Geometric data abounds, but our algorithms for geometry processing are failing. Whether from medical imagery, free-form architecture, self-driving cars, or 3D-printed parts, geometric data is often messy, riddled with “defects” that cause algorithms to crash or behave unpredictably. The traditional philosophy assumes geometry is given with 100% certainty and that algorithms can use whatever discretization is most convenient. As a result, geometric pipelines are leaky patchworks requiring esoteric training and involving many different people.

Instead, we adapt fundamental mathematics to work directly on messy geometric data. As an archetypical example, I will discuss how to generalize the classic formula for determining the inside from the outside of a curve to messy representations of a 3D surface. This work helps keep the geometry processing pipeline flowing, as validated on our large-scale geometry benchmarks.

Our long term vision is to replace the current leaky geometry processing pipeline with a robust workflow where processing operates directly on real geometric data found “in the wild”. To do this, we need to rethink how algorithms should gracefully degrade when confronted with imprecision and uncertainty. Our most recent work on differentiable rendering and geometry-based adversarial attacks on image classification demonstrates the potential power of this philosophy.