the same basis for  $  V $,
 +
differ only by a non-zero factor.
  
 +
A vector  $  x $
 +
belongs to a subspace  $  W $
 +
if the linear equations
  
 +
$$
 +
\sum _ {\alpha = 1 } ^ { p+1 } (- 1) ^ {\alpha - 1 } x ^ {i _  \alpha  } u ^
 +
{i _ {1} \dots i _ {\alpha - 1 }  i _ {\alpha + 1 }  \dots i _ {p} }  =  0,
 +
$$
 +
 +
with coefficients that are the Plücker coordinates for  $  W $,
 +
are fulfilled. In these equations  $  i _ {1} < \dots < i _ {p} $
 +
are all possible sets of numbers  $  1 \dots n $.
  
 
====Comments====
 
====Comments====
 
Relating the Plücker and Klein coordinates as above, the Plücker identity
 
Relating the Plücker and Klein coordinates as above, the Plücker identity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289041.png" /></td> </tr></table>
+
$$
 +
p _ {01} p _ {23} + p _ {02} p _ {31} + p _ {03} p _ {12}  = 0
 +
$$
  
 
becomes
 
becomes
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289042.png" /></td> </tr></table>
+
$$
 +
x _ {0}  ^ {2} + x _ {2}  ^ {2} + x _ {4}  ^ {2}  = \
 +
x _ {1}  ^ {2} + x _ {3}  ^ {2} + x _ {5}  ^ {2} .
 +
$$
  
The Plücker coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289043.png" />-dimensional subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289044.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289045.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289046.png" /> (over any field) define an imbedding of the Grassmann variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289047.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289048.png" />-dimensional projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289049.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289050.png" />. As a subvariety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289052.png" /> is given by quadratic relations, the Plücker relations, which look as follows:
+
The Plücker coordinates of p $-dimensional subspaces $  W $
 +
of an $  n $-dimensional space $  V $
 +
(over any field) define an imbedding of the Grassmann variety $  G _ {p} ( V) $
 +
into $  N $-dimensional projective space $  P  ^ {N} $
 +
with $  N = ( {} _ {p}  ^ {n} ) - 1 $.  
 +
As a subvariety of $  P  ^ {N} $,  
 +
$  G _ {p} ( V) $
 +
is given by quadratic relations, the Plücker relations, which look as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289053.png" /></td> </tr></table>
+
$$
 +
\sum _ { k=1 } ^ { p }  (- 1)  ^ {k} u ^ {i _ {1} \dots i _ {p-1} j _ {k} }
 +
u ^ {j _ {1} \dots \widehat{j _ {k} }  \dots j _ {p+1} }  = 0,
 +
$$
  
i.e. take <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289054.png" /> indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289055.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289056.png" /> and write down the relation above, using that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289057.png" /> if two of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289058.png" />'s are equal. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289060.png" />, there is just one relation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289061.png" />.
+
i.e. take $  2p $
 +
indices $  1 \leq  i _ {1} \dots i _ {p-1} $;  
 +
$  j _ {1} \dots j _ {p+1} \leq  n $
 +
and write down the relation above, using that $  u ^ {k _ {1} \dots k _ {p} } = 0 $
 +
if two of the $  k $'s are equal. If $  p = 2 $,  
 +
$  n = 4 $,  
 +
there is just one relation: $  u  ^ {12} u  ^ {34} - u  ^ {13} u  ^ {24} + u  ^ {14} u  ^ {23} = 0 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,  "Non-Euclidean geometry" , Univ. Toronto Press  (1965)  pp. 88–90</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.L. van der Waerden,  "Einführung in die algebraische Geometrie" , Springer  (1939)  pp. Chapt. 1</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,  "Non-Euclidean geometry" , Univ. Toronto Press  (1965)  pp. 88–90</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.L. van der Waerden,  "Einführung in die algebraische Geometrie" , Springer  (1939)  pp. Chapt. 1</TD></TR></table>

Latest revision as of 01:23, 21 January 2022


The coordinates of a straight line in three-dimensional space, the six numbers $ p _ {01} , p _ {02} , p _ {03} , p _ {23} , p _ {31} $, and $ p _ {12} $, of which the first three are the coordinates of the direction vector $ l $ for the straight line $ L $ and the second three are the moments of this vector about the origin. Let the line $ L $ pass through the points $ X $ and $ Y $ with projective coordinates $ ( x _ {0} : \dots : x _ {3} ) $ and $ ( y _ {0} : \dots : y _ {3} ) $, respectively; the Plücker coordinates for this line are the numbers

$$ p _ {ik} = x _ {i} y _ {k} - x _ {k} y _ {i} . $$

The Plücker coordinates are used in line geometry. They were first considered by J. Plücker (1869). Sometimes, instead of the Plücker coordinates one uses the Klein coordinates $ ( x _ {0} : \dots : x _ {5} ) $, which are related to the Plücker ones as follows:

$$ p _ {01} = x _ {0} + x _ {1} ,\ \ p _ {02} = x _ {2} + x _ {3} ,\ \ p _ {03} = x _ {4} + x _ {5} , $$

$$ p _ {23} = x _ {0} - x _ {1} ,\ p _ {31} = x _ {2} - x _ {3} ,\ p _ {12} = x _ {4} - x _ {5} . $$

