Open Access
Research Article
Space Curves Related by a Transformation of Combescure
*Çetin CamciCorresponding authorccamci@comu.edu.trDepartment of mathematicsFaculty of Sciences and ArtsOnsekiz Mart UniversityÇanakkale, TurkeyView full profile → , Ali Uçumaliucum05@gmail.comDepartment of MathematicsFaculty of Sciences and ArtsKirikkale UniversityKirikkale, TurkeyView full profile → , Kazim Ilarslankilarslan@yahoo.comDepartment of MathematicsFaculty of Sciences and ArtsKirikkale UniversityKirikkale, TurkeyView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 22 Aug 2021
- Accepted:
- 08 Nov 2021
- Published Online:
- 04 Jan 2022
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- 1726037X.2021.2011113
- Pages:
- 271–287
Abstract
In this paper, the curves associated with the Combescure transform are discussed. With the help of the fact that these curves have a common Frenet frame, an equivalence relation is defined. The equivalence classes obtained by this equivalence relation have been examined for some special curves and it has been obtained that all curves in the equivalence class of a helix curve are also helix curves. This is also true for k-slant helix curves. The important part of this paper consists of the useful construction method to obtain a curve from the given curve α with the help of Combescure transformation. With this method, some special curves such as Bertrand, Mannheim, Salkowski, anti-Salkowski or spherical curve can be obtained from any curve related by a Combescure transform. For example, we obtain an example of Mannheim curves explicitly obtained from an anti-Salkowski curve. It is not easy to find an example of Mannheim curves except circular helix in the literature. In general, the condition for being Bertrand anti-Salkowski or spherical curve of the curve β obtained from the given curve α with the help of Combescure transformation were obtained.
Keywords
Subject Classifications
53B3053A35
References
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