Difference between revisions of "Plücker coordinates"

Revision as of 14:52, 7 June 2020

The coordinates of a straight line in three-dimensional space, the six numbers , and , of which the first three are the coordinates of the direction vector for the straight line and the second three are the moments of this vector about the origin. Let the line pass through the points and with projective coordinates and , respectively; the Plücker coordinates for this line are the numbers

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 , which are related to the Plücker ones as follows:

More generally, one naturally considers the Plücker coordinates as coordinates of a -dimensional vector subspace of an -dimensional vector space . Then they are understood as the set of numbers equal to -subdeterminants of the -matrix with as columns , , the coordinate columns (in some basis for ) of the basis vectors of a subspace . If are the components of a column , , then the Plücker coordinates (or Grassmann coordinates) are the numbers

The Plücker coordinates are anti-symmetric in all indices. The number of significant Plücker coordinates is .

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

A vector belongs to a subspace if the linear equations

with coefficients that are the Plücker coordinates for , are fulfilled. In these equations are all possible sets of numbers .

Comments

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

becomes

The Plücker coordinates of -dimensional subspaces of an -dimensional space (over any field) define an imbedding of the Grassmann variety into -dimensional projective space with . As a subvariety of , is given by quadratic relations, the Plücker relations, which look as follows:

i.e. take indices ; and write down the relation above, using that if two of the 's are equal. If , , there is just one relation: .

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=48194

This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article

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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
-p0728901.png
-$#A+1 = 61 n = 0
-$#C+1 = 61 : ~/encyclopedia/old_files/data/P072/P.0702890 Pl\AGucker coordinates
-Automatically converted into TeX, above some diagnostics.
-Please remove this comment and the {{TEX|auto}} line below,
-if TeX found to be correct.
--->
-{{TEX|auto}}
+<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 coordinates of a straight line in three-dimensional space, the six numbers  $  p _ {01} , p _ {02} , p _ {03} , p _ {23} , p _ {31} $,
+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:
-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
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289012.png" /></td> </tr></table>
-p _ {ik}  =  x _ {i} y _ {k} - x _ {k} y _ {i} .
-$$
-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} ) $,
+<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>
-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} ,
-$$
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289027.png" /></td> </tr></table>
-p _ {23}  =  x _ {0} - x _ {1} ,\  p _ {31}  =  x _ {2} - x _ {3} ,\  p _ {12}  =  x _ {4} - x _ {5} .
-$$
-More generally, one naturally considers the Plücker coordinates as coordinates of a  $  p $-
+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" />.
-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 |
-The Plücker coordinates are anti-symmetric in all indices. The number of significant Plücker coordinates is  $  ( {} _ {p}  ^ {n} ) $.
+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
-When the basis of  $  W $
+<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>
-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 $
+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" />.
-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,
-$$
-with coefficients that are the Plücker coordinates for  $  W $,
-are fulfilled. In these equations  $  i _ {1} < \dots < i _ {p} $
-are all possible sets of numbers  $  1 \dots n $.
 ====Comments====
 Relating the Plücker and Klein coordinates as above, the Plücker identity
-$$
+<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
-$$
 becomes
-$$
+<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} .
-$$
-The Plücker coordinates of  $  p $-
+The Plücker coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289043.png" />-dimensional subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289044.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289045.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289046.png" /> (over any field) define an imbedding of the Grassmann variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289047.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289048.png" />-dimensional projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289049.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289050.png" />. As a subvariety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072890/p07289052.png" /> is given by quadratic relations, the Plücker relations, which look as follows:
-dimensional subspaces  $  W $
-of an  $  n $-
-dimensional space  $  V $(
-over any field) define an imbedding of the Grassmann variety  $  G _ {p} ( V) $
-into  $  N $-
-dimensional projective space  $  P  ^ {N} $
-with  $  N = ( {} _ {p}  ^ {n} ) - 1 $.
-As a subvariety of  $  P  ^ {N} $,
-$  G _ {p} ( V) $
-is given by quadratic relations, the Plücker relations, which look as follows:
-$$
+<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-} j j _ {k} }
-u ^ {j _ {1} \dots {j _ {k} } hat \dots j _ {p+} 1 }  =  0,
-$$
-i.e. take  $  2p $
+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" />.
-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====
 <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>

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Difference between revisions of "Plücker coordinates"

Revision as of 14:52, 7 June 2020

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