A new family of curves in space

TL;DR

This paper introduces a new family of space curves characterized by specific curvature and torsion conditions, providing explicit equations, examples, and applications to rectifying curves, expanding the understanding of spatial curve families.

Contribution

It defines a novel family of space curves with explicit intrinsic equations and demonstrates methods to generate diverse examples, including applications to rectifying curves.

Findings

Derived necessary and sufficient conditions for the new family.

Provided explicit position vectors with arc length parameter.

Showed how to generate a variety of examples and applications.

Abstract

This article defines a new family of curves in space, whose graphs generate shapes similar to whirls. An intrinsic equation is found, in terms of curvature and torsion, which gives necessary and sufficient conditions for the existence of this family. Its position vector is found, with arc length parameter and finally a way of generating a great variety of examples is shown; in particular an application to rectifying curves is given.