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)

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)