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A model that realizes the geometry of the three-dimensional projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p0729101.png" /> in the hyperbolic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p0729102.png" />. The Plücker interpretation is based on a special interpretation of the [[Plücker coordinates|Plücker coordinates]] of a straight line, which are defined for any straight line in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p0729103.png" />.
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Under projective transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p0729104.png" /> the Plücker coordinates transform linearly; the Plücker coordinates of straight lines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p0729105.png" /> give a one-to-one correspondence between the straight lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p0729106.png" /> and the points in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p0729107.png" /> whose coordinates are numerically equal to the Plücker coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p0729108.png" />.
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Straight lines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p0729109.png" /> are represented by the points of a non-singular quadric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291010.png" /> of index three.
+
A model that realizes the geometry of the three-dimensional projective space  $  P _ {3} $
 +
in the hyperbolic space  $  {}  ^ {3} S _ {5} $.  
 +
The Plücker interpretation is based on a special interpretation of the [[Plücker coordinates|Plücker coordinates]] of a straight line, which are defined for any straight line in $  P _ {3} $.
  
If one takes this quadric as the absolute and defines a projective (non-Euclidean) metric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291011.png" />, one gets the five-dimensional hyperbolic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291012.png" />. Under each [[Collineation|collineation]] and [[Correlation|correlation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291013.png" /> the Plücker coordinates transform linearly, i.e. each collineation and correlation is represented by a collineation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291014.png" /> that maps the absolute into itself. These collineations are thus displacements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291015.png" />. The displacements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291016.png" /> represent either collineations or correlations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291017.png" />.
+
Under projective transformations of $  P _ {3} $
 +
the Plücker coordinates transform linearly; the Plücker coordinates of straight lines in  $  P _ {3} $
 +
give a one-to-one correspondence between the straight lines of $  P _ {3} $
 +
and the points in the projective space  $  P _ {5} $
 +
whose coordinates are numerically equal to the Plücker coordinates in $  P _ {3} $.
  
Each line complex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291018.png" /> is put into correspondence with a point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291019.png" />. The projective geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291020.png" /> can be considered as a non-Euclidean geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291021.png" />. This interpretation of the geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291023.png" /> is called the Plücker interpretation, in connection with the role of the Plücker coordinates.
+
Straight lines in $  P _ {3} $
 +
are represented by the points of a non-singular quadric in $  P _ {5} $
 +
of index three.
  
If one takes a straight line as the basic object in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291024.png" />, the geometry of this space can be considered as the geometry on the absolute of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291025.png" />.
+
If one takes this quadric as the absolute and defines a projective (non-Euclidean) metric in $  P _ {5} $,
 +
one gets the five-dimensional hyperbolic space  $  {}  ^ {3} S _ {5} $.  
 +
Under each [[Collineation|collineation]] and [[Correlation|correlation]] of  $  P _ {3} $
 +
the Plücker coordinates transform linearly, i.e. each collineation and correlation is represented by a collineation of $  P _ {5} $
 +
that maps the absolute into itself. These collineations are thus displacements of $  {}  ^ {3} S _ {5} $.  
 +
The displacements of  $  {}  ^ {3} S _ {5} $
 +
represent either collineations or correlations in  $  P _ {3} $.
  
The group of projective transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291026.png" /> is isomorphic to the group of displacements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291027.png" />, and any involutory projective transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291028.png" /> corresponds to an involutory displacement in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291029.png" />. For example, a null system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291030.png" /> corresponds to a reflection in a point and its polar hyperplane in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291031.png" />; an involutory homology in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291032.png" /> corresponds to a hyperbolic paratactic displacement by a half-line in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291033.png" />, etc. Each connected component of the group of projective transformations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291034.png" /> corresponds to a connected component of the group of displacements for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291035.png" />.
+
Each line complex in  $  P _ {3} $
 +
is put into correspondence with a point in $  {}  ^ {3} S _ {5} $.  
 +
The projective geometry of  $  P _ {3} $
 +
can be considered as a non-Euclidean geometry of  $  {}  ^ {3} S _ {5} $.  
 +
This interpretation of the geometry of $  P _ {3} $
 +
in  $  {}  ^ {3} S _ {5} $
 +
is called the Plücker interpretation, in connection with the role of the Plücker coordinates.
  
