Bertrand and Mannheim curves of framed curves in the3-dimensional Euclidean space

Honda, Shun’ichi; Takahashi, Masatomo

doi:10.3906/mat-1905-63

Cited by 34 publications

(28 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Yaylı [3] investigated these curves in three dimensional Minkowski space in detail. Honda and Takahashi [4] stated the curvatures of the partner curve depending on the curvatures of the Mannheim curve as follows:…”

Section: Introduction and Material-methodsmentioning

confidence: 99%

Mannheim Curves in 3-Dimensional Euclidean Space

Öztürk¹

2020

International Scientific and Vocational Studies Journal

View full text Add to dashboard Cite

In this paper, we consider the Mannheim curve and the slant helix together. We called this curve as a Mannheim slant helix shortly. First we calculate the (first) curvature ( ), and the curvature of the tangent indicatrix of the Mannheim curve, in terms of the arc-lenght parameter of the curve. Also, we proved that if the Mannheim curve is also slant helix, i.e. if it is Mannheim slant helix, then the partner curve is general helix. Moreover, we show the striction curve of the ruled surface such that the base curve is Mannheim curve, and the rulings are the normal vector field of the Mannheim curve, is the Mannheim partner curve. Finally, we show the ruled surface such that the base curve is Mannheim curve, and the rulings are the normal vector field of the Mannheim curve is non-developable while the torsion of the Mannheim partner curve ( ) ≠ ±∞ for all s.

show abstract

Section: Introduction and Material-methodsmentioning

confidence: 99%

Mannheim Curves in 3-Dimensional Euclidean Space

Öztürk¹

2020

International Scientific and Vocational Studies Journal

View full text Add to dashboard Cite

show abstract

“…Then the parallels are normally flat and have the same Bishop frame with f . Parallels are basic and interesting objects to be studied in the cases of both hypersurfaces and curves [6,8].…”

Section: Parallels To Normally Flat Frontalsmentioning

confidence: 99%

Singularities of Developable Surfaces

Ishikawa

1999

Singularity Theory

View full text Add to dashboard Cite

It is known that the class of developable surfaces which have zero Gaussian curvature in three dimensional Euclidean space is preserved by the parallel transformations. A tangent developable surface is defined as a ruled developable surface by tangent lines to a space curve and it has singularities at least along the space curve, called the directrix or the the edge of regression. Also the class of tangent developable surfaces are invariant under the parallel deformations. In this paper the notions of tangent developable surfaces and their parallels are naturally generalized for frontal curves in general in Euclidean spaces of arbitrary dimensions. We study singularities appearing on parallels to tangent developable surfaces of frontal curves and give the classification of generic singularities on them for frontal curves in 3 or 4 dimensional Euclidean spaces.

show abstract

“…A lot of scholars are interested in geometric objects with symmetry in different spaces, and they study the different properties of them (cf. [7][8][9][10]).…”

Section: Introductionmentioning

confidence: 99%

Pedal Curves of Non-Lightlike Curves in Minkowski 3-Space

Yao

et al. 2022

Symmetry

View full text Add to dashboard Cite

We study the notions of pedal curves, contrapedal curves and B-Gauss maps of non-lightlike regular curves in Minkowski 3-space. Then we establish the relationships among the evolutes, the pedal and contrapedal curves. Moreover, we also investigate the singularities of these objects. Finally, we show some examples to comprehend the characteristics of the pedal and contrapedal curves in Minkowski 3-space.

show abstract

Bertrand and Mannheim curves of framed curves in the3-dimensional Euclidean space

Cited by 34 publications

References 12 publications

Mannheim Curves in 3-Dimensional Euclidean Space

Mannheim Curves in 3-Dimensional Euclidean Space

Singularities of Developable Surfaces

Pedal Curves of Non-Lightlike Curves in Minkowski 3-Space

Contact Info

Product

Resources

About