Parametric Representation of a Hypersurface Family With a Common Spatial Geodesic

TL;DR

This paper develops a parametric method to construct hypersurface families in four-dimensional space sharing a common spatial geodesic, utilizing the Frenet frame and providing explicit examples.

Contribution

It introduces a novel parametric representation for hypersurfaces with a shared geodesic in R4 based on the Frenet frame of the curve.

Findings

Derived necessary and sufficient conditions for the curve to be a geodesic.

Presented explicit examples illustrating the hypersurface construction.

Provided a linear combination framework for hypersurface parametrization.

Abstract

In this paper, we study the problem of finding a hypersurface family from a given spatial geodesic curve in R4. We obtain the parametric representation for a hypersurface family whose members have the same curve as a given geodesic curve. Using the Frenet frame of the given geodesic curve, we present the hypersurface as a linear combination of this frame and analyse the necessary and sufficient condition for that curve to be geodesic. We illustrate this method by presenting some examples.