Differential geometry of line of curvature on parametric hypersurfaces in $\mathbb{E}^4$

TL;DR

This paper presents an algorithm for computing lines of curvature on parametric hypersurfaces in four-dimensional Euclidean space and calculates their curvatures using the extended Darboux frame.

Contribution

It introduces a novel algorithm for lines of curvature on 4D hypersurfaces and applies the extended Darboux frame to determine their curvatures.

Findings

Effective algorithm for lines of curvature in $ extbf{E}^4$

Explicit formulas for curvatures using Darboux frame

Application to parametric hypersurfaces in 4D space

Abstract

The purpose of this paper is, first, to give an algorithm that enables to obtain the lines of curvature on parametric hypersurfaces in Euclidean 4-space, and then, to obtain the curvatures of such lines by using the extended Darboux frame along the curve.