
Journal for Geometry and Graphics 26 (2022), No. 2, 207216 Copyright Heldermann Verlag 2022 Construction of Hyperbolic Paraboloids According to a Prospective Outline in the Form of Hyperbola Switlana Botvinovska Kyiv National University of Construction and Architecture, Kyiv, Ukraine botvinovska@ua.fm Alla Zolotova Kyiv National University of Construction and Architecture, Kyiv, Ukraine zolotovaav1@gmail.com Alexander Mostovenko Kyiv National University of Construction and Architecture, Kyiv, Ukraine a.mostovenko25@gmail.com Hanna Sulimenko Kyiv National University of Construction and Architecture, Kyiv, Ukraine asulimenko@i.ua The presented research is oriented towards developing constructive surface modeling methods in 3Dgraphics 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 threeparameter 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 twoparameter set of closed spatial fourlink linear rings or a generator of a twoparameter set of parabolic crosssections. 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 wellknown 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 twoparameter set of pairs of tangent planes according to the number of pairs of points on the hyperbola branches. Each such pair defines a spatial fourlink linear ring having two vertices at selected points and two on the line of intersection of the tangent planes. Keywords: Secondorder surface, hyperbolic paraboloid, computer simulation, enveloping cone, described cone, contact line, outline. MSC: 51N05. [ Fulltextpdf (1125 KB)] 