展平单顶点折纸：非扩展情况
Flattening single-vertex origami: the non-expansive case
摘要：
单顶点折纸是一张纸，其内部或边界上的折叠顶点发出称为折痕的直线光线。单顶点折纸展平问题询问是否总是可以将有折痕的纸从与度量兼容的任何配置重新配置为平坦的、非重叠的位置，以使得纸不会被撕裂、拉伸，并且对于刚性折纸，除了沿着给定折痕之外，不会在任何地方弯曲。Streinu和Whiteley展示了如何减少问题的木匠的规则问题的球形多边形。利用球面扩张运动，他们解决了开< π和闭<= 2π球面多边形的情况。在这里，我们解决了总长度在[π，2π）之间的开多边形的情况，这需要非扩张运动。我们的运动规划算法工作在一个有限数量的离散步骤，我们给出精确的界限取决于链接的数量和角度赤字。
英文摘要：
A single-vertex origami is a piece of paper with straight-line rays called creases emanating from a fold vertex placed in its interior or on its boundary. The Single-Vertex Origami Flattening problem asks whether it is always possible to reconfigure the creased paper from any configuration compatible with the metric, to a flat, non-overlapping position, in such a way that the paper is not torn, stretched and, for rigid origami, not bent anywhere except along the given creases. Streinu and Whiteley showed how to reduce the problem to the carpenter's rule problem for spherical polygons. Using spherical expansive motions, they solved the cases of open < πand closed <= 2πspherical polygons. Here, we solve the case of open polygons with total length between [π, 2π), which requires non-expansive motions. Our motion planning algorithm works in a finite number of discrete steps, for which we give precise bounds depending on both the number of links and the angle deficit.
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