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