Concircular helices and concircular surfaces in Euclidean 3-space R3

TL;DR

This paper characterizes concircular helices and surfaces in Euclidean 3-space, providing differential equations and classifications, and shows how these surfaces relate to conical and spherical surfaces.

Contribution

It introduces a differential equation characterization of concircular helices and classifies concircular surfaces as specific ruled surfaces in R^3.

Findings

Concircular helices are characterized by a specific differential equation involving curvature and torsion.

Concircular surfaces are classified as either parallel to conical surfaces or as normal surfaces to spherical curves.

Concircular helices are geodesics on concircular surfaces.

Abstract

In this paper we characterize concircular helices in $R^3$ by means of a differential equation involving their curvature and torsion. We find a full description of concircular surfaces in $R^3$ as a special family of ruled surfaces, and we show that $M$ in $R^3$ is a proper concircular surface if and only if either $M$ is parallel to a conical surface or $M$ is the normal surface to a spherical curve. Finally, we characterize the concircular helices as geodesics of concircular surfaces.