Frontiers in Physics (Sep 2022)
A unified method for constructing developable surface pencils interpolating characteristic curves
Abstract
Developable surface is a simple and common surface in surface modeling. Geodesic, line of curvature, asymptotic curve, and D-type curve are important characteristic curves on the surfaces. This study gives a unified method for constructing developable surface pencils interpolating these four kinds of characteristic curves. Given a regular space curve R(r), we derive a new condition that a surface pencil P (r, t) interpolating R(r) is developable. The result shows that the condition completely depends on a univalent function λ and an angle θ. By choosing different λ and θ, we can not only control the shape of P (r, t), but also make R(r) become any kind of characteristic curve on P (r, t). Furthermore, we take natural and conjugate curve pairs as those characteristic curves to construct developable surface pairs. Finally, an example of a slant helix shows that the proposed unified method is more general than other methods, and has good interactivity and convenience.
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