Flexibility and Higher-order Rigidity of (certain) Geometric Constraint Systems

Geometric constraint systems are used to model a wide range of real-world objects whose movements are limited by natural, geometric constraints. These constraints are typically expressed by polynomial equations and, among other things, describe angles, incidence, distance, tangency, volume, and coplanarity. As a result, the class of admissible geometric objects ranges from polytopes and sphere packings to simplicial complexes and hydraulic platforms.

Deciding whether a geometric constraint system is flexible – that is, whether it admits continuous deformations beyond ambient Euclidean isometries – or rigid is computationally challenging. For that reason, one often relies on weaker but computationally tractable sufficient conditions for rigidity. In regular points, the dimension of the realization space is determined by the kernel of the constraint system’s Jacobian. Comparing this dimension with the dimension of the Lie group of Euclidean isometries yields a simple sufficient condition for rigidity called infinitesimal rigidity. However, this criterion is not necessary.

In this talk, we develop the theory of higher-order rigidity for geometric constraint systems and discuss corresponding algorithmic approaches grounded in real algebraic geometry. By constructing explicit examples of second-order rigid systems – such as the regular dodecahedron – we demonstrate that these conditions are not only theoretically relevant but have concrete applications to the real world. Moreover, we construct a robust numerical algorithm for approximating continuous motions of flexible geometric constraint systems based on the metric projection, which we compute using a combination of Riemannian optimization and homotopy continuation.