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The computational Complexity of Knot and Link Problems
 J. ACM
, 1999
"... We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting pr ..."
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Cited by 58 (8 self)
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We consider the problem of deciding whether a polygonal knot in 3dimensional Euclidean space is unknotted, capable of being continuously deformed without selfintersection so that it lies in a plane. We show that this problem, unknotting problem is in NP. We also consider the problem, unknotting problem of determining whether two or more such polygons can be split, or continuously deformed without selfintersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worstcase running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.
On Triangulating ThreeDimensional Polygons
 COMPUTATIONAL GEOMETRY: THEORY AND APPLICATIONS
, 1996
"... A threedimensional polygon is triangulable if it has a nonselfintersecting triangulation which defines a simplyconnected 2manifold. We show that the problem of deciding whether a 3dimensional polygon is triangulable is NPComplete. We then establish some necessary conditions and some sufficie ..."
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Cited by 28 (3 self)
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A threedimensional polygon is triangulable if it has a nonselfintersecting triangulation which defines a simplyconnected 2manifold. We show that the problem of deciding whether a 3dimensional polygon is triangulable is NPComplete. We then establish some necessary conditions and some sufficient conditions for a polygon to be triangulable, providing special cases when the decision problem may be answered in polynomial time.
Interactive Topological Drawing
, 1998
"... The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues ..."
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Cited by 21 (1 self)
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The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with the correct representation and display of these objects on a computer. By correctness we mean that the essential topological features of objects are maintained during interaction. We have chosen to limit the scope of topological drawing to knot theory, a domain that consists essentially of one class of object (embedded circles in threedimensional space) yet is rich enough to contain a wide variety of difficult problems of research interest. In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or selfintersections. This notion of equivalence may be thought of as the heart of knot theory. We present methods for the computer construction and interactive manipulation of a
The size of spanning disks for polygonal curves
 Discrete Comput. Geom
"... Abstract. For each integer n ≥ 0, there is a closed, unknotted, polygonal curve Kn in R 3 having less than 10n + 9 edges, with the property that any PiecewiseLinear triangulated disk spanning the curve contains at least 2 n−1 triangles. 1. Introduction. Let K be a closed polygonal curve in R3 consi ..."
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Cited by 9 (1 self)
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Abstract. For each integer n ≥ 0, there is a closed, unknotted, polygonal curve Kn in R 3 having less than 10n + 9 edges, with the property that any PiecewiseLinear triangulated disk spanning the curve contains at least 2 n−1 triangles. 1. Introduction. Let K be a closed polygonal curve in R3 consisting of n line segments. Assume that K is unknotted, so that it is the boundary of an embedded disk in R3. This paper considers the question: How many triangles are needed to triangulate a PiecewiseLinear (PL) spanning disk of K? The main result, Theorem 1 below,
The size of spanning disks for polygonal knots
, 1999
"... For each integer n ≥ 1 we construct a closed unknotted Piecewise Linear curve Kn in R 3 having less than 11n edges with the property that any Piecewise Linear triangluated disk spanning the curve contains at least 2 n−1 triangles. 1 Introduction. We show the existence of a sequence of unknotted simp ..."
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Cited by 6 (1 self)
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For each integer n ≥ 1 we construct a closed unknotted Piecewise Linear curve Kn in R 3 having less than 11n edges with the property that any Piecewise Linear triangluated disk spanning the curve contains at least 2 n−1 triangles. 1 Introduction. We show the existence of a sequence of unknotted simple closed curves Kn in R 3 having the following properties: • The curve Kn is a polygon with at most 11n edges. • Any Piecewise Linear (PL) embedding of a triangulated disk into R 3 with
Homeomorphisms and Metamorphosis of Polyhedral Models Using Fields of Directions Defined on Triangulations
 SPECIAL ISSUE ON COMPUTER GRAPHICS AND IMAGE PROCESSING
, 1997
"... Many approaches have been proposed to generate the shape interpolation or morphing of two polyhedral objects given in a facet based representation. Most of them focus only the correspondence problem, leaving the interpolation process to just an interpolation of corresponding vertices. In this arti ..."
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Cited by 2 (1 self)
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Many approaches have been proposed to generate the shape interpolation or morphing of two polyhedral objects given in a facet based representation. Most of them focus only the correspondence problem, leaving the interpolation process to just an interpolation of corresponding vertices. In this article we present a new approach which uses fields of directions defined on triangulations(FDTs) to treat both the problem of getting an homeomorphism between the models and that of morphing them. Consider that an scaled version(P 1 ) of one of the objects, has already been adequately placed in the interior of the other(P 2 ). The objective of the first part of the approach, is to obtain a field of unit vectors defined on a triangulation of the space between P 1 and P 2 . This field must have no singularities and the trajectories determined by it will be later used to get warping and morphing transformations between P 1 and P 2 . The morphing transformations obtained have the good property of being topology preserving ones but it can be hard to get an FDT defined on a triangulation of P 1  P 2 and the intermediate models can have a very large number of faces. To illustrate those aspects, transformations between simple models are presented.
