Научная визуализация, 2021, том 13, номер 1, страницы 1 - 14, DOI: 10.26583/sv.13.1.01
The Testing and Visualization of the Singularities of the Mutual Intersection of a Tetrahedron and a Quadric (Chasles' Theorem)
Автор: A. L. Kheyfets1
South Ural State University, Chelyabinsk, Russia
1 ORCID: 0000-0001-6490-359X, heifets@yandex.ru
Аннотация
This article presents the results of the experimental research and testing of Chasles’ historical theorem. The theorem shows the singularities of the intersection of an arbitrary tetrahedron and an arbitrary quadric (second-order surface). The need for testing is preconditioned by the absence of a proof of the theorem and the complexity of its perception in Chasles’ version.
The experiments included the construction, visualization, and study of 3D computer models using AutoCAD and SolidWorks. All forms of quadrics are considered in their different relative positions to the tetrahedron. The experimental procedure is considered in detail and the accuracy of the results is estimated. The author tested all the intersection variants given in the theorem: the edges intersect a quadric, the vertices belong to a quadric, the edges are tangent to a quadric, the faces are tangent to a quadric, etc. The experiments confirmed the scientific novelty of the theorem, which is that four intersecting straight lines drawn according to the algorithm of the theorem belong to the surface a single one-sheeted hyperboloid.
The paper investigated in detail the form of the theorem when the planes drawn through the edges of a tetrahedron are tangent and enclose the quadric. It shows that there are 4,096 combinations of plane positions. Only 64 of these combinations, obtained using AutoLisp, lead to the realization of the theorem. This conclusion supplements the theorem.
The results differ from the theorem in two forms. The paper presents a proof of one of the theorem forms, although a universal proof of the theorem has yet to be developed. The models and algorithms can be used when teaching computer geometric simulation.
Ключевые слова: 3D computer geometrical simulation, Michel Chasles, tetrahedron, quadrics, Pascal’s hexagon, Auto-CAD, AutoLisp.
