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Primitives for the manipulation of general subdivisions and the computations of Voronoi diagrams
 ACM Tmns. Graph
, 1985
"... The following problem is discussed: Given n points in the plane (the sites) and an arbitrary query point 4, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the given sites and then locating the query point in one of its regions. Two algorithms ar ..."
Abstract

Cited by 543 (11 self)
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of graphs in twodimensional manifolds. This structure represents simultaneously an embedding, its dual, and its mirror image. Furthermore, just two operators are sufficient for building and modifying arbitrary diagrams.
Planning Algorithms
, 2004
"... This book presents a unified treatment of many different kinds of planning algorithms. The subject lies at the crossroads between robotics, control theory, artificial intelligence, algorithms, and computer graphics. The particular subjects covered include motion planning, discrete planning, planning ..."
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Cited by 1108 (51 self)
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This book presents a unified treatment of many different kinds of planning algorithms. The subject lies at the crossroads between robotics, control theory, artificial intelligence, algorithms, and computer graphics. The particular subjects covered include motion planning, discrete planning, planning under uncertainty, sensorbased planning, visibility, decisiontheoretic planning, game theory, information spaces, reinforcement learning, nonlinear systems, trajectory planning, nonholonomic planning, and kinodynamic planning.
Surface Reconstruction by Voronoi Filtering
 Discrete and Computational Geometry
, 1998
"... We give a simple combinatorial algorithm that computes a piecewiselinear approximation of a smooth surface from a finite 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 ..."
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Cited by 418 (15 self)
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We give a simple combinatorial algorithm that computes a piecewiselinear approximation of a smooth surface from a finite 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 output is topologically valid and convergent (both pointwise and in surface normals) to the original surface. We describe an implementation of the algorithm and show example outputs. 1 Introduction The problem of reconstructing a surface from scattered sample points arises in many applications such as computer graphics, medical imaging, and cartography. In this paper we consider the specific reconstruction problem in which the input is a set of sample points S drawn from a smooth twodimensional manifold F embedded in three dimensions, and the desired output is a triangular mesh with vertex set equal to S that faithfully represen...
NOTE ON PETRIE DUALS AND HYPERCUBE EMBEDDINGS OF SEMIREGULAR POLYHEDRA
"... A nonPlatonic convex polyhedron in R 3 is semiregular if it is vertextransitive and its faces are regular polygons. The semiregular polyhedra consist of the 13 Archimedean solids and two infinite families of nsided prisms Prismn and antiprisms APrismn. The skeleton graphs (verticesedges) G(M) of ..."
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A nonPlatonic convex polyhedron in R 3 is semiregular if it is vertextransitive and its faces are regular polygons. The semiregular polyhedra consist of the 13 Archimedean solids and two infinite families of nsided prisms Prismn and antiprisms APrismn. The skeleton graphs (verticesedges) G
Diameter Rigidity Of Spherical Polyhedra
 Duke J. Math
, 1998
"... . We classify geodesically complete, compact 2dimensional spherical polyhedra X of diameter and injectivity radius ß. If X contains a point whose link has diameter ? ß then either (i) X is the spherical join of the finite set P of points whose link has diameter ? ß with the metric graph E = fx 2 X ..."
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Cited by 7 (1 self)
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. We classify geodesically complete, compact 2dimensional spherical polyhedra X of diameter and injectivity radius ß. If X contains a point whose link has diameter ? ß then either (i) X is the spherical join of the finite set P of points whose link has diameter ? ß with the metric graph E = fx 2
Spherical parametrization and remeshing
 ACM TRANS. GRAPH
, 2003
"... The traditional approach for parametrizing a surface involves cutting it into charts and mapping these piecewise onto a planar domain. We introduce a robust technique for directly parametrizing a genuszero surface onto a spherical domain. A key ingredient for making such a parametrization practica ..."
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Cited by 119 (2 self)
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The traditional approach for parametrizing a surface involves cutting it into charts and mapping these piecewise onto a planar domain. We introduce a robust technique for directly parametrizing a genuszero surface onto a spherical domain. A key ingredient for making such a parametrization
Radial Averages on Regular and Semiregular Graphs
, 909
"... In 1966, P. Günther proved the following result: Given a continuous function f on a compact surface M of constant curvature −1 and its periodic lift ˜ f to the universal covering, the hyperbolic plane, then the averages of the lift ˜ f over increasing spheres converge to the average of the function ..."
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f over the surface M. In this article, we prove similar results for functions on the vertices and edges of regular and semiregular graphs, with special emphasis on the convergence rate. However, we consider averages over more general sets, namely spherical arcs, which in turn imply results for tubes
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