Iterating Evolutes and Involutes

Arnold, Maxim; Fuchs, Dmitry; Izmestiev, Ivan; Tabachnikov, Serge; Tsukerman, Emmanuel

doi:10.1007/s00454-017-9890-y

Cited by 10 publications

(20 citation statements)

References 22 publications

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“…Thus γ(s, t) is evolving null Cartan curve parameterized by pseudo-arc s for all time t. Since γ s , N = 0 for all time t, Definition 3.3 implies that γ(s, t) is an evolving involute of α. From 7and (40) we get γ t = −τT − B = γ ss × γ sss , which means that γ(s, t) evolves according to null Betchov-Da Rios vortex filament equation (2). It generates timelike Hasimoto surface, since the plane spanned by {γ s , γ t } is a timelike (Fig.…”

Section: An Applicationmentioning

confidence: 96%

“…In this section, we give two examples of evolving involutes of the null Cartan curve α in E 3 1 , which evolve according to null Betchov-Da Rios vortex filament equation (2) and generate timelike Hasimoto surfaces.…”

Section: An Applicationmentioning

confidence: 99%

“…By using (7) and (38), we easily get β t = β ss × β sss . Hence β(s, t) evolves according to null Betchov-Da Rios vortex filament equation (2). Since the plane span{β s , β t } is a timelike, evolving curve β(s, t) generates a timelike Hasimoto surface which represents B-scroll ( Fig.…”

Section: An Applicationmentioning

confidence: 99%

“…In a simply isotropic space I (1) n and Galilean space G 4 , involutes of order k are introduced in [7,17]. For further properties of evolutes and involutes in Euclidean and Minkowski spaces, we refer to [2,12,14,24,25].…”

Section: Introductionmentioning

confidence: 99%

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On involutes of order k of a null Cartan curve in Minkowski spaces

Hanif

Zhong

Nešović

2019

Filomat

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In this paper, we define an involute and an evolving involute of order k of a null Cartan curve in Minkowski space E n 1 for n ≥ 3 and 1 ≤ k ≤ n − 1. In relation to that, we prove that if a null Cartan helix has a null Cartan involute of order 1 or 2, then it is Bertrand null Cartan curve and its involute is its Bertrand mate curve. In particular, we show that Bertrand mate curve of Bertrand null Cartan curve can also be a non-null curve and find the relationship between the Cartan frame of a null Cartan curve and the Frenet or the Cartan frame of its non-null or null Cartan involute of order 1 ≤ k ≤ 2. We show that among all null Cartan curves in E 3 1 , only the null Cartan cubic has two families of involutes of order 1, one of which lies on B-scroll. We also give some relations between involutes of orders 1 and 2 of a null Cartan curve in Minkowski 3-space. As an application, we show that involutes of order 1 of a null Cartan curve in E 3 1 , evolving according to null Betchov-Da Rios vortex filament equation, generate timelike Hasimoto surfaces.

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Section: An Applicationmentioning

confidence: 96%

Section: An Applicationmentioning

confidence: 99%

Section: An Applicationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On involutes of order k of a null Cartan curve in Minkowski spaces

Hanif

Zhong

Nešović

2019

Filomat

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“…We can recover this result by using the basis {h i k }. In fact, we shall prove that for any h ∈ C 0 , not necessarily in W 0 , S n h converges to 0, which means that the iteration of involutes of any closed curve γ ∈ H with dual length 0 converges to 0 (see [1] for the euclidean case).…”

Section: Convergence Of Involutes Iterationmentioning

confidence: 98%

Closed cycloids in a normed plane

2017

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Given a normed plane P, we call P-cycloids the planar curves which are homothetic to their double P-evolutes. It turns out that the radius of curvature and the support function of a P-cycloid satisfy a differential equation of Sturm-Liouville type. By studying this equation we can describe all closed hypocycloids and epicycloids with a given number of cusps. We can also find an orthonormal basis of C 0 (S 1 ) with a natural decomposition into symmetric and anti-symmetric functions, which are support functions of symmetric and constant width curves, respectively. As applications, we prove that the iterations of involutes of a closed curve converge to a constant and a generalization of the Sturm-Hurwitz Theorem. We also prove versions of the four vertices theorem for closed curves and six vertices theorem for closed constant width curves.Mathematics Subject Classification (2010). 52A10, 52A21, 53A15, 53A40.

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Discrete Cycloids from Convex Symmetric Polygons

Craizer

Teixeira²,

Balestro

2017

Discrete Comput Geom

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Cycloids, hipocycloids and epicycloids have an often forgotten common property: they are homothetic to their evolutes. But what if use convex symmetric polygons as unit balls, can we define evolutes and cycloids which are genuinely discrete? Indeed, we can! We define discrete cycloids as eigenvectors of a discrete double evolute transform which can be seen as a linear operator on a vector space we call curvature radius space. We are also able to classify such cycloids according to the eigenvalues of that transform, and show that the number of cusps of each cycloid is well determined by the ordering of those eigenvalues. As an elegant application, we easily establish a version of the four-vertex theorem for closed convex polygons. The whole theory is developed using only linear algebra, and concrete examples are given.

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Iterating Evolutes and Involutes

Cited by 10 publications

References 22 publications

On involutes of order k of a null Cartan curve in Minkowski spaces

On involutes of order k of a null Cartan curve in Minkowski spaces

Closed cycloids in a normed plane

Discrete Cycloids from Convex Symmetric Polygons

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