More generally, one naturally considers the Plücker coordinates as coordinates of a $ p $-dimensional vector subspace of an $ n $-dimensional vector space $ V $. Then they are understood as the set of numbers equal to $ ( p \times p) $-subdeterminants of the $ ( n \times p) $-matrix $ A( a _ {1} \dots a _ {p} ) $ with as columns $ a _ {i} $, $ 1 \leq i \leq p $, the coordinate columns (in some basis for $ V $) of the basis vectors of a subspace $ W $. If $ a _ {i} ^ {j} $ are the components of a column $ a _ {i} $, $ 1 \leq i \leq p $, then the Plücker coordinates (or Grassmann coordinates) are the numbers

$$ u ^ {i _ {1} \dots i _ {p} } = \left | \begin{array}{lll} a _ {1} ^ {i _ {i} } &\cdots &a _ {p} ^ {i _ {1} } \\ \vdots &\ddots &\vdots \\ a _ {1} ^ {i _ {p} } &\cdots &a _ {p} ^ {i _ {p} } \\ \end{array} \right | = \ p! a _ {1} ^ {[ i _ {1} } \dots a _ {p} ^ { {} i _ {p} ] } ,\ \ 1 \leq i _ \nu \leq n. $$

The Plücker coordinates are anti-symmetric in all indices. The number of significant Plücker coordinates is $ ( {} _ {p} ^ {n} ) $.

When the basis of $ W $ is changed and the basis for $ V $ is fixed, the Plücker coordinates are all multiplied by the same non-zero number. When the basis of $ V $ is changed and the basis for $ W $ is fixed, the Plücker coordinates transform as the components of a contravariant tensor of valency $ p $ (see Poly-vector). Two subspaces coincide if and only if their Plücker coordinates, calculated in the same basis for $ V $, differ only by a non-zero factor.

A vector $ x $ belongs to a subspace $ W $ if the linear equations

$$ \sum _ {\alpha = 1 } ^ { p+1 } (- 1) ^ {\alpha - 1 } x ^ {i _ \alpha } u ^ {i _ {1} \dots i _ {\alpha - 1 } i _ {\alpha + 1 } \dots i _ {p} } = 0, $$

with coefficients that are the Plücker coordinates for $ W $, are fulfilled. In these equations $ i _ {1} < \dots < i _ {p} $ are all possible sets of numbers $ 1 \dots n $.

Comments

Relating the Plücker and Klein coordinates as above, the Plücker identity

$$ p _ {01} p _ {23} + p _ {02} p _ {31} + p _ {03} p _ {12} = 0 $$

becomes

$$ x _ {0} ^ {2} + x _ {2} ^ {2} + x _ {4} ^ {2} = \ x _ {1} ^ {2} + x _ {3} ^ {2} + x _ {5} ^ {2} . $$

The Plücker coordinates of $ p $-dimensional subspaces $ W $ of an $ n $-dimensional space $ V $ (over any field) define an imbedding of the Grassmann variety $ G _ {p} ( V) $ into $ N $-dimensional projective space $ P ^ {N} $ with $ N = ( {} _ {p} ^ {n} ) - 1 $. As a subvariety of $ P ^ {N} $, $ G _ {p} ( V) $ is given by quadratic relations, the Plücker relations, which look as follows:

$$ \sum _ { k=1 } ^ { p } (- 1) ^ {k} u ^ {i _ {1} \dots i _ {p-1} j _ {k} } u ^ {j _ {1} \dots \widehat{j _ {k} } \dots j _ {p+1} } = 0, $$

i.e. take $ 2p $ indices $ 1 \leq i _ {1} \dots i _ {p-1} $; $ j _ {1} \dots j _ {p+1} \leq n $ and write down the relation above, using that $ u ^ {k _ {1} \dots k _ {p} } = 0 $ if two of the $ k $'s are equal. If $ p = 2 $, $ n = 4 $, there is just one relation: $ u ^ {12} u ^ {34} - u ^ {13} u ^ {24} + u ^ {14} u ^ {23} = 0 $.

References

[a1] H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1965) pp. 88–90
[a2] B.L. van der Waerden, "Einführung in die algebraische Geometrie" , Springer (1939) pp. Chapt. 1
How to Cite This Entry:
Plücker coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pl%C3%BCcker_coordinates&oldid=23454
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article