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1 Introduct ion The problem of computing a piecewise linear approximation to a surface from a set of sample points on the surface has been a focus of research in solid modeling and graphics due to its many applications. The input to this surface reconstruct ion problem consists of the three dimensional coordinates of the sampled points. The crust algorithm(More)
We prove an O(n(k + 1)1/3) upper bound for planar k-sets. This is the first considerable improvement on this bound after its early solution approximately 27 years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of k-levels in the arrangement of line segments, k convex polygons in the union of n lines,(More)
Surface reconstruction from unorganized sample points is an important problem in computer graphics, computer aided design, medical imaging and solid modeling. Recently a few algorithms have been developed that have theoretical guarantee of computing a topologically correct and geometrically close surface under certain condition on sampling density.(More)
Many applications in geometric modeling, computer graphics, visualization and computer vision benefit from a reduced representation called curve-skeletons of a shape. These are curves possibly with branches which compactly represent the shape geometry and topology. The lack of a proper mathematical definition has been a bottleneck in developing and applying(More)
Geometric shapes are identified with their features. For computational purposes a concrete mathematical definition of features is required. In this paper we use a topological approach, namely dynamical systems, to define features of shapes. To exploit this definition algorithmically we assume that a point sample of the shape is given as input from which(More)
A silver is a tetrahedon whose four vertices lie close to a plane and whose orthogonal projection to that plane is a convex quadrilateral with no short edge. Silvers are notoriously common in 3-dimensional Delaunay triangulations even for well-spaced point sets. We show that, if the Delaunay triangulation has the ratio property introduced in Miller et al.(More)
Medial axis as a compact representation of shapes has evolved as an essential geometric structure in a number of applications involving 3D geometric shapes. Since exact computation of the medial axis is difficult in general, efforts continue to approximate them. One line of research considers the point cloud representation of the boundary surface of a solid(More)