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21
Topology guaranteeing manifold reconstruction using distance function to noisy data, Research Report 429 (2005), available at http://math.ubourgogne.fr/topo/chazal/publications.htm
"... Given a smooth compact codimension one submanifold S of R k and a compact approximation K of S, we prove that it is possible to reconstruct S and to approximate the medial axis of S with topological guarantees using unions of balls centered on K. We consider two notions of noisyapproximation that g ..."
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Cited by 18 (2 self)
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Given a smooth compact codimension one submanifold S of R k and a compact approximation K of S, we prove that it is possible to reconstruct S and to approximate the medial axis of S with topological guarantees using unions of balls centered on K. We consider two notions of noisyapproximation that generalize sampling conditions introduced by Amenta & al. and Dey & al. Our results are based upon critical point theory for distance functions. For the two approximation conditions, we prove that the connected components of the boundary of unions of balls centered on K are isotopic to S. Our results allow to consider balls of different radii. For the first approximation condition, we also prove that a subset (known as the λmedial axis) of the medial axis of R k \ K is homotopy equivalent to the medial axis of S. We obtain similar results for smooth compact submanifolds S of R k of any codimension.
Computational topology for reconstruction of surfaces with boundary: integrating experiments and theory
 Proceedings of the IEEE International Conference on Shape Modeling and Applications, June 15 17, 2005
, 2005
"... Abstract. This paper presents new mathematical foundations for topologically correct surface reconstruction techniques that are applicable to 2manifolds with boundary, where provable techniques previously had been limited to surfaces without boundary. This is done by an intermediate construction of ..."
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Cited by 9 (4 self)
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Abstract. This paper presents new mathematical foundations for topologically correct surface reconstruction techniques that are applicable to 2manifolds with boundary, where provable techniques previously had been limited to surfaces without boundary. This is done by an intermediate construction of the envelope (as defined herein) of the original surface. For any compact C 2 manifold M it is shown that its envelope is C 1,1 and this envelope can be reconstructed with topological guarantees. Then it is shown that there exists a piecewise linear (PL) subset of the reconstruction of the envelope that is ambient isotopic to M, whenever M is orientable. The emphasis of this paper is upon the mathematical proofs needed for these extensions, where more practical applications and examples are presented in a companion paper.
Preserving computational topology by subdivision of quadratic & cubic Bézier curves
 Computing
"... Nonselfintersection is both a topological and a geometric property. It is known that nonselfintersecting regular Bézier curves have nonselfintersecting control polygons, after sufficiently many uniform subdivisions. Here a sufficient condition is given within R 3 for a nonselfintersecting, r ..."
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Cited by 5 (2 self)
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Nonselfintersection is both a topological and a geometric property. It is known that nonselfintersecting regular Bézier curves have nonselfintersecting control polygons, after sufficiently many uniform subdivisions. Here a sufficient condition is given within R 3 for a nonselfintersecting, regular C 2 cubic Bézier curve to be ambient isotopic to its control polygon formed after sufficiently many subdivisions. The benefit of using the control polygon as an approximant for scientific visualization is presented in this paper.
Computational topology for geometric design and molecular design
 in Mathematics for Industry: Challenges and Frontiers
, 2005
"... The nascent field of computational topology holds great promise for resolving several longstanding industrial design modeling challenges. Geometric modeling has become commonplace in industry as manifested by the critical use of Computer Aided Geometric Design (CAGD) systems within the automotive, ..."
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Cited by 5 (2 self)
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The nascent field of computational topology holds great promise for resolving several longstanding industrial design modeling challenges. Geometric modeling has become commonplace in industry as manifested by the critical use of Computer Aided Geometric Design (CAGD) systems within the automotive, aerospace, shipbuilding and consumer product industries. Commercial CAGD packages depend upon complementary geometric and topological algorithms. The emergence of geometric modeling for molecular simulation and pharmaceutical design presents new challenges for supportive topological software within Computer Aided Molecular Design (CAMD) systems. For both CAGD and CAMD systems, splines provide relatively mature geometric technology. However, there remain pernicious issues regarding the ‘topology ’ of these models, particularly for support of robust simulations which rely upon the topological characteristics of adjacency, connectivity and nonselfintersection. This paper presents current challenges and frontiers for reliable simulation and approximation of topology for geometric models. The simultaneous consideration of CAGD and CAMD is important to provide unifying abstractions to benefit both domains. In engineering applications it is a common requirement that topological equivalence be preserved during geometric modifications, but in molecular simulations attention is focused upon where topological changes have occurred as indications of important chemical changes. The methods presented here are supportive of both these disciplinary approaches.
Computational topology for isotopic surface reconstruction
 Theoretical Computer Science 365 (3) (2006) 184
, 2006
"... Abstract. New computational topology techniques are presented for surface reconstruction of 2manifolds with boundary, while rigorous proofs have previously been limited to surfaces without boundary. This is done by an intermediate construction of the envelope (as defined herein) of the original sur ..."
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Cited by 4 (1 self)
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Abstract. New computational topology techniques are presented for surface reconstruction of 2manifolds with boundary, while rigorous proofs have previously been limited to surfaces without boundary. This is done by an intermediate construction of the envelope (as defined herein) of the original surface. For any compact C 2 manifold M embedded in R 3, it is shown that its envelope is C 1,1. Then it is shown that there exists a piecewise linear (PL) subset of the reconstruction of the envelope that is ambient isotopic to M, whenever M is orientable. The emphasis of this paper is upon the formal mathematical proofs needed for these extensions. (Practical application examples have already been published in a companion paper.) Possible extensions to nonorientable manifolds are also discussed. The mathematical exposition relies heavily on known techniques from differential geometry and topology, but the specific new proofs are intended to be sufficiently specialized to prompt further algorithmic discoveries.
