Plane Fields’ Quadratic Cremona Correspondence Set By Imaginary F-Points

Abstract

Birational (Cremona) correspondences of two planes П, П' or Cremona transformations on the combined plane П = П' represent an effective tool for design of smooth dynamic curves and two-dimensional lines. The simplest birational correspondence is a quadratic map Ω of plane fields on one another. In the projective definition of the quadratic correspondence can participate two pairs of imaginary complex conjugate F-points set as double points of elliptic involutions on the lines associated with the third pair of F-points. In this case, the imaginary projective F-bundles cannot be used for generation of points corresponding in Ω. A generic constructive algorithm for design of corresponding points in a quadratic mapping Ω(П ↔ П'), set both by real and imaginary F-points is proposed in this paper. The algorithm is based on the use of auxiliary projective correspondence Δ between the points of the planes П, П' and Hirst’s transformation Ψ with the center in the one of real F-points. A theorem on the existence of an invariant conic common to Ω and Δ mappings has been proved. Has been demonstrated a possibility for quadratic mapping’s presentation as a product of collinearity and Hirst’s transformation: Ω = ΔΨ. Has been considered a solution for auxiliary problems arising during the generic constructive algorithm’s implementation: buildup a conic section, that is incident to imaginary points, and plotting the corresponding points in collinearity set by imaginary points. It has been demonstrated that there are two or four possible versions of collinearity for plane fields П, П', set by with participation of the imaginary corresponding points, due to an uncertainty related to the order of their relative correspondence. Have been completely solved the problem of mapping a straight line in a conic section within the quadratic Cremona correspondance of fields П ≠ П', set by a pair of real fundamental points, and two pairs of imaginary ones. It has been demonstrated that in general case the problem has two solutions. The obtained results are useful for the development of the geometric theory related to imaginary elements and this theory’s application in linear and non-linear descriptive geometry, operating projective images of first and second orders.