Tools for Reliable Geometry

Reliability in geometry processing and physical simulation is a critical requirement for scientific research, manufacturing, and safety-critical engineering applications. Typical implementations of geometric queries rely on floating-point arithmetic and discrete sampling of quantities, which can lead to undetected geometric degeneracies, non-physical states, and solver failures. These problems are even more severe when working with high-order curved geometry, which is often preferred to piecewise linear representations thanks to its better fidelity. This thesis presents a comprehensive stack of tools designed to ensure provably conservative results for geometric computations without sacrificing computational efficiency. The first contribution is an algorithm for continuously checking and enforcing the geometrical validity of high-order finite elements as they deform over time. This method employs a robust branch-and-bound approach to approximate the earliest time of element inversion. When integrated into an elastodynamic simulation framework, the method effectively prevents non-physical states and ensures the stability of the solver under extreme deformations. The second contribution, MiSo (Minimize+Solve), is a domain-specific language (DSL) and compiler that automates the generation of robust, interval-based solvers for a wide class of geometric queries, including collision detection and primitive intersections. MiSo abstracts the mathematical definition of a query from its numerical implementation, using a hybrid approach that combines Natural Interval Extensions with Bézier-based inclusion functions to achieve high performance. The generated solvers provide strict guarantees of correctness and are competitive with, or faster than, hand-optimized implementations for several geometric queries. The final contribution is TIGHT, a C++ library for fast and correctly rounded interval arithmetic. TIGHT provides the foundational numerical primitives required for reliable geometric computation, ensuring bit-by-bit reproducibility and supporting transcendental functions with minimal interval width. Together, these tools facilitate the development of simulation pipelines where geometric reliability is a hard constraint rather than a heuristic compromise.