This paper provides computational tools for the modeling and design of quad mesh mechanisms, which are meshes allowing continuous flexions under the assumption of rigid faces and hinges in the edges. We combine methods and results from different areas, namely differential geometry of surfaces, rigidity and flexibility of bar and joint frameworks, algebraic geometry, and optimization. The basic idea to achieve a time-continuous flexion is time-discretization justified by an algebraic degree argument. We are able to prove computationally feasible bounds on the number of required time instances we need to incorporate in our optimization. For optimization to succeed, an informed initialization is crucial. We present two computational pipelines to achieve that: one based on remeshing isometric surface pairs, another one based on iterative refinement. A third manner of initialization proved very effective: We interactively design meshes which are close to a narrow known class of flexible meshes, but not contained in it. Having enjoyed sufficiently many degrees of freedom during design, we afterwards optimize towards flexibility.

中文翻译：

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 四网格机构


本文提供了用于四边网格机构建模和设计的计算工具，四边网格机构是在假设边上的刚性面和铰链的情况下允许连续弯曲的网格。我们结合了来自不同领域的方法和结果，即表面的微分几何、杆和关节框架的刚度和柔韧性、代数几何和优化。实现时间连续弯曲的基本思想是时间离散化，由代数度论证证明。我们能够证明我们需要在优化中纳入所需时间实例数量的计算上可行的边界。要使优化成功，明智的初始化至关重要。我们提出了两个计算管道来实现这一目标：一个基于重新划分等距表面对的网格，另一个基于迭代细化。第三种初始化方式被证明非常有效：我们交互式地设计网格，这些网格接近已知的一类狭窄的柔性网格，但不包含在其中。在设计过程中享受了足够多的自由度后，我们随后会针对灵活性进行优化。