Surfaces of osculating circles in Euclidean space

TL;DR

This paper introduces a new class of surfaces in Euclidean 3-space called surfaces of osculating circles, classifies those that are canal or Weingarten surfaces, and explores their geometric properties.

Contribution

It defines surfaces of osculating circles in Euclidean space and classifies the subclass that are canal or Weingarten surfaces, expanding geometric understanding.

Findings

Surfaces of osculating circles contain a uniparametric family of planar lines of curvature.

Classification of these surfaces as canal or Weingarten surfaces.

Identification of geometric properties specific to these classes.

Abstract

We introduce a new class of surfaces in Euclidean $3$-space, called surfaces of osculating circles, using the concept of osculating circle of a regular curve. These surfaces contain a uniparametric family of planar lines of curvature. In this paper, we classify those ones that are canal surfaces and Weingarten surfaces.