1) A collineation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291036.png" /> with positive determinant, including the identity transformation, corresponds to a displacement in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291037.png" /> with determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291038.png" /> (identity transformations are included here).
+
If one takes a straight line as the basic object in  $  P _ {3} $,
 +
the geometry of this space can be considered as the geometry on the absolute of  $  {}  ^ {3} S _ {5} $.
  
2) Any correlation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291039.png" /> with positive determinant (including the null system) corresponds to a displacement in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291040.png" /> with determinant equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291041.png" /> that transforms the proper and ideal domains, respectively, into themselves (including reflections in a point).
+
The group of projective transformations of  $  P _ {3} $
 +
is isomorphic to the group of displacements of  $  {}  ^ {3} S _ {5} $,
 +
and any involutory projective transformation of  $  P _ {3} $
 +
corresponds to an involutory displacement in $  {}  ^ {3} S _ {5} $.  
 +
For example, a null system in  $  P _ {3} $
 +
corresponds to a reflection in a point and its polar hyperplane in  $  {}  ^ {3} S _ {5} $;
 +
an involutory homology in  $  P _ {3} $
 +
corresponds to a hyperbolic paratactic displacement by a half-line in $  {}  ^ {3} S _ {5} $,
 +
etc. Each connected component of the group of projective transformations for  $  P _ {3} $
 +
corresponds to a connected component of the group of displacements for  $  {}  ^ {3} S _ {5} $.
  
3) Any collineation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291042.png" /> having negative determinant corresponds to a displacement in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291043.png" /> with determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291044.png" /> that transforms the proper domain into the ideal domain and vice versa, and this component contains a hyperbolic displacement by a half-line.
+
1) A collineation of $  P _ {3} $
 +
with positive determinant, including the identity transformation, corresponds to a displacement in $  {}  ^ {3} S _ {5} $
 +
with determinant $  + 1 $(
 +
identity transformations are included here).
  
4) Any correlation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291045.png" /> having negative determinant corresponds to a displacement in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291046.png" /> with determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291047.png" /> that transforms the proper domain into the ideal domain and vice versa.
+
2) Any correlation in $  P _ {3} $
 +
with positive determinant (including the null system) corresponds to a displacement in $  {}  ^ {3} S _ {5} $
 +
with determinant equal to  $  - 1 $
 +
that transforms the proper and ideal domains, respectively, into themselves (including reflections in a point).
  
The images under symmetries that correspond to one another in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291049.png" /> are put into correspondence with numerical invariants, between which there are certain relationships.
+
3) Any collineation of  $  P _ {3} $
 +
having negative determinant corresponds to a displacement in $  {}  ^ {3} S _ {5} $
 +
with determinant  $  + 1 $
 +
that transforms the proper domain into the ideal domain and vice versa, and this component contains a hyperbolic displacement by a half-line.
  
The Plücker interpretation is used in research on the displacement groups for the three-dimensional non-Euclidean spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291052.png" />, which are isomorphic to certain subgroups of the displacement group for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291053.png" />. There is also a relation between the groups of motions for these three-dimensional spaces (elliptic and hyperbolic) and groups of displacements of spaces of lower dimensions (see [[Fubini model|Fubini model]]; [[Kotel'nikov interpretation|Kotel'nikov interpretation]]). The Plücker interpretation is also used in examining the interpretation of the three-dimensional symplectic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291054.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291055.png" />.
+
4) Any correlation in  $  P _ {3} $
 +
having negative determinant corresponds to a displacement in  $  {}  ^ {3} S _ {5} $
 +
with determinant  $  - 1 $
 +
that transforms the proper domain into the ideal domain and vice versa.
 +
 
