Helices associated to helical curves, relatively normal-slant helices and isophote curves

TL;DR

This paper introduces associated helices linked to special surface curves like helical, relatively normal-slant, and isophote curves, deriving their parametric forms via Darboux frames and surface curvatures.

Contribution

It defines a new class of associated helices and provides a method to determine their parametric forms based on surface curvatures and differential equations.

Findings

Derived differential equations for associated helices

Explicit parametric forms for helices related to specific surface curves

Framework applicable to various surface curves for helix construction

Abstract

This study introduces a new type of general helix called associated helix which is associated to a special surface curve. The basic idea is to determinate the parametric form of an associated helix by means of Darboux frame and surface curvatures of a special surface curve such as helical curve, relatively normal-slant helix or isophote curve. For each surface curve, a differential equation system is obtained and by solving this system, parametric form of an associated helix is introduced.