Locus of the centers of Meusnier spheres in Euclidean 3-space

TL;DR

This paper explores the geometric properties of Meusnier spheres in Euclidean 3-space, establishing their centers' locus as an evolute curve for principal lines and relating it to focal curves and special types of curves.

Contribution

It provides a new geometric interpretation of the locus of Meusnier sphere centers and links it to known curve types like evolutes, helices, and slant helices.

Findings

Locus of Meusnier sphere centers forms an evolute for principal lines.

Relations established between the locus and focal curves.

Connections made with helices and slant helices.

Abstract

In this study, we investigate the locus of the centers of the Meusnier spheres. Just as focal curve is the locus of the centers of the osculating spheres, we investigate the geometrical interpretation on the locus of the centers of the Meusnier spheres. We proved that if the curve is a principal line, the locus of the centers of the Meusnier spheres of the curve is an evolute curve. Then, we give the relations between this evolute curve and the focal curve. Also, we give some relations between helices, slant helices and the locus of the centers of the Meusnier spheres of the curve.