Application of ambient isotopy to surface approximation and interval solids
 CAD
, 2004
"... Given a nonsingular compact 2manifold ¦ without boundary, we present methods for establishing a family of surfaces which can approximate ¦ so that each approximant is ambient isotopic to ¦. The methods presented here offer broad theoretical guidance for a rich class of ambient isotopic approximatio ..."
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Cited by 3 (2 self)
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Given a nonsingular compact 2manifold ¦ without boundary, we present methods for establishing a family of surfaces which can approximate ¦ so that each approximant is ambient isotopic to ¦. The methods presented here offer broad theoretical guidance for a rich class of ambient isotopic approximations, for applications in graphics, animation and surface reconstruction. They are also used to establish sufficient conditions for an interval solid to be ambient isotopic to the solid it is approximating. Furthermore, the normals of the approximant are compared to the normals of the original surface, as these approximating normals play prominent roles in many graphics algorithms. The methods are based on global theoretical considerations and are compared to existing local methods. Practical implications of these methods are also presented. For the global case, a differential surface analysis is performed to find a positive number § so that the offsets ¦©¨�����§� � of ¦ at distances �� § are nonsingular. In doing so, a normal tubular neighborhood, ¦���§� � , of ¦ is constructed. Then, each approximant of ¦ lies inside ¦���§� �. Comparisons between these global and local constraints are given. Key words: Ambient isotopy; computational topology; surface reconstruction; interval solids; offsets and deformations; reverse engineering Preprint submitted to Elsevier Science 30 July 2003 1
Topological neighborhoods for spline curves : practice & theory, preprint
, 2006
"... The unresolved subtleties of floating point computations in geometric modeling become considerably more difficult in animations and scientific visualizations. Some emerging solutions based upon topological considerations will be presented. A novel geometric seeding algorithm for Newton’s method was ..."
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Cited by 3 (1 self)
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The unresolved subtleties of floating point computations in geometric modeling become considerably more difficult in animations and scientific visualizations. Some emerging solutions based upon topological considerations will be presented. A novel geometric seeding algorithm for Newton’s method was used in experiments to determine feasible support for these visualization applications. 1 Computing the pipe surface radius Parametric curves have been shown to have a particular neighborhood whose boundary is nonselfintersecting [5]. It has also been shown that specified movements of the curve within this neighborhood preserve the topology of the curve [9, 8], as is desired in visualization. This neighborhood is defined by a single value, which is the radius of a pipe surface, where that radius depends on curvature and the minimum length over those line segments which are normal to the curve at both endpoints of the line segment [5]. Since computation of curvature is a welltreated problem, the focus of this paper is efficient and accurate floating point techniques to compute the other dependeancy for that radius.
Computational topology for regular closed sets (within the ITANGO project), invited article
 Topology Atlas
"... Abstract. The Boolean algebra of regular closed sets is prominent in topology, particularly as a dual for the StoneČech compactification. This algebra is also central for the theory of geometric computation, as a representation for combinatorial operations on geometric sets. However, the issue of c ..."
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Cited by 2 (1 self)
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Abstract. The Boolean algebra of regular closed sets is prominent in topology, particularly as a dual for the StoneČech compactification. This algebra is also central for the theory of geometric computation, as a representation for combinatorial operations on geometric sets. However, the issue of computational approximation introduces unresolved subtleties that do not occur within “pure ” topology. One major effort towards reconciling this mathematical theory with computational practice is our ongoing ITANGO project. The acronym ITANGO is an abbreviation for “Intersections—Topology, Accuracy and Numerics for Geometric Objects”. The longrange goals and initial progress of the ITANGO team in development of computational topology are presented.
SelfDelaunay Meshes for Surfaces
, 2010
"... In the Euclidean plane, a Delaunay triangulation can be characterized by the requirement that the circumcircle of each triangle be empty of vertices of all other triangles. For triangulating a surface S in R3, the Delaunay paradigm has typically been employed in the form of the restricted Delaunay t ..."
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Cited by 1 (0 self)
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In the Euclidean plane, a Delaunay triangulation can be characterized by the requirement that the circumcircle of each triangle be empty of vertices of all other triangles. For triangulating a surface S in R3, the Delaunay paradigm has typically been employed in the form of the restricted Delaunay triangulation, where the empty circumcircle property is defined by using the Euclidean metric in R3 to measure distances on the surface. More recently, the intrinsic (geodesic) metric of S has also been employed to define the Delaunay condition. In either case the resulting mesh M is known to approximate S with increasing accuracy as the density of the sample points increases. However, the use of the reference surface S to define the Delaunay criterion is a serious limitation. In particular, in the absence of the original reference surface, there is no way of verifying if a given mesh meets the criterion. We define a selfDelaunay mesh as a triangle mesh that is a Delaunay triangulation of its vertex set with respect to the intrinsic metric of the mesh itself. This yields a discrete surface representation criterion that can be validated by the properties of the mesh alone, independent of any reference surface the mesh is supposed to represent. The intrinsic Delaunay