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In this paper we present an efficient algorithm to test if two given paths are homotopic; that is, whether they wind around obstacles in the plane in the same way. For simple paths specified by <i>n</i> line segments with obstacles described by <i>n</i> points, our algorithm runs in <i>O</i>(<i>n</i> log <i>n</i>) time, which we show is tight. For… (More)
A common geometric problem in computer graphics and geographic information systems is to compute the arrangement of a set of n segments that can be colored red and blue so that there are no red/red or blue/blue crossings. We give a sweep algorithm that uses the minimum arithmetic precision and runs in optimal O(n log n + k) time and O(n) space to output an… (More)
We show that every set of n points in general position has a minimum pseudo-triangula-tion whose maximum vertex degree is five. In addition, we demonstrate that every point set in general position has a minimum pseudo-triangulation whose maximum face degree is four (i.e. each interior face of this pseudo-triangulation has at most four vertices). Both degree… (More)
Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex poly-hedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be… (More)
Finding a combinatorial test for rigidity in 3D is an open problem. We prove that vertex connectivity cannot be used to construct such a test by describing a class of mechanisms that increase the vertex connectivity of flexible graphs to 5. Our result is tight, as minimally rigid graphs in 3D can be at most 5-connected.
In this paper, we show how standard GIS operations like the complement, union, intersection, and buffering of maps can be made more flexible by using fuzzy set theory. In particular, we present a variety of algorithms for operations on fuzzy raster maps, focusing on buffer operations for such maps. Furthermore, we show how widely-available special-purpose… (More)
We present a triangulated closed polyhe-dron that has no edge unfolding, and a triangulated open polyhedron that has no unfolding whatsoever.