 +
The images under symmetries that correspond to one another in  $  P _ {3} $
 +
and  $  {}  ^ {3} S _ {5} $
 +
are put into correspondence with numerical invariants, between which there are certain relationships.
 +
 
 +
The Plücker interpretation is used in research on the displacement groups for the three-dimensional non-Euclidean spaces $  S _ {3} $,  
 +
$  {}  ^ {1} S _ {3} $
 +
and $  {}  ^ {2} S _ {3} $,  
 +
which are isomorphic to certain subgroups of the displacement group for $  {}  ^ {3} S _ {5} $.  
 +
There is also a relation between the groups of motions for these three-dimensional spaces (elliptic and hyperbolic) and groups of displacements of spaces of lower dimensions (see [[Fubini model|Fubini model]]; [[Kotel'nikov interpretation|Kotel'nikov interpretation]]). The Plücker interpretation is also used in examining the interpretation of the three-dimensional symplectic space $  \mathop{\rm Sp} _ {3} $
 +
in $  {}  ^ {3} S _ {5} $.
  
 
The Plücker interpretation was proposed by J. Plücker [[#References|[1]]].
 
The Plücker interpretation was proposed by J. Plücker [[#References|[1]]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Plücker,  "Neue Geometrie des Raumes gegründet auf die Betrachtung der geraden Linie als Raumelement" , Teubner  (1868)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Non-Euclidean geometry" , Moscow  (1955)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F. Klein,  "Vorlesungen über höhere Geometrie" , Springer  (1926)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Plücker,  "Neue Geometrie des Raumes gegründet auf die Betrachtung der geraden Linie als Raumelement" , Teubner  (1868)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Non-Euclidean geometry" , Moscow  (1955)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F. Klein,  "Vorlesungen über höhere Geometrie" , Springer  (1926)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The quadric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291056.png" /> whose points represent the lines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072910/p07291057.png" /> is often referred to as the Plücker quadric.
+
The quadric in $  P _ {5} $
 +
whose points represent the lines in $  P _ {3} $
 +
is often referred to as the Plücker quadric.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.W.P. Hirschfeld,  "Finite projective spaces of three dimensions" , Oxford Univ. Press  (1985)</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><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.M.Y. Sommerville,  "Analytical geometry of three dimensions" , Cambridge Univ. Press  (1934)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.W.P. Hirschfeld,  "Finite projective spaces of three dimensions" , Oxford Univ. Press  (1985)</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><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.M.Y. Sommerville,  "Analytical geometry of three dimensions" , Cambridge Univ. Press  (1934)</TD></TR></table>

Latest revision as of 08:06, 6 June 2020


A model that realizes the geometry of the three-dimensional projective space $ P _ {3} $ in the hyperbolic space $ {} ^ {3} S _ {5} $. The Plücker interpretation is based on a special interpretation of the Plücker coordinates of a straight line, which are defined for any straight line in $ P _ {3} $.

Under projective transformations of $ P _ {3} $ the Plücker coordinates transform linearly; the Plücker coordinates of straight lines in $ P _ {3} $ give a one-to-one correspondence between the straight lines of $ P _ {3} $ and the points in the projective space $ P _ {5} $ whose coordinates are numerically equal to the Plücker coordinates in $ P _ {3} $.

Straight lines in $ P _ {3} $ are represented by the points of a non-singular quadric in $ P _ {5} $ of index three.

If one takes this quadric as the absolute and defines a projective (non-Euclidean) metric in $ P _ {5} $, one gets the five-dimensional hyperbolic space $ {} ^ {3} S _ {5} $. Under each collineation and correlation of $ P _ {3} $ the Plücker coordinates transform linearly, i.e. each collineation and correlation is represented by a collineation of $ P _ {5} $ that maps the absolute into itself. These collineations are thus displacements of $ {} ^ {3} S _ {5} $. The displacements of $ {} ^ {3} S _ {5} $ represent either collineations or correlations in $ P _ {3} $.

