Origami has shown the potential to approximate three-dimensional curved surfaces by folding through designed crease patterns on flat materials. The Miura-ori tessellation is a widely used pattern in engineering and tiles the plane when partially folded. Based on constrained optimization, this paper presents the construction of generalized Miura-ori patterns that can approximate three-dimensional parametric surfaces of varying curvatures while preserving the inherent properties of the standard Miura-ori, including developability, flat-foldability and rigid-foldability. An initial configuration is constructed by tiling the target surface with triangulated Miura-like unit cells and used as the initial guess for the optimization. For approximation of a single target surface, a portion of the vertexes on the one side is attached to the target surface; for fitting of two target surfaces, a portion of vertexes on the other side is also attached to the second target surface. The parametric coordinates are adopted as the unknown variables for the vertexes on the target surfaces whilst the Cartesian coordinates are the unknowns for the other vertexes. The constructed generalized Miura-ori tessellations can be rigidly folded from the flat state to the target state with a single degree of freedom.

中文翻译：

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为曲面构建刚性可折叠广义 Miura-ori 镶嵌

折纸展示了通过在平面材料上设计的折痕图案折叠来近似三维曲面的潜力。Miura-ori 镶嵌是工程中广泛使用的图案，在部分折叠时可以平铺平面。基于约束优化，本文提出了广义 Miura-ori 模式的构建，该模式可以近似不同曲率的三维参数曲面，同时保留标准 Miura-ori 的固有特性，包括可展开性、平面折叠性和刚性折叠性。初始配置是通过用三角化的 Miura 样单位单元平铺目标表面来构建的，并用作优化的初始猜测。对于单个目标表面的逼近，一侧顶点的一部分附着在目标表面上；为了拟合两个目标面，另一侧的部分顶点也附着在第二个目标面上。参数坐标被用作目标表面上顶点的未知变量，而笛卡尔坐标是其他顶点的未知变量。构建的广义 Miura-ori 镶嵌可以从平坦状态刚性折叠到具有单个自由度的目标状态。