Curved origami featuring curved tiles can store elastic energy and bias the structure toward a desired shape when subjected to appropriate constraints. Without constraints, however, a curved origami structure generally has infinitely many degrees of freedom. For engineering applications in which a particular folding motion is desired, a great many constraints have to be introduced. One natural strategy to reduce degrees of freedom is to discretize curved origami into piecewise linear origami while closely approximating the desired curved folding motion. Then, as we show, the degrees of freedom are vastly reduced (typically to one DoF in cases where the folding angle at the crease is not trivial, and excluding overall rigid body motions). In this paper we present a Lagrangian approach for constructing curved origami structures which is complementary to our approach in Liu and James (2024). Then, we generalize this approach to allow quite general curved creases. We outline a way to discretize curved crease patterns and present two optimization methods to refine these patterns. Our approach ensures that the deformed piecewise linear origami closely matches the folding angles and crease configurations of the curved origami. Piecewise linear origami structures that approximate curved origami offer promising potential for architectural and robotic design applications. They also give an efficient way to design desired smooth structures by optimizing finite-dimensional piecewise linear structures.

中文翻译：

---

可折叠的分段线性折纸，近似于弯曲的瓷砖折纸


以弯曲瓷砖为特色的弯曲折纸可以储存弹性能，并在受到适当约束时将结构偏向所需的形状。然而，如果没有约束，弯曲的折纸结构通常具有无限多的自由度。对于需要特定折叠运动的工程应用，必须引入大量约束。减少自由度的一种自然策略是将弯曲折纸离散化为分段线性折纸，同时紧密接近所需的弯曲折叠运动。然后，正如我们所展示的，自由度大大减少（在折痕处的折叠角度不重要的情况下，通常减少到 1 DoF，并且不包括整体刚体运动）。在本文中，我们提出了一种构建弯曲折纸结构的拉格朗日方法，该方法与我们在 Liu 和 James （2024） 中的方法相辅相成。然后，我们将这种方法推广到允许相当普遍的弯曲折痕。我们概述了一种离散化弯曲折痕图案的方法，并提出了两种优化方法来优化这些图案。我们的方法确保变形的分段线性折纸与弯曲折纸的折叠角度和折痕配置紧密匹配。近似于弯曲折纸的分段线性折纸结构为建筑和机器人设计应用提供了广阔的潜力。它们还通过优化有限维分段线性结构，提供了一种有效的方法来设计所需的平滑结构。