An approach for designing a surface pencil through a given asymptotic curve

TL;DR

This paper introduces a new method for constructing surfaces that interpolate a given curve as an asymptotic curve, analyzing conditions for ruled and developable surfaces, with illustrative examples.

Contribution

It presents a novel approach for designing surfaces with a specified asymptotic curve, including conditions for ruled and developable surfaces.

Findings

No non-planar developable surface can have a given curve as an asymptotic curve.

The method successfully constructs surfaces with the prescribed asymptotic curve.

Examples demonstrate the applicability of the proposed approach.

Abstract

Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve has attracted much interest. In the present paper, we propose a new method to construct a surface interpolating a given curve as the asymptotic curve of it. Also, we analyze the conditions when the resulting surface is a ruled surface. Furthermore, we prove that there exists no developable surface possessing a given curve as an asymptotic curve except plane. Finally, we illustrate this method by presenting some examples.