On the polyhedra of graceful spheres and circular geodesics.(English)Zbl 1370.68295
Summary: We construct a polyhedral surface called agraceful surface, which provides best possible approximation to a given sphere regarding certain criteria. In digital geometry terms, the graceful surface is uniquely characterized by its minimality while guaranteeing the connectivity of certain discrete (polyhedral) curves defined on it. The notion of “gracefulness” was first proposed in [V. E. Brimkov andR. P. Barneva, Lect. Notes Comput. Sci. 1568, 53–64 (1999;Zbl 0933.68139)] and shown to be useful for triangular mesh discretization through graceful planes and graceful lines. In this paper we extend the considerations to a nonlinear object such as a sphere. In particular, we investigate the properties of a discrete geodesic path between two voxels and show that discrete 3D circles, circular arcs, and Mobius triangles are all constructible on a graceful sphere, with guaranteed minimum thickness and the desired connectivity in the discrete topological space.
MSC:
| 68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |
Keywords:
graceful surface;graceful sphere;polyhedral surface;discrete geometry;discrete spherical geodesics;3D imagingCitations:
Zbl 0933.68139Cite
Full Text:DOI
References:
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