On Computer Modeling of Quadratic Surface Intersections

The article is dedicated to the problem of finding intersection curves of algebraic surfaces of the second order, which remains relevant for modern three-dimensional modeling and problems of applied geometry. This paper proposes a somewhat new algorithm based on linear algebra methods. It is a modification of the method proposed by Levin, implemented using the GeoGebra computer algebra system. The idea of the proposed method involves finding a coordinate system in which the equation of one of the second-order surfaces is simplified to a so-called normal (or canonical) form. The surface for which the canonical coordinate system is determined will be referred to as the control surface. Next, the equation of the inter-section curve is found in the new coordinate system by solving, where possible, the system of second-order algebraic equations. If the control surface is degenerate, transitioning to its convenient parameterization allows solving the mentioned system of algebraic equations in the general case. Finally, using the inverse transformation of three-dimensional space, the equation of the intersection curve is obtained in the original coordinate system. An important stage of this study was testing the developed method for various cases of surfaces, one of which is degenerate. The results demonstrated that the proposed algorithms enable finding parametric equations of curves with high accuracy and provide clear visualization for three-dimensional modeling. The proposed method is compared with existing algorithms, and its advantages for the specified case are shown. The analysis of the results indicates the potential applicability of the method in specialized modeling and computation systems.