Namespaces
Variants
Actions

Difference between revisions of "Kotel'nikov interpretation"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
An interpretation of the manifold of straight lines in the three-dimensional [[Lobachevskii space|Lobachevskii space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055820/k0558201.png" /> on the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055820/k0558202.png" /> (or on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055820/k0558203.png" />). With every straight line in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055820/k0558204.png" /> one associates its [[Plücker coordinates|Plücker coordinates]], which are defined in this case up to sign. Using these line coordinates one establishes a correspondence between the straight lines and their polars in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055820/k0558205.png" />, and also defines vectors of lines and their polars. One of two mutually-polar lines is represented by a vector of unit length, and the other by a vector of imaginary unit length. The manifold of pairs of mutually-polar straight lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055820/k0558206.png" /> is represented by the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055820/k0558207.png" /> with radius of curvature 1 or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055820/k0558208.png" />, and this correspondence is continuous. Isotropic straight lines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055820/k0558209.png" /> are represented by points of the absolute in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055820/k05582010.png" />. The connected group of motions of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055820/k05582011.png" /> is isomorphic to the group of motions of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055820/k05582012.png" />.
+
<!--
 +
k0558201.png
 +
$#A+1 = 12 n = 0
 +
$#C+1 = 12 : ~/encyclopedia/old_files/data/K055/K.0505820 Kotel\AApnikov interpretation
 +
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}}
 +
{{TEX|done}}
 +
 
 +
An interpretation of the manifold of straight lines in the three-dimensional [[Lobachevskii space|Lobachevskii space]] $  {}  ^ {1} S _ {3} $
 +
on the complex plane $  S _ {2} ( i) $(
 +
or on $  {}  ^ {1} S _ {2} ( i) $).  
 +
With every straight line in $  {}  ^ {1} S _ {3} $
 +
one associates its [[Plücker coordinates|Plücker coordinates]], which are defined in this case up to sign. Using these line coordinates one establishes a correspondence between the straight lines and their polars in $  {}  ^ {1} S _ {3} $,  
 +
and also defines vectors of lines and their polars. One of two mutually-polar lines is represented by a vector of unit length, and the other by a vector of imaginary unit length. The manifold of pairs of mutually-polar straight lines of $  {}  ^ {1} S _ {3} $
 +
is represented by the plane $  S _ {2} ( i) $
 +
with radius of curvature 1 or $  i $,  
 +
and this correspondence is continuous. Isotropic straight lines in $  {}  ^ {1} S _ {3} $
 +
are represented by points of the absolute in $  S _ {2} ( i) $.  
 +
The connected group of motions of the space $  {}  ^ {1} S _ {3} ( i) $
 +
is isomorphic to the group of motions of the plane $  S _ {2} ( i) $.
  
 
The Kotel'nikov interpretation is sometimes understood in a broader sense, as the interpretation of manifolds of straight lines in three-dimensional spaces as complex or other two-dimensional planes (see [[Fubini model|Fubini model]]).
 
The Kotel'nikov interpretation is sometimes understood in a broader sense, as the interpretation of manifolds of straight lines in three-dimensional spaces as complex or other two-dimensional planes (see [[Fubini model|Fubini model]]).
Line 7: Line 30:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.P. Kotel'nikov,  "Projective theory of vectors" , Kazan'  (1899)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Study,  "Geometrie der Dynamen" , Teubner  (1903)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.P. Kotel'nikov,  "Projective theory of vectors" , Kazan'  (1899)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Study,  "Geometrie der Dynamen" , Teubner  (1903)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 22:15, 5 June 2020


An interpretation of the manifold of straight lines in the three-dimensional Lobachevskii space $ {} ^ {1} S _ {3} $ on the complex plane $ S _ {2} ( i) $( or on $ {} ^ {1} S _ {2} ( i) $). With every straight line in $ {} ^ {1} S _ {3} $ one associates its Plücker coordinates, which are defined in this case up to sign. Using these line coordinates one establishes a correspondence between the straight lines and their polars in $ {} ^ {1} S _ {3} $, and also defines vectors of lines and their polars. One of two mutually-polar lines is represented by a vector of unit length, and the other by a vector of imaginary unit length. The manifold of pairs of mutually-polar straight lines of $ {} ^ {1} S _ {3} $ is represented by the plane $ S _ {2} ( i) $ with radius of curvature 1 or $ i $, and this correspondence is continuous. Isotropic straight lines in $ {} ^ {1} S _ {3} $ are represented by points of the absolute in $ S _ {2} ( i) $. The connected group of motions of the space $ {} ^ {1} S _ {3} ( i) $ is isomorphic to the group of motions of the plane $ S _ {2} ( i) $.

The Kotel'nikov interpretation is sometimes understood in a broader sense, as the interpretation of manifolds of straight lines in three-dimensional spaces as complex or other two-dimensional planes (see Fubini model).

Kotel'nikov interpretations were first proposed by A.P. Kotel'nikov (see [1]) and independently by E. Study (see [2]).

References

[1] A.P. Kotel'nikov, "Projective theory of vectors" , Kazan' (1899) (In Russian)
[2] E. Study, "Geometrie der Dynamen" , Teubner (1903)
[3] B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)

Comments

One also encounters Kotel'nikov model instead of interpretation.

References

[a1] B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)
How to Cite This Entry:
Kotel'nikov interpretation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kotel%27nikov_interpretation&oldid=47523
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article