New Plane Curves with Curvature Depending on Distance from the Origin

Castro, Ildefonso; Castro-Infantes, Ildefonso; Castro-Infantes, Jesús

doi:10.1007/s00009-017-0912-z

Cited by 10 publications

(15 citation statements)

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“…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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Section: Introductionmentioning

confidence: 99%

A new approach to rotational Weingarten surfaces

Carretero¹,

Castro²

2021

Preprint

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“…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%

“…(ii) and (iii) in Remark 2.1, eliminating ds, we arrive atIn this way, we deduce the pseudopolar equations of this family given by that combined with (2.14), taking into account Remark 2.1, provide us the curves of this family. Accordingly to Section 7.1 in[6], we will refer these curves as Lorentzian sinusoidal spirals. Thus, we deduce the following characterization of this wide family of curves in 2 .Corollary 5.1.…”

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“…Spacelike (blue) and timelike (red) curves in 2 such that ( Taking into account what happens in the Euclidean case (see Remark 7.1 in[6]), the Lorentzian sinusoidal spirals include the Lorentzian versions of some very interesting plane curves, even some conics. In particular, up to dilations, we emphasize the following curves:(i) = n 2: the Lorentzian Bernoulli pseudolemniscate defined byFigure 12); (iv) = − n 2: the Lorentzian equilateral pseudohyperbolas defined by…”

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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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“…The recent literature devoted to the study of particular cases of Singer's posed problem consisting on determining plane curves α = (x, y) given κ = κ(r), r = x 2 + y 2 , includes several papers like [DVM09], [VDM09], [MHDV10], [MHDV11a], [MHDV11b], [MHM14] or [MMH14]. In addition, the authors have studied the cases κ = κ(y) and κ = κ(r) in [CCI16] and [CCIs17] respectively, for a large number of prescribed curvature functions.…”

Section: Introductionmentioning

confidence: 99%

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

Castro¹,

Castro-Infantes²,

Castro-Infantes³

2018

Preprint

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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 Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in 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's 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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New Plane Curves with Curvature Depending on Distance from the Origin

Cited by 10 publications

References 10 publications

A new approach to rotational Weingarten surfaces

A new approach to rotational Weingarten surfaces

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

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

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