ON CURVES AND TUBES IN $\mathbb{R}^n$

TL;DR

This paper derives explicit formulas for the curvature of regular curves in any dimension and introduces the concept of tubes around curves, calculating their volume and generalizing classical theorems.

Contribution

It provides a new explicit curvature formula for curves in Euclidean space and generalizes Pappus's theorems for tubes with arbitrary cross sections.

Findings

Explicit curvature formulas in $\\mathbb{R}^n$

Volume calculations for tubes with arbitrary cross sections

Generalization of Pappus's second theorem

Abstract

In this paper, we present a new explicit formula for the curvatures of a regular curve with an arbitrary parameter in the Euclidean space $\mathbb{R}^n$, $n\geq 2$, expressed only in terms of its derivatives. We introduce also the notion of tube with arbitrary cross sections around a curve for which we calculate the volume and give a generalization for the second theorem of Pappus. The first theorem of Pappus is obtained for sphere tubes in arbitrary dimension.