DESIGN PROPERTIES OF HYPERBOLIC PARABOLOID AND THEIR APPLICATION IN COMPUTER MODELING

Abstract

Setting and solving the problem presented in the article is a relevant topic in computer modeling. In particular, to create special models for building quadrics and solve problems associated with analyzing the shape of a surface and switching from one surface determinant to another (surface determinant change problems)

The object of the presented study is a hyperbolic paraboloid, as one of the surfaces widely used in architecture as a coating shell for large-span structures. The main goal of the work is to transition from representing the surface of a hyperbolic paraboloid with four segments that form a spatial closed broken to its "canonical" form, that is, to finding its vertex, axis, symmetry planes and shape parameters of the hyperbolic paraboloid.

In the work, the position is proved: if the hyperbolic paraboloid Γ is given by a closed spatial broken line of four segments (determinant), then the line passing through the middle of the segments connecting the opposite vertices of this broken line is parallel to the axis of the given hyperbolic paraboloid Γ.

Algorithms for solving three problems are presented. By one of the algorithms, you can find the direction of the axis of the hyperbolic paraboloid specified by an arbitrary determinant. The second shows how, by means of computer graphics, an arbitrary determinant can be designed onto a plane by a parallelogram. According to the third algorithm, you can find the "canonical" form of a hyperbolic paraboloid given by an arbitrary determinant.

Examples are presented and the purpose of further development of the work is indicated, namely modeling the surface of a hyperbolic paraboloid along a given line of outline.

Key words: quadric; hyperbolic paraboloid; tetrahedron; computer simulation.