Concircular helices and concircular surfaces in Euclidean 3-space R3

TL;DR

This work introduces concircular vector fields in $\mathbb{R}^3$ via $Y(p)=\mu p+v$ to unite classical curve classes (generalized helices, slant helices, and rectifying curves) under a single framework. It establishes a differential equation relating curvature, torsion, and their ratio $\rho=\tau/\kappa$ that characterizes proper concircular helices, with $m=-\mu/\lambda$ and $\rho'=0$ recovering known cases; it then classifies concircular surfaces as a special family of ruled surfaces, proving that a nontrivial surface is concircular iff it is either parallel to a conical surface or the normal surface to a spherical curve. The paper further shows that geodesics on concircular surfaces are concircular helices and provides explicit parametrizations and conditions for these geodesics, including a concrete example family with constant precession. Collectively, the results unify and extend geometric structures around concircularity, offering constructive representations and linking surface geometry to helix-like curve behavior via tangency and distance-to-origin properties.

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.

Figures (1)

• Figure 1: Concircular helices "of constant precession"

Theorems & Definitions (14)

• definition 1
• theorem 1
• proposition 1
• proof
• proposition 2
• proof
• theorem 2
• proof
• theorem 3
• proof