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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. Unfortunately, this sampling condition is not always met in practice due to noise, non-smoothness or simply due to inadequate sampling. This leads to undesired holes and other artifacts in the output surface. Certain CAD applications such as creating a prototype from a model boundary require a water-tight surface, i.e., no hole should be allowed in the surface. In this paper we describe a simple algorithm called Tight Cocone that works on an initial mesh generated by a popular surface reconstruction algorithm and fills up all holes to output a water-tight surface. In doing so, it does not introduce any extra points and produces a triangulated surface interpolating the input sample points. In support of our method we present experimental results with a number of difficult data sets.

Abstract. 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, parametric minimum spanning trees, and parametric matroids in general.

A sliver is a tetrahedron whose four vertices lie close to a plane and whose orthogonal projection to that plane is a convex quadrilateral with no short edge. Slivers 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. [1995], then there is an assignment of weights so the weighted Delaunay triangulation contains no slivers. We also give an algorithm to compute such a weight assignment.