Construction of Hyperbolic Paraboloids According to a Prospective Outline in the Form of Hyperbola

S. Botvinovska ; A. Zolotova ; A. Mostovenko ; H. Sulimenko

Journal for geometry and graphics, Tome 26 (2022) no. 2, pp. 207-216

Résumé

The presented research is oriented towards developing constructive surface modeling methods in 3D-graphics program environment. First, the problem of hyperbolic paraboloid construction based on its perspective outline is considered. The point of view and the outlined line defines a cone encircling a three-parameter set of hyperbolic paraboloid surfaces. Based on ideas and algorithms proposed in previous studies, the following conclusion was made: An arbitrary hyperbola, considered as the line of contact of a hyperbolic paraboloid with a cone whose vertex is chosen arbitrarily, is a determinant of a hyperbolic paraboloid. This determinant can be used to generate a two-parameter set of closed spatial four-link linear rings or a generator of a two-parameter set of parabolic cross-sections. They all belong to the same hyperbolic paraboloid. The axis of this paraboloid is parallel to the line connecting the cone’s vertex with the center of the contact hyperbola. Redefinition of the determinant of a hyperbolic paraboloid is based on well-known theoretical principles: Each pair of intersecting generators determines the tangent plane of the paraboloid at the point of their intersection; the asymptotes of the hyperbola are parallel to the planes of parallelism of the hyperbolic paraboloid. The described cone specifies a two-parameter set of pairs of tangent planes according to the number of pairs of points on the hyperbola branches. Each such pair defines a spatial four-link linear ring having two vertices at selected points and two on the line of intersection of the tangent planes.

Détail

Classification : 51N05
Mots-clés : Second-order surface, hyperbolic paraboloid, computer simulation, enveloping cone, described cone, contact line, outline