Hypersurface family with a common isoasymptotic curve

TL;DR

This paper develops a method to construct hypersurface families in four-dimensional space that share a common asymptotic curve, using the Frenet frame and analyzing conditions for asymptoticity.

Contribution

It introduces a novel approach to generate hypersurfaces with a prescribed asymptotic curve in R^4 based on Frenet frame analysis.

Findings

Derived necessary and sufficient conditions for a curve to be asymptotic on the hypersurface.

Presented explicit examples demonstrating the construction method.

Established a framework for hypersurface design with shared asymptotic curves.

Abstract

In the present paper, we handle the problem of finding a hypersurface family from a given asymptotic curve in R^4. Using the Frenet frame of the given asymptotic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be asymptotic. We illustrate this method by presenting some examples.