Scientific Visualization, 2018, volume 10, number 1, pages 56 - 68, DOI: 10.26583/sv.10.1.04

Computer visualization of conic curve passing through the imaginary point and the imaginary concerning direct

Author: V. A. Korotkiy

South Ural State University, Department of engineering and computer graphics,

Chelyabinsk, Russian Federation

ORCID: 0000-0002-5266-4701, Ospolina@mail.ru

__Abstract__

The proposed algorithms graphical computer simulation of the second order curves, given an arbitrary set of real and imaginary points and tangents. Developed a set of software for geo-metrically accurate construction of the main axes, asymptotes, foci and subsequent visualization of the conic sections, given both real and imaginary elements. To solve the problem used projec-tive synthetic geometric algorithms, totally eliminating the need to perform any algebraic calcula-tions. An example visualization of a conic passing through the valid point relating to the four imaginary straight lines.
Developed visualization tools used to study quadratic Cremona convert the specified imagi-nary fundamental points (F-points). Birational (cremonesi) conversion is an effective machine design the smooth curves and dynamic two-dimensional lines. The simplest birational mapping is a quadratic mapping of flat fields with each other, which is determined by using two pairs of projective bundles of straight lines with vertices in the fundamental points. In the projective def-inition of the quadratic conformity can participate two pairs of imaginary complex conjugate of F-points as double points of elliptic involutions on the lines associated with a valid pair of F-points. In this case, the imaginary projective bundles cannot be used to construct points corre-sponding to a quadratic transformation. The paper proposes a generic constructive algorithm for corresponding points in a quadratic transformation, given both real and imaginary F-points. The algorithm is based on the use of assisted projective matching and transformation Girst centered in one of the valid F-points. The possibility of representation for the quadratic map with imagi-nary F-points is the product of collineation and Girst transformation.

**Keywords**: elliptic involution, harmonic homology, imaginary complex conjugate elements, birational quad-ratic compliance, inversion, conversion Hirst.