Representations of Rectifying Isotropic Curves and Their Centrodes in Complex 3-Space

Jinhua Qian,Pei Yin,Hongzeng Wang,Xueshan Fu

doi:10.3390/math9121451

Abstract

In this work, the rectifying isotropic curves are investigated in three-dimensional complex space C3. The conclusion that an isotropic curve is a rectifying curve if and only if its pseudo curvature is a linear function of its pseudo arc-length is achieved. Meanwhile, the rectifying isotropic curves are expressed by the Bessel functions explicitly. Last but not least, the centrodes of rectifying isotropic curves are explored in detail.

Highlights

Chen first proposed the notion of rectifying curves, which represent a class of space curves whose position vector always lies in its rectifying plane in Euclidean 3-space [1]
The rectifying curves are generalized into the ones in Minkowski 3-space, which are divided into space-like rectifying curves
The rectifying curves are defined in Euclidean 4-space, i.e., the position vector of a space curve lies in the orthogonal complement of its principal normal vector field [7]

Summary

Introduction

The position vector of a rectifying isotropic curve r(s) can be expressed as follows: r(s) = λ(s)e1(s) + μ(s)e3(s), where λ(s) and μ(s) are nonzero analytic functions. R(s) is congruent to a rectifying isotropic curve if, and only if, the pseudo curvature κ(s) of r(s) is a non-constant linear function of the pseudo arc length s, i.e., the following: κ(s) = c1s + c2, (0 = c1, c2 ∈ C). R(s) is a rectifying isotropic curve if, and only if, one of the following statements holds: (1) r(s), r(s) = 2a(s + b); (2) the tangent component r(s), e1(s) = a; (3) the binormal component r(s), e3(s) = s + b, where a = 0 and b are constants.

Gamma functions

Bessel functions