Each line complex in $ P _ {3} $ is put into correspondence with a point in $ {} ^ {3} S _ {5} $. The projective geometry of $ P _ {3} $ can be considered as a non-Euclidean geometry of $ {} ^ {3} S _ {5} $. This interpretation of the geometry of $ P _ {3} $ in $ {} ^ {3} S _ {5} $ is called the Plücker interpretation, in connection with the role of the Plücker coordinates.

If one takes a straight line as the basic object in $ P _ {3} $, the geometry of this space can be considered as the geometry on the absolute of $ {} ^ {3} S _ {5} $.

The group of projective transformations of $ P _ {3} $ is isomorphic to the group of displacements of $ {} ^ {3} S _ {5} $, and any involutory projective transformation of $ P _ {3} $ corresponds to an involutory displacement in $ {} ^ {3} S _ {5} $. For example, a null system in $ P _ {3} $ corresponds to a reflection in a point and its polar hyperplane in $ {} ^ {3} S _ {5} $; an involutory homology in $ P _ {3} $ corresponds to a hyperbolic paratactic displacement by a half-line in $ {} ^ {3} S _ {5} $, etc. Each connected component of the group of projective transformations for $ P _ {3} $ corresponds to a connected component of the group of displacements for $ {} ^ {3} S _ {5} $.

1) A collineation of $ P _ {3} $ with positive determinant, including the identity transformation, corresponds to a displacement in $ {} ^ {3} S _ {5} $ with determinant $ + 1 $( identity transformations are included here).

2) Any correlation in $ P _ {3} $ with positive determinant (including the null system) corresponds to a displacement in $ {} ^ {3} S _ {5} $ with determinant equal to $ - 1 $ that transforms the proper and ideal domains, respectively, into themselves (including reflections in a point).

3) Any collineation of $ P _ {3} $ having negative determinant corresponds to a displacement in $ {} ^ {3} S _ {5} $ with determinant $ + 1 $ that transforms the proper domain into the ideal domain and vice versa, and this component contains a hyperbolic displacement by a half-line.

4) Any correlation in $ P _ {3} $ having negative determinant corresponds to a displacement in $ {} ^ {3} S _ {5} $ with determinant $ - 1 $ that transforms the proper domain into the ideal domain and vice versa.

The images under symmetries that correspond to one another in $ P _ {3} $ and $ {} ^ {3} S _ {5} $ are put into correspondence with numerical invariants, between which there are certain relationships.

The Plücker interpretation is used in research on the displacement groups for the three-dimensional non-Euclidean spaces $ S _ {3} $, $ {} ^ {1} S _ {3} $ and $ {} ^ {2} S _ {3} $, which are isomorphic to certain subgroups of the displacement group for $ {} ^ {3} S _ {5} $. There is also a relation between the groups of motions for these three-dimensional spaces (elliptic and hyperbolic) and groups of displacements of spaces of lower dimensions (see Fubini model; Kotel'nikov interpretation). The Plücker interpretation is also used in examining the interpretation of the three-dimensional symplectic space $ \mathop{\rm Sp} _ {3} $ in $ {} ^ {3} S _ {5} $.

The Plücker interpretation was proposed by J. Plücker [1].

References

[1] J. Plücker, "Neue Geometrie des Raumes gegründet auf die Betrachtung der geraden Linie als Raumelement" , Teubner (1868)
[2] B.A. Rozenfel'd, "Non-Euclidean geometry" , Moscow (1955) (In Russian)
[3] F. Klein, "Vorlesungen über höhere Geometrie" , Springer (1926)

Comments

The quadric in $ P _ {5} $ whose points represent the lines in $ P _ {3} $ is often referred to as the Plücker quadric.

References

[a1] J.W.P. Hirschfeld, "Finite projective spaces of three dimensions" , Oxford Univ. Press (1985)
[a2] B.L. van der Waerden, "Einführung in die algebraische Geometrie" , Springer (1939) pp. Chapt. 1
[a3] D.M.Y. Sommerville, "Analytical geometry of three dimensions" , Cambridge Univ. Press (1934)
How to Cite This Entry:
Plücker interpretation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pl%C3%BCcker_interpretation&oldid=22907
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article