On the invariants of ruled surfaces generated by the dual involute frenet trihedron

Bi̇li̇ci̇, Mustafa

doi:10.1501/commua1_0000000801

Cited by 4 publications

(1 citation statement)

References 6 publications

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“…Bilici has examined ruled surfaces produced by a Frenet trihedron of closed dual involute for a specific dual curve. He has specifically focused on the relations between the pitch, the angle of the pitch, and the drall of these surfaces [20]. Some ruled surfaces produced using the Frenet trihedron, Blaschke frame, and the surface family are studied in [21][22][23][24].…”

Section: Introduction (Compulsory)mentioning

confidence: 99%

Characterizations of Unit Darboux Ruled Surface with Quaternions

Çalışkan

2023

Journal of New Theory

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This paper presents a quaternionic approach to generating and characterizing the ruled surface drawn by the unit Darboux vector. The study derives the Darboux frame of the surface and relates it to the Frenet frame of the base curve. Moreover, it obtains the quaternionic shape operator and its matrix representation using the normal and geodesic curvatures to provide a more detailed analysis. To illustrate the concepts discussed, the paper offers a clear example that will help readers better understand the concepts and showcases the quaternionic shape operator, Gauss curvature, mean curvature, and rotation matrix. Finally, it emphasizes the need for further research on this topic.

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Section: Introduction (Compulsory)mentioning

confidence: 99%

Characterizations of Unit Darboux Ruled Surface with Quaternions

Çalışkan

2023

Journal of New Theory

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Solutions of localized induction equation associated with involute–evolute curve pair

Eren,

Myrzakulova,

Ersoy

et al. 2023

Soft Comput

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In this paper, we derive the surfaces, which are called Hasimoto surfaces, corresponding to solutions of the localized induction equation for involute evolute curves. We investigate the differential geometric properties of these surfaces associated with the involute evolute curves. Moreover, we calculate the Gaussian and mean curvatures of Hasimoto surfaces in terms of curvatures of the curve pairs. Then, we determine the conditions for these surfaces' parameter curves to be the geodesics, asymptotics, and/or lines of curvature of the surface. We have demonstrated the evolutions of involute evolute curves in examples.

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The Dual Expression of Parallel Equidistant Ruled Surfaces in Euclidean 3-Space

2022

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In this study, we examine the dual expression of Valeontis’ concept of parallel p-equidistant ruled surfaces well known in Euclidean 3-space, according to the Study mapping. Furthermore, we show that the dual part of the dual angle on the unit dual sphere corresponds to the p-distance. We call these ruled surfaces we obtained “dual parallel equidistant ruled surfaces” and we briefly denote them with “DPERS”. Furthermore, we find the Blaschke vectors, the Blaschke invariants and the striction curves of these DPERS and we give the relationships between these elements. Moreover, we show the relationships between the Darboux screws, the instantaneous screw axes, the instantaneous dual Pfaff vectors and dual Steiner rotation vectors of these surfaces. Finally, we give an example, which we reinforce this article, and we explain all of these features with the figures on the example. Furthermore, we see that the corresponding dual curves on the dual unit sphere to these DPERS are such that one of them is symmetric with respect to the imaginary symmetry axis of the other.

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On the invariants of ruled surfaces generated by the dual involute frenet trihedron

Cited by 4 publications

References 6 publications

Characterizations of Unit Darboux Ruled Surface with Quaternions

Characterizations of Unit Darboux Ruled Surface with Quaternions

Solutions of localized induction equation associated with involute–evolute curve pair

The Dual Expression of Parallel Equidistant Ruled Surfaces in Euclidean 3-Space

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