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We give a simple combinatorial algorithm that computes a piecewise-linear approximation of a smooth surface from a nite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled surfaces, where density depends on \local feature size", the… (More)

We construct a graph on a planar point set, which captures its shape in the following sense: if a smooth curve is sampled densely enough, the graph on the samples is a polygonalization of the curve, with no extraneous edges. The required sampling density v aries with the Local Feature Size on the curve, so that areas of less detail can besampled less… (More)

We describe our experience with a new algorithm for the reconstruction of surfaces from unorganized sample points in I R 3. The algorithm is the first for this problem with provable guarantees. Given a " good sample " from a smooth surface, the output is guaranteed to be topologically correct and convergent to the original surface as the sampling density… (More)

We study several versions of the problem of generating triangular meshes for finite element methods. We show how to triangulate a planar point set or polygonally bounded domain with triangles of bounded aspect ratio; how to triangulate a planar point set with triangles having no obtuse angles; how to triangulate a point set in arbitrary dimension with… (More)

We survey the computational geometry relevant to nite element mesh generation. We especially focus on optimal triangulations of geometric domains in two-and three-dimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We… (More)

Manual selection of single particles in images acquired using cryo-electron microscopy (cryoEM) will become a significant bottleneck when datasets of a hundred thousand or even a million particles are required for structure determination at near atomic resolution. Algorithm development of fully automated particle selection is thus an important research… (More)

Marshall Bern * We study a basic problem in mathematical origami: determine if a given crease pattern can be folded to a flat origami. We show that assigning mountain and valley folds is NP-hard. We also show that determining a suitable overlap order for flaps is NP-hard, even assuming a valid mountain and valley assignment.

A simple but important special case of the hidden surface removal problem is one in which the scene consists of <italic>n</italic> rectangles with sides parallel to the <italic>x</italic> and <italic>y</italic>-axes, with viewpoint at <italic>z</italic> = ∞ (that is, an orthographic projection). This special case has application to overlapping… (More)