Origami and Kirigami, the famous Japanese art forms of paper folding and cutting, have inspired the design of novel materials & structures utilizing their geometry. In this article, we explore the geometry of the lesser known popup art, which uses the facilities of both origami and kirigami via appropriately positioned folds and cuts. The simplest popup-unit resembles a four-bar mechanism, whose cut-fold pattern can be arranged on a sheet of paper to produce different shapes upon deployment. Each unit has three parameters associated with the length and height of the cut, as well as the width of the fold. We define the mean and Gaussian curvature of the popup structure via the discrete surface connecting the fold vertices and develop a geometric description of the structure. Using these definitions, we arrive at a design pipeline that identifies the cut-fold pattern required to create popup structure of prescribed shape which we test in experiments. By introducing splay to the rectangular unit-cell, a single cut-fold pattern is shown to take multiple shapes along the trajectory of deployment, making possible transitions from negative to positive curvature surfaces in a single structure. We demonstrate application directions for these structures in drag-reduction, packaging, and architectural facades.

翻译：折纸与剪纸作为日本著名的纸艺形式，其几何原理启发了新型材料与结构的设计。本文探讨了相对鲜为人知的弹跳式立体纸艺的几何学，该艺术形式通过合理布置折痕与切口，综合运用了折纸与剪纸的技法。最基本的弹跳单元类似于四连杆机构，其切-折图案可排列在纸面上，在展开时形成不同形状。每个单元具有三个参数，分别对应切口的长度与高度，以及折痕的宽度。我们通过连接折痕顶点的离散曲面定义了弹跳结构的平均曲率与高斯曲率，并建立了该结构的几何描述。基于这些定义，我们构建了一套设计流程，可识别生成指定形状弹跳结构所需的切-折图案，并通过实验进行了验证。通过对矩形单元引入展开变形，单个切-折图案可沿展开轨迹呈现多种形态，使得单一结构能够实现从负曲率曲面到正曲率曲面的过渡。我们展示了此类结构在减阻、包装和建筑立面等领域的应用方向。