Geometric Algorithms for Constructing the Algebraic Surfaces on the Basis of Grassmann’s Ideas and Their Extensive Equations

Abstract

We consider a problem of constructing algebraic surfaces on the basis of Grassmann’s ideas. We consider the surfaces of the second and higher orders with the purpose of obtaining their Grassmann’s extensive equations and their images in Java environment. The works (Banai and Manevich in Proceedings of the 6th International Conference on Engineering Computer Graphics and Descriptive Geometry, Tokyo, Japan, 1994; Manevich and Sluzkin in Linear Constructions of Grassmann, Kwant, 9 (in Russian). Moscow, pp. 19–22, 1977) [1, 2] were devoted to construction of flat curves of high order on the basis of Grassmann’s ideas. It should be noted that in the catalog of mathematical models of surfaces (Schilling in Catalog Mathematischer Modelle Mit 106 Abbildungen. Siebente Auflage. Leipzig 1911) [3], and in Pabler and Lordick (J Geom Graph 21:263–271, 2017), Velichova (J Geom Graph 19:13–29, 2015) [4, 5] one can see the physical models and the computer images of high degree surfaces. For the construction of the algebraic surfaces of high degrees according to Grassmann points, planes and lines are given as a geometric basis. Linear operations such as the joining of two points by a straight line and the construction of intersection point of straight line with a plane are conducted sequentially. The last operation in this sequence—the intersection of a straight line with a plane generally gives the point of the desired surface. It is shown that on certain straight lines in the process of constructing a surface, projective rows of points are formed. The surface equation is seen from the record of Grassmann extensive equations. The proof that the resulting set of points in space belongs to a high degree surfaces for a number of cases is given by the method of projective geometry. Most of the constructions are based on the G. Grassmann statement:

Let there be a certain number of fixed points, straight lines and planes in space. An arbitrary point X is associated with these elements by means of connections and intersections. If as a result of such constructions it turns out that certain four points lie in the same plane or certain two straight lines are intersected, then the point X belongs to an algebraic surface whose order is equal to the number indicating how many times the point X participated in the constructions. (Klein in Linear Constructions of Grassmann, pp. 220–224) [6]

On the basis of this theorem we write a computation algorithm for finding images of surfaces of high degrees. It should be emphasized that all the extensive equations remain invariant when the coordinate system is replaced. Usage of the proposed constructions makes it possible to understand more deeply the various methods for the formation of algebraic surfaces, which can be useful in the study of various sections of analytic and projective geometry.