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A non-Newtonian some partner curves in multiplicative Euclidean space

Aykut Has, Beyhan Yılmaz

Abstract

The aim of this article is to characterize pairs of curves within multiplicative (non-Newtonian) spaces. Specifically, we investigate how famous curve pairs such as Bertrand partner curves, Mannheim partner curves, which are prominent in differential geometry, are transformed under the influence of multiplicative analysis. By leveraging the relationships between multiplicative Frenet vectors, we introduce multiplicative versions of Bertrand, Mannheim curve pairs. Subsequently, we characterize these curve pairs using multiplicative arguments. Examples are provided, and multiplicative graphs are presented to enhance understanding of the subject matter. Through this analysis, we aim to elucidate the behavior and properties of these curve pairs within the context of multiplicative geometry.

A non-Newtonian some partner curves in multiplicative Euclidean space

Abstract

The aim of this article is to characterize pairs of curves within multiplicative (non-Newtonian) spaces. Specifically, we investigate how famous curve pairs such as Bertrand partner curves, Mannheim partner curves, which are prominent in differential geometry, are transformed under the influence of multiplicative analysis. By leveraging the relationships between multiplicative Frenet vectors, we introduce multiplicative versions of Bertrand, Mannheim curve pairs. Subsequently, we characterize these curve pairs using multiplicative arguments. Examples are provided, and multiplicative graphs are presented to enhance understanding of the subject matter. Through this analysis, we aim to elucidate the behavior and properties of these curve pairs within the context of multiplicative geometry.
Paper Structure (7 sections, 6 theorems, 87 equations, 7 figures)

This paper contains 7 sections, 6 theorems, 87 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Multiplicative Calculus and Multiplicative Space
  3. Differential Geometry of Curves in Multiplicative Space
  4. Multiplicative partner curves
  5. Multiplicative Bertrand partner curves
  6. Multiplicative Mannheim partner curves
  7. Conclusion

Key Result

Proposition 4.2

Let $\mathbf x$ and $\mathbf y$ form a multiplicative Bertrand partner curve in the multiplicative Euclidean space. In this case, $\lambda$ in Eq. 8 is a multiplicative constant.

Figures (7)

  • Figure 1: Multiplicative orthogonal system
  • Figure 2: Multiplicative unit circle and sphere
  • Figure 3: Multiplicative plane
  • Figure 4: Multiplicative Bertrand partner curve and their multiplicative Frenet vectors.
  • Figure 5: Multiplicative Bertrand partner curve.
  • ...and 2 more figures

Theorems & Definitions (18)

  • Definition 4.1
  • Remark 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Theorem 4.4
  • proof
  • Example 4.5
  • Definition 4.2
  • ...and 8 more