On construction of Mannheim curves with Myller configuration

Authors

• ZEHRA İŞBİLİRFollow
• BAHAR DOĞAN YAZICIFollow
• MURAT TOSUNFollow

Author ORCID Identifier

ZEHRA İŞBİLİR: 0000-0001-5414-5887

BAHAR DOĞAN YAZICI: 0000-0001-5690-4840

MURAT TOSUN: 0000-0002-4888-1412

Abstract

In this study, our aim is to strengthen the results of the existing literature on the concept of Mannheim curves by using the natural and idiosyncratic structure of Myller configuration. For this purpose, we examine Mannheim curves with Frenet-type frame in Myller configuration for Euclidean 3-space E₃ and Euclidean 4-space E₄. Since the classical Frenet frames in E₃ and E₄ are special cases of Frenet-type frames with Myller configuration in E₃ and E₄, and because of the special and natural structure of the Myller configuration, the Mannheim curves with Frenet frames in E₃ and E₄ are also special cases of the Mannheim curves with Frenet-type frames in E₃ and E₄. We derive several relations among the Frenet-type frame elements of Mannheim curves with Myller configuration. Subsequently, numerical examples illustrating the presented concepts are constructed, together with corresponding figures. We also introduce a new perspective that yields a fairly large family of curves for Mannheim curves with Myller configuration in E₃. Indeed, this family contains several distinct special cases, and we exemplify only one of them by focusing on the most famous case, the Frenet frame. The main reason for this is that this study opens a new perspective for Mannheim curves, since the values a_i(s) are smooth functions and the parameter s ∈ I is the arc-length parameter, where i ∈ {1, 2, 3} (and i ∈ {1, 2, 3, 4} for E₄). Moreover, the condition ∑³_i=1 a_i²(s) = 1 (and ∑⁴_i=1 a_i²(s) = 1 for E₄) defining these special functions arises from the Myller configuration in E₃ (and E₄). Consequently, we present several graphs illustrating the relationships between special cases and the general setting studied in this paper and in the existing literature.

DOI

10.55730/1300-0098.3672

Keywords

Frenet-type frame, Mannheim curve, Myller configuration, versor field

First Page

613

Last Page

635

Publisher

The Scientific and Technological Research Council of Türkiye (TÜBİTAK)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Recommended Citation

İŞBİLİR, Z, DOĞAN YAZICI, B, & TOSUN, M (2026). On construction of Mannheim curves with Myller configuration. Turkish Journal of Mathematics 50 (4): 613-635. https://doi.org/10.55730/1300-0098.3672