Plane curves with curvature depending on distance to a line

Castro, Ildefonso; Castro-Infantes, Ildefonso

doi:10.1016/j.difgeo.2015.11.002

Cited by 15 publications

(17 citation statements)

References 2 publications

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“…It is straightforward to check that both curves satisfy the translating-type soliton equation = (( ) ) κ g N 1, 1 , . Hence, we have obtained in this section (see also Section 7.1 in [5]) certain Lorentzian versions of the grim-reaper curves of the Euclidean plane. We will simply call them Lorentzian grimreapers.…”

Section: Case =mentioning

confidence: 94%

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Curves in the Lorentz-Minkowski plane with curvature depending on their position

Castro

Castro-Infantes

2020

Open Mathematics

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Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in the Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in {{\mathbb{L}}}^{2} whose curvature depends on the Lorentzian pseudodistance from the origin, and those ones whose curvature depends on the Lorentzian pseudodistance through the horizontal or vertical geodesic to a fixed lightlike geodesic. Making use of the notions of geometric angular momentum (with respect to the origin) and geometric linear momentum (with respect to the fixed lightlike geodesic), respectively, we get two abstract integrability results to determine such curves through quadratures. In this way, we find out several new families of Lorentzian spiral, special elastic and grim-reaper curves whose intrinsic equations are expressed in terms of elementary functions. In addition, we provide uniqueness results for the generatrix curve of the Enneper surface of second kind and for Lorentzian versions of some well-known curves in the Euclidean setting, like the Bernoulli lemniscate, the cardioid, the sinusoidal spirals and some non-degenerate conics. We are able to get arc-length parametrizations of them and they are depicted graphically.

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Section: Case =mentioning

confidence: 94%

“…Some recent literature studies [3][4][5][6] are devoted to the study of particular cases of Singer's posed problem: determine plane curves = ( ) α…”

Section: Introductionmentioning

confidence: 99%

Curves in the Lorentz-Minkowski plane with curvature depending on their position

Castro

Castro-Infantes

2020

Open Mathematics

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“…These may be good reasons why rotational surfaces are probably one of the main classes of Weingarten surfaces and continue to deserve attention. We propose in this paper a new approach for their study, inspired mainly by [CCI16].…”

Section: Introductionmentioning

confidence: 99%

“…κ(x, z) = c x for curves in the xz-plane. Motivated by the above question and by the classical elasticae, the authors studied in [CCI16] the plane curves whose curvature depends on the distance to a line (say the z-axis and so κ = κ(x)) and in [CCIs17] the plane curves whose curvature depends on the distance from a point (say the origin, and so κ = κ(r), r = √ x 2 + z 2 ) requiring in both cases the computation of three quadratures too. They also considered the analogous problems in Lorentz-Minkowski plane in [CCIs18] and [CCIs20a].…”

Section: Introductionmentioning

confidence: 99%

A new approach to rotational Weingarten surfaces

Carretero¹,

Castro²

2021

Preprint

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Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new approach to the study of rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing any type of Weingarten condition on a rotational surface to a first order differential equation on the momentum of the generatrix curve. In this line, we provide two new classification results involving a cubic and an hyperbola in the Wdiagram of the surface characterizing, respectively, the non-degenerated quadric surfaces of revolution and the elasticoids, defined as the rotational surfaces generated by the rotation of the Euler elastic curves around their directrix line.As another application of our approach, we deal with the problem of prescribing mean or Gauss curvature on rotational surfaces in terms of arbitrary continuous functions depending on distance from the surface to the axis of revolution. As a consequence, we provide simple new proofs of some classical results concerning rotational surfaces, like Euler's theorem about minimal ones, Delaunay's theorem on constant mean curvature ones, and Darboux's theorem about constant Gauss curvature ones.

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“…In 1740, Bernoulli proposed a simple geometric model for an elastic curve in E 2 ; according to which an elastic curve or elastica is a critical point of the elastic energy functional R Ä 2 : Elastic curves in E 2 were already classified by Euler in 1743 but it was not until 1928 that they were also studied in E 3 by Radon, who derived the Euler-Lagrange equations and showed that they can be integrated explicitly. The elastica problem in real space forms has been recently considered using different approaches (see [1][2][3][4][5] and [6]). Are there other interesting elastic curves?…”

Section: Introductionmentioning

confidence: 99%