Existence and uniqueness for Legendre curves

Fukunaga, Tomonori; Takahashi, Masatomo

doi:10.1007/s00022-013-0162-6

Cited by 72 publications

(93 citation statements)

References 8 publications

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“…We also can not use the Frenet-Serret type formula to study the properties of the original curve. In order to overcome this difficulty, we take advantage of the way developed by T. Fukunaga and M. Takahashi in [2] instead of the classical way. We give the detailed descriptions about this way as follows.…”

Section: The Frontals In the Spherementioning

confidence: 99%

“…Unfortunately, if the curve is not regular at a point, then we can not define the pedal curve at this point as the classical way. In [2] , T. Fukunaga and M. Takahashi firstly define frontals (or fronts) in Euclidean plane and Legendrian curves (or Legendrian immersions) in the unit tangent bundle of R 2 . The differential geometric properties of the frontal is studied in [3].…”

Section: Introductionmentioning

confidence: 99%

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Pedal curves of fronts in the sphere

Li¹,

Pei²

2016

J. Nonlinear Sci. Appl.

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Notions of the pedal curves of regular curves are classical topics. T. Nishimura [T. Nishimura, Demonstratio Math., 43 (2010), 447-459] has done some work associated with the singularities of pedal curves of regular curves. But if the curve has singular points, we can not define the Frenet frame at these singular points. We also can not use the Frenet frame to define and study the pedal curve of the original curve. In this paper, we consider the differential geometry of pedal curves of singular curves in the sphere. We define the pedal curve of a front and give properties of such pedal curve by using a moving frame along a front. At last, we give the classification of singularities of the pedal curves of fronts.

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Section: The Frontals In the Spherementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Pedal curves of fronts in the sphere

Li¹,

Pei²

2016

J. Nonlinear Sci. Appl.

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“…These are generalisations of evolutes and involutes of regular plane curves. In order to define an evolute and an involute of the front, we review on Legendre curves in the unit tangent bundle, the Frenet formula and the curvature of the Legendre curve ( [8]). …”

Section: Example 22mentioning

confidence: 99%

“…We have the existence and the uniqueness for Legendre curves in the unit tangent bundle like as regular plane curves, see in [8]. In fact, the Legendre curve whose associated curvature of the Legendre curve is (ℓ, β), is given by the form…”

Section: Example 22mentioning

confidence: 99%

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Involutes of fronts in the Euclidean plane

Fukunaga

Takahashi

2015

Beitr Algebra Geom

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The notions of involutes (also known as evolvents) and evolutes were studied by C. Huygens. For a regular plane curve, an involute of it is the trajectory described by the end of stretched string unwinding from a point of the curve. Even if a regular curve, the involute of the curve have singularities. By using a moving frame of the front and the curvature of the Legendre immersion in the unit tangent bundle, we define an involute of the front in the Euclidean plane and discuss properties of them. We also consider about relationship between evolutes and involutes of fronts without inflection points. As a result, we observe that the evolutes and the involutes of fronts without inflection points are corresponding to the differential and the integral in classical calculus.

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One‐parameter families of Legendre curves and plane line congruences

Kabata

Takahashi

2022

Mathematische Nachrichten

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Families of curves in the Euclidean plane naturally contain singular curves, where the frame of classical differential geometry does not work well. We introduce the notions of one-parameter family of Legendre curves in the Euclidean plane, congruent equivalence and curvature. Especially, a one-parameter family of Legendre curves can contain singular curves, and is determined by the curvature up to congruence. We also give properties of one-parameter families of Legendre curves. As applications, we give a relation between one-parameter families of Legendre curves and Legendre surfaces. Moreover, we study plane line congruences (one-parameter families of lines in plane) in terms of the curvatures as one-parameter families of Legendre curves.

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Existence and uniqueness for Legendre curves

Cited by 72 publications

References 8 publications

Pedal curves of fronts in the sphere

Pedal curves of fronts in the sphere

Involutes of fronts in the Euclidean plane

One‐parameter families of Legendre curves and plane line congruences

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