Imaginary Straight Lines in Cartesian Coordinate System

Abstract

A geometric model of imaginary conjugate straight lines a~b, allowing symbolic representation of these lines on the real coordinate plane xy is considered. In order to connect the algebraic and geometric representations of imaginary straight lines, it is proposed to use the “mark” formed by orthogonal d1 ⊥ d2 and main g1~g2 directions of the elliptic involution σ in the pencil V. The specification of two pairs of pulling apart each other real straight lines d1~d2, g1~g2 passing through V, uniquely defines the elliptic involution σ in the pencil V, therefore, the V(d1 ⊥ d2, g1~g2) mark completely defines a pair of imaginary double straight lines a~b of elliptic involution σ(V), that allows consider the mark as an “image” of these imaginary straight lines. When using a mark, it is required to establish a one-to-one correspondence between complex coefficients of imaginary double straight lines equations and a graphically given mark. The direct and inverse problems are solved in this paper. The direct one is creation a mark representing imaginary straight lines, given by its own equations. The inverse one is determination of coefficients for the equations of imaginary lines defined by the mark. The essence of the direct and inverse problems consists in establishing a oneto-one correspondence between the equations of imaginary double straight elliptic involutions σ in the pencil V, and a graphically given mark containing the orthogonal and main directions of this involution. To solve both the direct and inverse problems, the Hirsch theorem (A.G. Hirsch) is used, which establishes a one-to-one correspondence between the complex Cartesian coordinates for a pair of imaginary conjugated points and real coordinates of a special “marker” symbolically representing these points. Have been considered examples of solution for geometric problems involving imaginary lines. In particular, has been solved the problem of constructing a circle passing through a given point and touching imaginary lines defined by its mark V(d1 ⊥ d2, g1~g2). Has been proposed a graphical and analytical algorithm for determining the coefficients of equations of imaginary tangents, traced to a conic section from its inner point.