# Kotel'nikov interpretation

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)