View source for Kotel'nikov interpretation

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" />.

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>