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On non-Newtonian Helices in Multiplicative Euclidean Space

Aykut Has, Beyhan Yılmaz

Abstract

In this article, spherical indicatrices of a curve and helices are re-examined using both the algebraic structure and the geometric structure of non-Newtonian (multiplicative) Euclidean space. Indicatrices of a multiplicative curve on the multiplicative sphere in multiplicative space are obtained. In addition, multiplicative general helix, multiplicative slant helix and multiplicative clad and multiplicative g-clad helix characterizations are provided. Finally, examples and drawings are given.

On non-Newtonian Helices in Multiplicative Euclidean Space

Abstract

In this article, spherical indicatrices of a curve and helices are re-examined using both the algebraic structure and the geometric structure of non-Newtonian (multiplicative) Euclidean space. Indicatrices of a multiplicative curve on the multiplicative sphere in multiplicative space are obtained. In addition, multiplicative general helix, multiplicative slant helix and multiplicative clad and multiplicative g-clad helix characterizations are provided. Finally, examples and drawings are given.
Paper Structure (6 sections, 13 theorems, 86 equations, 9 figures)

This paper contains 6 sections, 13 theorems, 86 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Multiplicative Calculus and Multiplicative Space
  3. Differential Geometry of Curves in Multiplicative Space
  4. Main Results
  5. Multiplicative Spherical Indicatries
  6. Multiplicative Helices

Key Result

Proposition 4.1

Let $\mathbf x_t$ be the multiplicative tangent indicatrix of a multiplicative naturally parameterized curve $\mathbf x$. The multiplicative naturally parameter $s_t$ of $\mathbf x_t$ is given by where $s$ is multiplicative naturally parameter of $\mathbf x$ and $\kappa(s)$ is multiplicative curvature of $\mathbf x$

Figures (9)

  • Figure 1: Multiplicative orthogonal vectors $\mathbf u=(e^{\frac{1}{2}},e^{-\frac{3}{4}},e^{\frac{3}{2}})$ and $\mathbf v=(e^{\frac{3}{4}},e,e^{\frac{1}{4}})$.
  • Figure 2: Multiplicative orthogonal system $\mathbf u=(e^{\frac{1}{2}},e^{-\frac{3}{4}},e^{\frac{3}{2}})$, $\mathbf v=(e^{\frac{3}{4}},e,e^{\frac{1}{4}})$ and $\mathbf u\times_*\mathbf v=(e^{-\frac{27}{16}},e,e^{\frac{17}{16}})$.
  • Figure 3: A multiplicative sphere with centered at multiplicative origin $O(0_*,0_*,0_*)$ and radius $1_*$.
  • Figure 4: A multiplicative circle in the plane $z=e^{\frac{\sqrt{3}}{2}}$ with centered at $(0_*,0_*,0_*)$, radius $r=1/e^2$ and $0_*<s<e^{2\pi}$.
  • Figure 5: Multiplicative curve and its multiplicative Frenet frame.
  • ...and 4 more figures

Theorems & Definitions (28)

  • Proposition 4.1
  • proof
  • Theorem 4.2
  • proof
  • Proposition 4.3
  • proof
  • Proposition 4.4
  • Theorem 4.5
  • Proposition 4.6
  • Proposition 4.7
  • ...and 18 more