The relation between parameter curves and lines of curvature on canal surfaces

TL;DR

This paper explores the conditions under which parameter curves on canal surfaces also serve as lines of curvature, providing new insights into their geometric properties and specific radius functions for special spine curves.

Contribution

It investigates the relationship between parameter curves and lines of curvature on canal surfaces and derives radius functions for special spine curves.

Findings

Parameter curves can be lines of curvature under certain conditions.

Explicit radius functions are obtained for specific spine curves.

The geometric relationship enhances understanding of canal surface properties.

Abstract

A canal surface is the envelope of a moving sphere with varying radius, defined by the trajectory C(t) (spine curve) of its center and a radius function r(t). In this paper, we investigate when parameter curves of the canal surface are also lines of curvature. Last of all, for special spine curves we obtain the radius function of canal surfaces.