View source for Kotel'nikov interpretation

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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]]).

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

====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>

====Comments====
One also encounters Kotel'nikov model instead of interpretation.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.A. [B.A. Rozenfel'd] Rosenfel'd,   "A history of non-euclidean geometry" , Springer  (1988)  (Translated from Russian)</TD></TR></table>