Elastic Sturmian spirals in the Lorentz-Minkowski plane

Uçum, Ali; İlarslan, Kazım; Mladenov, Iväılo M.

doi:10.1515/math-2016-0103

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…and we will show its close relationship with a certain class of elastic curves of 2 . Recall that a unit speed spacelike or timelike curve γ in 2 is said to be an elastica under tension σ (see [8,13]) if it satisfies the differential equation…”

Section: )mentioning

confidence: 99%

“…But now, concerning the Lorentzian version of Singer's problem, our knowledge is much more restricted in comparison with the Euclidean case. In fact, we can only mention the articles [7,8] in this line, both devoted to Sturmian spiral curves. The authors initiated in [9] the study of the spacelike and timelike curves in 2 satisfying = ( ) κ κ y or = ( ) κ κ x .…”

Section: Introductionmentioning

confidence: 99%

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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: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Castro

Castro-Infantes

2020

Open Mathematics

View full text Add to dashboard Cite

show abstract

“…But now, referred to the Lorentzian version of Singer's problem, our knowledge is much more restricted in comparison with the Euclidean case. In fact, we can only mention the articles [IUM15] and [UIM16] in this line, both devoted to Sturmian spiral curves. The authors initiated in [CCIs18] the study of the spacelike and timelike curves in L 2 satisfying κ = κ(y) or κ = κ(x), both conditions geometrically interpreted as that the curvature is expressed in terms of the Lorentzian pseudodistance to timelike or spacelike fixed geodesics.…”

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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Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers

Castro

Castro-Infantes

2018

Open Mathematics

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This article is motivated by a problem posed by David A. Singer in 1999 and by the classical Euler elastic curves. We study spacelike and timelike curves in the Lorentz-Minkowski plane 𝕃2 whose curvature is expressed in terms of the Lorentzian pseudodistance to fixed geodesics. In this way, we get a complete description of all the elastic curves in 𝕃2 and provide the Lorentzian versions of catenaries and grim-reaper curves. We show several uniqueness results for them in terms of their geometric linear momentum. In addition, we are able to get arc-length parametrizations of all the aforementioned curves and they are depicted graphically.

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Elastic Sturmian spirals in the Lorentz-Minkowski plane

Cited by 3 publications

References 9 publications

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

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

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

Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers

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