Surfaces Family With Common Null Asymptotic

TL;DR

This paper develops a method to construct a family of surfaces sharing a common null asymptotic curve using Cartan frames, providing parametric representations and conditions for asymptoticity.

Contribution

It introduces a new approach to generate surfaces with a shared null asymptotic curve via Cartan frames, including explicit parametric forms and necessary conditions.

Findings

Derived parametric representations for the surface family.

Established necessary and sufficient conditions for the asymptotic curve.

Provided illustrative examples demonstrating the method.

Abstract

We analyzed the problem of finding a surfaces family through an asymptotic curve with Cartan frame. We obtain the parametric representation for surfaces family whose members have the same as an asymptotic curve. By using the Cartan frame of the given null curve, we present the surface as a linear combination of this frame and analysed the necessary and sufficient condition for that curve to satisfy the asymptotic requirement. We illustrate the method by giving some examples.