AUTHOR=Wang Jun , Chen Miaochao , Wang Dongyin 

TITLE=A unified method for constructing developable surface pencils interpolating characteristic curves

JOURNAL=Frontiers in Physics

VOLUME=10

YEAR=2022

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2022.979094

DOI=10.3389/fphy.2022.979094

ISSN=2296-424X

ABSTRACT=<p>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 <bold><italic>R</italic></bold>(<italic>r</italic>), we derive a new condition that a surface pencil <bold><italic>P</italic></bold> (<italic>r</italic>, <italic>t</italic>) interpolating <bold><italic>R</italic></bold>(<italic>r</italic>) is developable. The result shows that the condition completely depends on a univalent function <italic>λ</italic> and an angle <italic>θ</italic>. By choosing different <italic>λ</italic> and <italic>θ</italic>, we can not only control the shape of <bold><italic>P</italic></bold> (<italic>r</italic>, <italic>t</italic>), but also make <bold><italic>R</italic></bold>(<italic>r</italic>) become any kind of characteristic curve on <bold><italic>P</italic></bold> (<italic>r</italic>, <italic>t</italic>). 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.</p>