• Journal of Korea Multimedia Society
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• v.3 no.3
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• pp.298-308
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• 2000

We present an efficient algorithm for finding the sequence of extreme vortices of a moving regular convex polyhedron of with respect to a fixed plane H.. The algorithm utilizes the spherical Voronoi diagram that results from the outward unit normal vectors nF$_{i}$ 's of faces of P. It is well-known that the Voronoi diagram of n sites in the plane can be computed in 0(nlogn) time, and this bound is optimal. However. exploiting the convexity of P, we are able to construct the spherical Voronoi diagram of nF$_{i}$ ,'s in O(n) time. Using the spherical Voronoi diagram, we show that an extreme vertex problem can be transformed to a spherical point location problem. The extreme vertex problem can be solved in O(logn) time after O(n) time and space preprocessing. Moreover, the sequence of extreme vertices of a moving regular convex polyhedron with respect to H can be found in (equation omitted) time, where m$^{j}$ $_{k}$ (1$\leq$j$\leq$s) is the number of edges of a spherical Voronoi region sreg(equation omitted) such that (equation omitted) gives one or more extreme vertices.

• Journal of KIISE:Computer Systems and Theory
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• v.31 no.3_4
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• pp.247-256
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• 2004

This paper presents an efficient collision detection algorithm the performance of which is independent of animation speed. Most of the previous collision detection algorithms are incremental and discrete methods, which find out the neighborhood of the extreme vertex at the previous time instance in order to get an extreme vertex at each time instance. However, if an object collides with another one with a high torque, then the angular speed becomes faster. Hence, the candidate by the incremental algorithms may be farther from the real extreme vertex at this time instance. Therefore, the worst time complexity nay be $O(n^2)$, where n is the number of faces. Moreover, the total time complexity of incremental algorithms is dependent on the time step size of animation because a smaller time step yields more frequent evaluation of Euclidean distance. In this paper, we propose a new method to overcome these drawbacks. We construct a spherical extreme vertex diagram on Gauss Sphere, which has geometric properties, and then generate the distance function of a polyhedron and a plane by using this diagram. In order to efficiently compute the exact collision time, we apply the interval Newton method to the distance function.