Characterization of ideal knots

Schuricht, Friedemann; Mosel, Heiko von der

doi:10.1007/s00526-003-0216-y

Cited by 36 publications

(37 citation statements)

References 20 publications

(46 reference statements)

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“…But by (18), (23) The following two lemmas and the theorem that follows are essential in determining the structure of the right-hand side g and therefore of the measure f . The main argument is the following.…”

Section: Van Der Heijden Peletier and Planquémentioning

confidence: 99%

“…If one of the ends of this interval equals 0 or T there is nothing to prove; we therefore assume that min{x 0 , T − x 1 − 1} ≥ d > 0. Now multiply (18) with the function…”

Section: Symmetrymentioning

confidence: 99%

“…Parallel advances have been made on the highly related ideal knots and Gehring links, where ropelength is minimized instead of elastic energy [4,18,3].…”

Section: Introductionmentioning

confidence: 99%

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Self-Contact for Rods on Cylinders

Heijden

Peletier

Planquè

2006

Arch Rational Mech Anal

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C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationSelf-contact for rods on cylinders G.H.M. van der Heijden, M.A. Peletier, R. Planqué Self-contact for rods on cylinders ABSTRACT We study self-contact phenomena in elastic rods that are constrained to lie on a cylinder. By choosing a particular set of variables to describe the rod centerline the variational setting is made particularly simple: the strain energy is a second-order functional of a single scalar variable, and the self-contact constraint is written as an integral inequality. Using techniques from ode theory (comparison principles) and variational calculus (cut-and-paste arguments) we fully characterize the structure of constrained minimizers. An important auxiliary result states that the set of self-contact points is continuous, a result that contrasts with known examples from contact problems in free rods. Abstract. We study self-contact phenomena in elastic rods that are constrained to lie on a cylinder. By choosing a particular set of variables to describe the rod centerline the variational setting is made particularly simple: the strain energy is a second-order functional of a single scalar variable, and the self-contact constraint is written as an integral inequality. REPORT MAS-E0411 AUGUST 2004Using techniques from ode theory (comparison principles) and variational calculus (cut-and-paste arguments) we fully characterize the structure of constrained minimizers. An important auxiliary result states that the set of selfcontact points is continuous, a result that contrasts with known examples from contact problems in free rods.

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Section: Van Der Heijden Peletier and Planquémentioning

confidence: 99%

“…If one of the ends of this interval equals 0 or T there is nothing to prove; we therefore assume that min{x 0 , T − x 1 − 1} ≥ d > 0. Now multiply (18) with the function…”

Section: Symmetrymentioning

confidence: 99%

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Self-Contact for Rods on Cylinders

Heijden

Peletier

Planquè

2006

Arch Rational Mech Anal

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“…In the past decade, there has been a great deal of interest in exploring the geometry of tight knots; the definition of thickness has been refined and fully understood [10], it has been shown that C 1,1 minimizers exist in each knot type [5,8,9], some minimizing links have been found [5], and a theory of ropelength criticality has started to emerge [4,21]. The development of this theory has been fueled by a steady stream of numerical data on ropelength minimizers, from Pieranski's original SONO algorithm [15] and Rawdon's TOROS [16], to second-generation efforts such as Smutny and Maddocks' biarc computations [6,19] and the RIDGERUNNER project of Cantarella, Piatek, and Rawdon.…”

Section: Introductionmentioning

confidence: 99%

A Fast Octree-Based Algorithm for Computing Ropelength

Ashtonand¹,

Cantarella²

2005

Physical and Numerical Models in Knot Theory

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The ropelength of a space curve is usually defined as the quotient of its length by its thickness: the diameter of the largest embedded tube around the knot. This idea was extended to space polygons by Eric Rawdon, who gave a definition of ropelength in terms of doubly-critical self-distances (local minima or maxima of the distance function on pairs of points on the polygon) and a function of the turning angles of the polygon. A naive algorithm for finding the doubly-critical self-distances of an n-edge polygon involves comparing each pair of edges, and so takes O(n 2 ) time. In this paper, we describe an improved algorithm, based on the notion of octrees, which runs in O(n log n) time. The speed of the ropelength computation controls the performance of ropelength-minimizing programs such as Rawdon and Piatek's TOROS. An implementation of our algorithm is freely available under the GNU Public License.

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“…An exciting recent development in this field of mathematics has been the formulation of a kind of Euler-Lagrange equation describing length-critical knots in terms of the set of self-contacts of their tubes [7,16,38]. This theory has allowed us to make some conjectures about the tightening process, but these conjectures have * e-mail: cantarel@math.uga.edu, supported by NSF DMS #0204862 † e-mail: piatek@cs.washington.edu, supported by NSF DMS #0311010 ‡ e-mail: rawdon@mathcs.duq.edu, supported by NSF DMS #0311010 Figure 1: A tight 7.2 knot as computed using our method.…”

Section: Introductionmentioning

confidence: 99%

Visualizing the tightening of knots

Cantarella

Piatek

Rawdon

IEEE Visualization 2005 - (VIS'05)

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The study of physical models for knots has recently received much interest in the mathematics community. In this paper, we consider the ropelength model, which considers knots tied in an idealized rope. This model is interesting in pure mathematics, and has been applied to the study of a variety of problems in the natural sciences as well.Modeling and visualizing the tightening of knots in this idealized rope poses some interesting challenges in computer graphics. In particular, self-contact in a deformable rope model is a difficult problem which cannot be handled by standard techniques. In this paper, we describe a solution based on reformulating the contact problem and using constrained-gradient techniques from nonlinear optimization. The resulting animations reveal new properties of the tightening flow and provide new insights into the geometric structure of tight knots and links.

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Characterization of ideal knots

Cited by 36 publications

References 20 publications

Self-Contact for Rods on Cylinders

Self-Contact for Rods on Cylinders

A Fast Octree-Based Algorithm for Computing Ropelength

Visualizing the tightening of knots

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