Construction of a surface pencil with a common special surface curve

TL;DR

This paper introduces D-type curves on surfaces, characterizes their properties, and constructs surface pencils with these curves, including special cases like helices and planar curves, supported by examples and visualizations.

Contribution

It defines D-type curves based on Darboux vectors, establishes conditions for these curves on surface pencils, and explores special cases with explicit examples.

Findings

D-type curves are characterized by a constant inner product between surface normal and Darboux vector.

Necessary and sufficient conditions for D-type curves on surface pencils are derived.

Examples and visualizations demonstrate the theoretical results.

Abstract

In this study, we introduce a new type of surface curves called D-type curve. This curve is defined by the property that the unit Darboux vector W0 of a space curve r(s) and unit surface normal n along the curve r(s) satisfy the condition <n,W0>=constant. We point out that a D-type curve is a geodesic curve or an asymptotic curve in some special cases. Then, by using the Frenet vectors and parametric representation of a surface pencil as a linear combination of the Frenet vectors, we investigate necessary and sufficient condition for a curve to be a D-type curve on a surface pencil. Moreover, we introduce some corollaries by considering D-type curve as a helix, a Salkowski curve or a planar curve. Finally, we give some examples for obtained results and plot the surfaces by using Mapple.