Year: 1997
Type: article
Abstract: We present a gallery of simple curvature continuous surfaces that possess the topological structure of the Platonic solids_ These spherelike surfaces consist of ODe cubic triangular or biquadratic quadrilateral patch per vertex of the solid and interpolate the vertices of the dual solid. §l. Polynomial curvature continuous surfaces Constructing low degree polynomial curvature continuous surfaces is a difficult problem. Existing parametric solutions [3, 6, 5] require both a large number of patches and high degree. A recent implicit curvature continuous construction requires only algebraic degree 4, but consists of many pieces [4]. On the other hand the existence of low degree (rational) curvature continues representations of shapes such as the sphere hints at the existence of elegant solutions for restricted geometric shapes. This paper describes a small family of polynomial surfaces that approximate the sphere and have the symmetry structure of Platonic polyhedra. These spheroids embody a remarkably simple construction principle: pick a Platonic polyhedron and its dual; associate with each 3-valent vertex of the dual a triangular patch of degree 3, or with each 4-valent vertex a quadrilateral patch of degree bi-2. The Bernstein-Bezier control points of the spheroid patches are the face centroids, edge midpoints and scaled vertices of the dual. By symmetry, the spheroid interpolates the vertices of the Platonic and by scaling those of the dual Platonic. Using their simple, explicit parametrization the two higWy symmetric platonic spheroids can be proven strictly convex. The table below collects basic spheroid properties. Table 1 patches interpolates max Gauss name curvature Tetroid 4 Tetrahedron 3.24 Hexoid 6 Cube 1.92 Octoid 8 Octahedron 1.69 Dodecoid 20 Icosahedron 1.25 2 J. Peters, L. Kobbelt Construction Principle Gauss curvature curvature lines, needle plot vertices ofprogenitor solid are unit size Fig. 2. Synopsis of patch transitions and curvature used in Sections 3·6. §2. Guide to the gallery Each of the following four sections gives the synopsis of one platonic spheroid. Pick a Platonic solid. Its dual will be interpolated and gives the particular spheroid its name. The part closer to a vertex V of the original Platonic solid is covered by a single patch (there is no dodecoid since that would require a 5-sided patch). The complete surface is obtained by applying the operations of the solid's symmetry group (see e.g. [2]) to this one patch. The patch associated with V has quadratic boundary curves and one interior control point. Each boundary curve connects the centroids of two edge-adjacent facets attached to V and uses the midpoint of their common edge as middle coefficient of the quadratic curve in Bezier form. If V is three-valent} the Bezier representation of the boundaries arc degree-raised. The center control point of the biquadratic or cubic patch is always a multiple of V. Besides the representative patch, each synopsis shows} in the upper left sketch, the facet centroids, edge midpoints and the two vertices that determine the tangent plane common to two patches} say p and q. Tangent continuity of the surface follows from the existence of a linear scalar polynomial) such that the versal derivative DIP along the common edge of p and q is a linear combination of the transversal derivatives D2P and D 2q across the common edge, here parametrized by u. In symbols p(u, 0) = q(u, 0) and A(u)D,P(u, 0) = D2P(u, 0) + D2q(u, 0). In the synopsis} the coefficients of DiP are collected in a matrix. The upper right figure shows the spheroid textured by Gauss curvature, curvature lines and a needle plot proportional to the curvature. The normalization is chosen so that the vertices of the name-giving Platonic solid are on the unit sphere. Scale and maximal curvature are displayed on the right. Since the surfaces are symmetric across the edges and vertices} first order smoothness implies that the transitions are also curvature continuous. Finally, the positivity of all Bezier coefficients in numerator and denominator of the expression for Gauss curvature establishes convexity of octoid and icosoid.
Authors: Jörg Peters, Leif Kobbelt
Institution: Purdue University West Lafayette
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W1571105655