Graphics, Geometry, and Computing
"... ABSTRACT Computational geometry is a recent discipline with foundations in many branches of mathematics, and which is supposed to serve many applied areas. In this talk, after defining its subject matter, and reviewing its most important tools and paradigms, I will present a quick survey of some rec ..."
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ABSTRACT Computational geometry is a recent discipline with foundations in many branches of mathematics, and which is supposed to serve many applied areas. In this talk, after defining its subject matter, and reviewing its most important tools and paradigms, I will present a quick survey of some recent results that lie right at the frontiers of computational geometry. I hope this survey will give some idea of how computational geometry relates to those disciplines, and what we can and cannot expect from it. 1 What is computational geometry? Computational geometry (CG) is usually defined as &quot;the mathematical study of algorithms for solving geometric problems.&quot; Like most definitions, this one seems fine, but only until one really needs it. When I started preparing this talk I soon realized that this definition was too vague, and could easily be stretched to include most of mathematics an computer science. So, to make my task feasible, I must &quot;clarify &quot; that definition, by imposing four additional conditions. First, consider the phrase &quot;geometric problem. &quot; Obviously, any mathematical problem can be reformulated as a geometric problem in rather trivial ways (&quot;How many triangles is two triangles plus two triangles?&quot;). So, Condition One is that geometry must play an essential rolein the statement of the problem, or in the description of the algorithm, or in the accompanying proofs and analyses. Thus, for example, &quot;compute the convex hull of n given points &quot; easily passes this criterion; &quot;sort n given points in lefttoright order &quot; probably does not. Second, we need to clarify the meaning of &quot;algorithm. &quot; You may have noticed that in its origins, and throughout most of its history, geometry was primarily a computational discipline. To the ancient Greeks, ruler and compass were not artists ' tools, but rather computing
The size of spanning disks for PL Knots.
, 1998
"... For each integer n ? 1 we construct a closed unknotted PL curve Kn in R 3 having less than 33n edges with the property that any PL triangluated disk spanning the curve contains at least 2 n triangles. 1 Introduction. We show the existence of a sequence of unknotted simple closed curves Kn in ..."
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For each integer n ? 1 we construct a closed unknotted PL curve Kn in R 3 having less than 33n edges with the property that any PL triangluated disk spanning the curve contains at least 2 n triangles. 1 Introduction. We show the existence of a sequence of unknotted simple closed curves Kn in R 3 having the following properties: ffl The curve Kn is a polygon with at most 33n edges. ffl Any PL embedding of a triangulated disk into R 3 with boundary Kn contains at least 2 n triangular faces. The existence of such disks has implications to the complexity of geometric algorithms. For example, it shows that algorithms to test knot triviality that search for embedded disks in the complement need to deal with disks containing exponentially many triangles. Thus the exponential bounds on the size of the normal disks that are analyzed in [1],[3],[4],[5], and [6] cannot be replaced with polynomial bounds. Approaches to other problems, such as the word problem for 3manifold groups,...
1 Homeomorphisms and Metamorphosis of Polyhedral Models Using Fields of Directions Defined on
"... Abstract Many approaches have been proposed to generate the shape interpolation or morphing of two polyhedral objects given in a facet based representation. Most of them focus only the correspondence problem, leaving the interpolation process to just an interpolation of corresponding vertices. In th ..."
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Abstract Many approaches have been proposed to generate the shape interpolation or morphing of two polyhedral objects given in a facet based representation. Most of them focus only the correspondence problem, leaving the interpolation process to just an interpolation of corresponding vertices. In this article we present a new approach which uses fields of directions defined on triangulations(FDTs) to treat both the problem of getting an homeomorphism between the models and that of morphing them. Consider that an scaled version(P1) of one of the objects, has already been adequately placed in the interior of the other(P2). The objective of the first part of the approach, is to obtain a field of unit vectors defined on a triangulation of the space between P1 and P2. This field must have no singularities and the trajectories determined by it will be later used to get warping and morphing transformations between P1 and P2. The morphing transformations obtained have the good property of being topology preserving ones but it can be hard to get an FDT defined on a triangulation of P1 P2 and the intermediate models can have a very large number of faces. To illustrate those aspects, transformations between simple models are presented.