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74
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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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.
Almost Normal Heegaard Splittings
, 2001
"... The study of threemanifolds via their Heegaard splittings was initiated by Poul Heegaard in 1898 in his thesis. Our approach to the subject is through almost normal surfaces, as introduced by Hyam Rubinstein [28] and distance, as introduced by John Hempel [12]. Among the results presented... ..."
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Cited by 8 (4 self)
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The study of threemanifolds via their Heegaard splittings was initiated by Poul Heegaard in 1898 in his thesis. Our approach to the subject is through almost normal surfaces, as introduced by Hyam Rubinstein [28] and distance, as introduced by John Hempel [12]. Among the results presented...
Protein similarity from knot theory and geometric convolution
 J Comput Biol
, 2004
"... interpreted as representing the official policies, either expressed or implied, of the Pennsylvania Department ..."
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interpreted as representing the official policies, either expressed or implied, of the Pennsylvania Department
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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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.
Computing Linking Numbers of a Filtration
 In Algorithms in Bioinformatics (LNCS 2149
, 2001
"... We develop fast algorithms for computing the linking number of a simplicial complex within a filtration. We give experimental results in applying our work toward the detection of nontrivial tangling in biomolecules, modeled as alpha complexes. ..."
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We develop fast algorithms for computing the linking number of a simplicial complex within a filtration. We give experimental results in applying our work toward the detection of nontrivial tangling in biomolecules, modeled as alpha complexes.
Motion Planning for Knot Untangling
 Int. J. of Robotics Research
, 2002
"... When given a very tangled but unknotted circular piece of string it is usually quite easy to move it around and tug on parts of it until it untangles. However solving this problem by computer, either exactly or heuristically, is challenging. Various approaches have been tried, employing ideas from a ..."
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When given a very tangled but unknotted circular piece of string it is usually quite easy to move it around and tug on parts of it until it untangles. However solving this problem by computer, either exactly or heuristically, is challenging. Various approaches have been tried, employing ideas from algebra, geometry, topology and optimization. This paper investigates the application of motion planning techniques to the untangling of mathematical knots. Such an approach brings together robotics and knotting at the intersection of these fields: rational manipulation of a physical model. In the past, simulated annealing and other energy minimization methods have been used to find knot untangling paths for physical models. Using a probabilistic planner, we have untangled some standard benchmarks described by over four hundred variables much more quickly than has been achieved with minimization. We also show how to produce candidates with minimal number of segments for a given knot. We discuss novel motion planning techniques that were used in our algorithm and some possible applications of our untangling planner in computational topology and in the study of DNA rings.
Area inequalities for embedded disks spanning unknotted curves
 2003, arXiv:math.DG/0306313. EFFICIENTLY BOUND 4MANIFOLDS 43
"... We show that a smooth unknotted curve in R 3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length ..."
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We show that a smooth unknotted curve in R 3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giving the upper bound on the area grows exponentially in 1/r 2. In the direction of lower bounds, we give a sequence of length one curves with r→0for which the area of any spanning disk is bounded from below by a function that grows exponentially with 1/r. In particular, given any constant A, there is a smooth, unknotted length one curve for which the area of a smallest embedded spanning disk is greater than A. 1
Emerging Challenges in Computational Topology
 Results of the NFS Workshop on Computational Topology
, 1999
"... Here we present the results of the NSFfunded Workshop on Computational Topology, which met on June 11 and 12 in Miami Beach, Florida. This report identifies important problems involving both computation and topology. Author affiliations: Marshall Bern, Xerox Palo Alto Research Ctr., bern@parc. ..."
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Here we present the results of the NSFfunded Workshop on Computational Topology, which met on June 11 and 12 in Miami Beach, Florida. This report identifies important problems involving both computation and topology. Author affiliations: Marshall Bern, Xerox Palo Alto Research Ctr., bern@parc.xerox.com. David Eppstein, Univ. of California, Irvine, Dept. of Information & Computer Science, eppstein@ics.uci.edu. Pankaj K. Agarwal, Duke Univ., Dept. of Computer Science, pankaj@cs.duke.edu. Nina Amenta, Univ. of Texas, Austin, Dept. of Computer Sciences, amenta@cs.utexas.edu. Paul Chew, Cornell Univ., Dept. of Computer Science, chew@cs.cornell.edu. Tamal Dey, Ohio State Univ., Dept. of Computer and Information Science, tamaldey@cis.ohiostate.edu. David P. Dobkin, Princeton Univ., Dept. of Computer Science, dpd@cs.princeton.edu. Herbert Edelsbrunner, Duke Univ., Dept. of Computer Science, edels@cs.duke.edu. Cindy Grimm, Brown Univ., Dept. of Computer Science, cmg@cs.brown.edu. Leonid...
OPTIMIZING THE DOUBLE DESCRIPTION METHOD FOR NORMAL SURFACE ENUMERATION
"... Abstract. Many key algorithms in 3manifold topology involve the enumeration of normal surfaces, which is based upon the double description method for finding the vertices of a convex polytope. Typically we are only interested in a small subset of these vertices, thus opening the way for substantial ..."
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Abstract. Many key algorithms in 3manifold topology involve the enumeration of normal surfaces, which is based upon the double description method for finding the vertices of a convex polytope. Typically we are only interested in a small subset of these vertices, thus opening the way for substantial optimization. Here we give an account of the vertex enumeration problem as it applies to normal surfaces and present new optimizations that yield strong improvements in both running time and memory consumption. The resulting algorithms are tested using the freely available software package Regina. 1.
The minimal number of triangles needed to span a polygon embedded in R d
 J. GoodmanR. Pollack Festscrift Volume
, 2003
"... Given a closed polygon P having n edges, embedded in R d, we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface embedded in R d having P as its geometric boundary. More generally we obtain such bounds for a triangulated (locally flat) PL surfac ..."
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Given a closed polygon P having n edges, embedded in R d, we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface embedded in R d having P as its geometric boundary. More generally we obtain such bounds for a triangulated (locally flat) PL surface having P as its boundary which is immersed in R d and whose interior is disjoint from P. The most interesting case is dimension 3, where the polygon may be knotted. We use the Seifert surface construction to show that for any polygon embedded in R 3 there exists an embedded orientable triangulated PL surface having at most 7n 2 triangles, whose boundary is a subdivision of P. We complement this with a construction of families of polygons with n vertices for which any such embedded surface requires at least 1 2 n2 − O(n) triangles. We also exhibit families of polygons in R 3 for which Ω(n 2) triangles are required in any immersed PL surface of the above kind. In contrast, in dimension 2 and in dimensions d ≥ 5 there always exists an embedded locally flat PL disk having P as boundary that contains at most n triangles. In dimension 4 there always exists an immersed locally flat PL disk of the above kind that contains at most 3n triangles. An unresolved case is that of embedded PL surfaces in dimension 4, where we establish only an O(n 2) upper bound. These results can be viewed as providing qualitative discrete analogues of the isoperimetric inequality for piecewise linear (PL) manifolds. In dimension 3 they imply that the (asymptotic) discrete isoperimetric constant lies between 1/2 and 7. Keywords: isoperimetric inequality, Plateau’s problem, computational complexity