Matthias Adrian-Himmelmann

arXiv2026

Deformations and second-order rigidity of polytopes

We study deformations of polytopes that preserve edge lengths and face coplanarities. This gives rise to a notion of rigidity, for which we develop a second-order theory together with an effective algorithm for testing second-order rigidity symbolically. The strength of these new tools is demonstrated on several polytope classes, including the previously intractable regular dodecahedron, which we find to be rigid. We also find that a single tested polytope evades our techniques and will therefore provide a simple test case for future developments of tools of even higher order. This paper also contains a study of so-called edge-length perturbations. We point out connections between rigidity and the ability to realize polytopes with slightly perturbed edge lengths. To explore these connections in practice, we dedicate a section to another case study of the regular dodecahedron. We investigate the local behavior of its realization space with a view towards edge-length perturbations, singularities and generic global rigidity.

math.CO↗

arXiv2026

Approximating Continuous Motions of Geometric Constraint Systems

The realization space of geometric constraint systems is given by the vanishing locus of polynomials corresponding to natural geometric constraints. Such geometric constraint systems arise in many real-world scenarios such as structural engineering and soft matter physics. When a geometric constraint system is flexible, it admits continuous deformations. The ability to explicitly compute such continuous motions is essential for analyzing the constraint system's quasistatic or elastic properties. However, this task is computationally challenging, even for comparatively simple geometric constraint systems, making numerical strategies attractive. In this article, we present a general numerical framework for approximating continuous motions of geometric constraint systems given by quadratic polynomials. Our approach combines Riemannian optimization with numerical algebraic geometry to construct continuous motions via the metric projection onto the constraint set. By using homotopy continuation, we ensure that the computed motions correspond to genuine solutions of the constraint system and avoid numerical artifacts such as path-jumping. To handle singularities and over-determined systems, we introduce theoretical enhancements including randomization, adaptive step size control and a second-order analysis. These methods are implemented in the Julia package DeformationPaths.jl, which supports a broad class of geometric constraint systems and demonstrates its robust and effective performance across a wide range of test cases.

math.MG↗