Discrete Hashimoto Surfaces and a Doubly Discrete Smoke-Ring Flow

Hoffmann, Tim

doi:10.1007/978-3-7643-8621-4_5

Cited by 33 publications

(62 citation statements)

References 9 publications

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“…In the theory of discrete integrable systems, it seems clear due to work of Hoffmann and others that 2 tan(θ/2) is the right notion of curvature density for equilateral polygons. See [HK04,Hof08].…”

Section: Curvature Densitymentioning

confidence: 99%

Curves of Finite Total Curvature

Sullivan

2008

Discrete Differential Geometry

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Abstract. We consider the class of curves of finite total curvature, as introduced by Milnor. This is a natural class for variational problems and geometric knot theory, and since it includes both smooth and polygonal curves, its study shows us connections between discrete and differential geometry. To explore these ideas, we consider theorems of Fáry/Milnor, Schur, Chakerian and Wienholtz.Keywords. Curves, finite total curvature, Fáry/Milnor theorem, Schur's comparison theorem, distortion.Here we introduce the ideas of discrete differential geometry in the simplest possible setting: the geometry and curvature of curves, and the way these notions relate for polygonal and smooth curves. The viewpoint has been partly inspired by work in geometric knot theory, which studies geometric properties of space curves in relation to their knot type, and looks for optimal shapes for given knots.After reviewing Jordan's definition of the length of a curve, we consider Milnor's analogous definition [Mil50] of total curvature. In this unified treatment, polygonal and smooth curves are both contained in the larger class of FTC (finite total curvature) curves. We explore the connection between FTC curves and BV functions. Then we examine the theorems of Fáry/Milnor, Schur and Chakerian in terms of FTC curves. We consider relations between total curvature and Gromov's distortion, and then we sketch a proof of a result by Wienholtz in integral geometry. We end by looking at ways to define curvature density for polygonal curves.A companion article [DS08] examines more carefully the topology of FTC curves, showing that any two sufficiently nearby FTC graphs are isotopic. The article [Sul08], also in this volume, looks at curvatures of smooth and discrete surfaces; the discretizations are chosen to preserve various integral curvature relations.Our whole approach in this survey should be compared to that of Alexandrov and Reshetnyak [AR89], who develop much of their theory for curves having one-sided tangents everywhere, a class somewhat more general than FTC.

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Section: Curvature Densitymentioning

confidence: 99%

Curves of Finite Total Curvature

Sullivan

2008

Discrete Differential Geometry

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“…Leider Abbildung 6. Evolution eines Ovals unter dem Rauchring-Fluss, aus [10]. In diesem Beispiel ist die Form der Kurve so gewählt, dass sie sich unter periodischer Formänderung vorwärtsbewegt.…”

Section: Diskrete Elastische Kurvenunclassified

“…Rauchring-Fluss, aus [10] ist das Integral (17) dort divergent. Man kann sich dadurch behelfen, dass man gleichzeitig mit dem Schlauchradius auch die Wirbelstärke gegen Null gehen lässt.…”

Section: Abbildung 7 Evolution Eines Polygons Unter Dem Doppelt Diskunclassified

Diskretisierung in Geometrie und Dynamik - Elastische Stäbe und Rauchringe

Alexander

Springborn

2013

Mitteilungen Der Deutschen Mathematiker-Vereinigung

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“…3. Among these, the first notion is the one considered by Steiner, the second corresponds to a finite element discretization using piecewise linear functions, and the third also arises in the theory of discrete integrable systems [4,16].…”

Section: Discrete Curvatures Of "Steiner-type"mentioning

confidence: 99%

“…To treat the first term, we use (16) and ∇I ≡ 0, where ∇ denotes covariant differentiation with respect to the metric induced by I, to derive…”

Section: Lemma 13 Let P Be a Vertex Of A Discrete Net Of Curvature LImentioning

confidence: 99%