Intrinsic representation curves

TL;DR

This paper introduces a family of curves parametrized by arc length, characterized by an angular function and an intrinsic fraction function, with explicit formulas for curvature, torsion, and geodetic curvature, applicable to helices.

Contribution

It provides explicit formulas for curvature, torsion, and geodetic curvature of a new family of curves based on intrinsic functions, extending the understanding of special curves.

Findings

Explicit formulas for curvature, torsion, and geodetic curvature.

Application to general helices and slant helices.

Characterization of curves via intrinsic fraction and angular functions.

Abstract

The purpose of this article is to find a family of curves parametrized by arc length and that depend on an angular function and an intrinsic fraction function, which is defined as the quotient between torsion and curvature. We find for this family of curves explicit formulas of curvature, torsion and geodetic curvature, in terms of the angular function and the intrinsic fraction function. Applications are found for the case of the general helices and slant helices.