Computing the minimum enclosing sphere of free-form hypersurfaces in arbitrary dimensions

Authors:
Ramanathan Muthuganapathy;Gershon Elber;Gill Barequet;Myung-Soo Kim
Affiliations:
Center for Graphics and Geometric Computing, Department of Computer Science, The Technion-Israel Institute of Technology, Haifa 32000, Israel;Center for Graphics and Geometric Computing, Department of Computer Science, The Technion-Israel Institute of Technology, Haifa 32000, Israel;Center for Graphics and Geometric Computing, Department of Computer Science, The Technion-Israel Institute of Technology, Haifa 32000, Israel;School of Computer Science and Engineering, Seoul National University, Seoul 151-744, South Korea
Venue:
Computer-Aided Design
Year:
2011

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Abstract

The problem of computing the minimum enclosing sphere (MES) of a point set is a classical problem in Computational Geometry. As an LP-type problem, its expected running time on the average is linear in the number of points. In this paper, we generalize this approach to compute the minimum enclosing sphere of free-form hypersurfaces, in arbitrary dimensions. This paper makes the bridge between discrete point sets (for which indeed the results are well-known) and continuous curves and surfaces, showing that the general solution for the former can be adapted for the latter. To compute the MES of a pair of hypersurfaces, each one having a contact point (a point at which the sphere touches the hypersurface), antipodal constraints are employed. For more than a pair, equidistance constraints along with tangency constraints are applied. These constraints yield a finite set of solution points which are used to identify the minimum enclosing sphere. The algorithm uses the LP-characteristic of the problem to process the input set. Furthermore, an optimization procedure that uses the convex hull of sampled points from the hypersurfaces is also described. Finally, results from our implementation are presented.