Pedal curves of frontals in the Euclidean plane

A pedal curve (a contrapedal curve) of a regular plane curve is the locus of the feet of the perpendiculars from a point to the tangents (normals) to the curve. These curves can be parametrized by using the Frenet frame of the curve. Yet provided that the curve has some singular points, the Frenet frame at these singular points is not well‐defined. Thus, we cannot use the Frenet frame to examine pedal or contrapedal curves. In this paper, pedal and contrapedal curves of plane curves, which have singular points, are investigated. By using the Legendrian Frenet frame along a front, the pedal and contrapedal curves of a front are introduced and properties of these curves are given. Then, the condition for a pedal (and a contrapedal) curve of a front to be a frontal is obtained. Furthermore, by considering the definitions of the evolute, the involute, and the offset of a front, some relationships are given. Finally, some illustrated examples are presented.

“…The following proposition has been obtained for pedal curves of frontals in Li and Pei . Similarly, it can be given for pedal curves of fronts.…”

Section: Pedal and Contrapedal Curves Of Fronts In The Euclidean Planementioning

confidence: 82%

Section: Pedal and Contrapedal Curves Of Fronts In The Euclidean Planementioning

confidence: 99%

See 1 more Smart Citation

Notes on pedal and contrapedal curves of fronts in the Euclidean plane

Tuncer

Ceyhan

Gök

et al. 2018

“…Along with the moving frame, they defined a pair of smooth functions like as the curvature of a regular curve and called the pair the curvature of the Legendrian curve. By using the moving frame, they can give a new definition of evolute of the front . In this paper, we proceed with this way to investigate the evolutes of smooth curves in the Minkowski plane.…”

Section: Introductionmentioning

confidence: 99%

“…In their study, 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 the previous studies . If a curve has singular points, we can not construct its moving frame.…”

Section: Introductionmentioning

confidence: 99%

Evolutes of fronts in the Minkowski plane

Sun

2018

Self Cite

We consider the differential geometry of evolutes of singular curves and give the definitions of spacelike fronts and timelike fronts in the Minkowski plane. We also give the notions of moving frames along the non‐lightlike fronts in the Minkowski plane. By using the moving frames, we define the evolutes of non‐lightlike fronts and investigate the geometric properties of these evolutes. We obtain that the evolute of a spacelike front is a timelike front and the evolute of a timelike front is a spacelike front. Since the evolute of a non‐lightlike front is also a non‐lightlike front, we can take evolute again. We study the Minkowski Zigzag number of non‐lightlike fronts and give the n‐th evolute of the non‐lightlike front. Finally, we give an example to illustrate our results.

Slant helix of order n and sequence of Darboux developables of principal‐directional curves

Wang

Zhao

2020

Self Cite

In this paper, we consider the sequence of the principal-directional curves of a curve and define the slant helix of order n (n-SLH) of the curve in Euclidean 3-space. The notion is an extension of the notion of slant helix. We present an important formula that determines if the nth principal-directional curve of can be the slant helix of order n (n ≥ 1). As an application of singularity theory, we study the singularities classifications of the Darboux developable of nth principal-directional curve of. It is demonstrated that the formula plays a key role in characterizing the singularities of the Darboux developables of the nth principal-directional curve of a curve .