Alese Leonardo
Department of Mathematics, Institute of Geometry, TU Graz, Kopernikusgasse 24, 8010 Graz, Austria.
Beitr Algebra Geom. 2022;63(1):19-43. doi: 10.1007/s13366-021-00568-1. Epub 2021 Mar 16.
In this paper we consider developable surfaces which are isometric to planar domains and which are piecewise differentiable, exhibiting folds along curves. The paper revolves around the longstanding problem of existence of the so-called folded annulus with multiple creases, which we partially settle by building upon a deeper understanding of how a curved fold propagates to additional prescribed foldlines. After recalling some crucial properties of developables, we describe the local behaviour of curved folding employing normal curvature and relative torsion as parameters and then compute the very general relation between such geometric descriptors at consecutive folds, obtaining novel formulae enjoying a nice degree of symmetry. We make use of these formulae to prove that any proper fold can be propagated to an arbitrary finite number of rescaled copies of the first foldline and to give reasons why problems involving infinitely many foldlines are harder to solve.
在本文中，我们考虑与平面区域等距且分段可微的可展曲面，这些曲面沿曲线呈现褶皱。本文围绕所谓具有多条折痕的折叠环面的存在这一长期存在的问题展开，我们通过更深入地理解弯曲折痕如何传播到额外规定的折线上，部分解决了该问题。在回顾了可展曲面的一些关键性质之后，我们将法曲率和相对挠率作为参数来描述弯曲折叠的局部行为，然后计算连续折痕处这些几何描述符之间非常一般的关系，得到具有良好对称度的新公式。我们利用这些公式证明任何适当的折痕都可以传播到第一条折线的任意有限数量的缩放副本，并给出涉及无限多条折线的问题更难解决的原因。
