Original scientific paper

https://doi.org/10.31896/k.25.5

Polyhedrons the Faces of which are Special Quadric Patches

Milena Stavrić orcid.org/0000-0002-8682-2026 ; University of Technology Graz, Graz, Austria
Albert Wiltsche ; University of Technology Graz, Graz, Austria
Gunter Weiss ; University of Technology Vienna, Vienna, Austria

Abstract

We seize an idea of Oswald Giering (see [1] and [2]), who replaced pairs of faces of a polyhedron by patches of hyperbolic paraboloids and link up edge-quadrilaterals of a polyhedron with the pencil of quadrics determined by that quadrilateral. Obviously only ruled quadrics can occur. There is a simple criterion for the existence of a ruled hyperboloid of revolution through an arbitrarily given quadrilateral. Especially, if a (not planar) quadrilateral allows one symmetry, there exist two such hyperboloids of revolution through it, and if the quadrilateral happens to be equilateral, the pencil of quadrics through it contains even three hyperboloids of revolution with pairwise orthogonal axes. To mention an example, for right double pyramids, as for example the octahedron, the axes of the hyperboloids of revolution are, on one hand, located in the plane of the regular guiding polygon, and on the other, they are parallel to the symmetry axis of the double pyramid. Not only for platonic solids, but for all polyhedrons, where one can define edge-quadrilaterals, their pairs of face-triangles can be replaced by quadric patches, and by this one could generate roong of architectural relevance. Especially patches of hyperbolic paraboloids or, as we present here, patches of hyperboloids of revolution deliver versions of such roong, which are also practically simple to realize.

Keywords

polyhedron; quadric; hyperboloid of revolution; Bézier patch

Hrčak ID:

269195

URI

https://hrcak.srce.hr/269195

Publication date:

27.12.2021.

Article data in other languages: croatian

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