Quad mesh mechanisms

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.