We present a novel approach to generate developable strips along a space curve. The key idea of the new method is to use the rotation angle between the Frenet frame of the input space curve, and its Darboux frame of the curve on the resulting developable strip as a free design parameter, thereby revolving the strip around the tangential axis of the input space curve. This angle is not restricted to be constant but it can be any differentiable function defined on the curve, thereby creating a large design space of developable strips that share a common directrix curve. The range of possibilities for choosing the rotation angle is diverse, encompassing constant angles, linearly varying angles, sinusoidal patterns, and even solutions derived from initial value problems involving ordinary differential equations. This enables the potential of the proposed method to be used for a wide range of practical applications, spanning fields such as architectural design, industrial design, and papercraft modeling. In our computational and physical examples, we demonstrate the flexibility of the method by constructing, among others, toroidal and helical windmill blades for papercraft models, curved foldings, triply orthogonal structures, and developable strips featuring a log-aesthetic directrix curve.

中文翻译：

---

您所需要的只是旋转：构建可展开的条带


我们提出了一种沿空间曲线生成可显影条带的新方法。新方法的关键思想是将输入空间曲线的 Frenet 框架与生成的可展开条带上曲线的 Darboux 框架之间的旋转角度作为自由设计参数，从而围绕输入空间曲线的切向轴旋转条带。这个角度不限于常数，但它可以是曲线上定义的任何可微函数，从而创建一个由共享公共 directrix 曲线的可展开条带组成的大设计空间。选择旋转角度的可能性范围多种多样，包括常角、线性变化角、正弦模式，甚至是从涉及常微分方程的初值问题中得出的解。这使得所提出的方法的潜力可以用于广泛的实际应用，跨越建筑设计、工业设计和纸艺建模等领域。在我们的计算和物理示例中，我们通过构建用于纸艺模型的环形和螺旋风车叶片、弯曲的折叠、三重正交结构以及具有对数美学准线曲线的可展开条带等来展示该方法的灵活性。