# Geometry

AlphaGeometry is a cutting-edge AI system for solving geometry problems that surpasses existing technologies. It combines the predictive capability of neural language models with the reasoning power of a rule-driven inference engine to tackle complex geometrical challenges. Utilizing a neuro-symbolic approach, AlphaGeometry consists of a neural language model and a symbolic reasoning engine that work together to discover proofs for intricate geometrical theorems. By generating one billion random geometrical object graphs and deriving all relationships from them, AlphaGeometry ultimately yields 100 million unique training samples, 9 million of which include additional constructions. AlphaGeometry's language model can provide insightful suggestions when faced with geometry problems from international mathematics competitions. This system marks the world's first AI model capable of reaching the bronze medal level at the International Math Olympiad.

NeROIC is a novel method for obtaining object representations from online image collections. It can capture high-quality geometric and material properties of any object in photographs with varying cameras, lighting, and backgrounds. It can be used for novel view synthesis, relighting, and harmonious background synthesis, among other object-centric rendering applications. Building upon the multi-stage approach of neural radiance fields, we first infer surface geometry and refine rough initial camera parameters. Simultaneously, we leverage a coarse foreground object mask to enhance training efficiency and geometric quality. We also introduce a robust normal estimation technique that mitigates the impact of geometric noise while preserving crucial details. Finally, we extract surface material properties and environmental lighting, represented using spherical harmonics, and handle transient elements such as sharp shadows. The combination of these components forms a highly modular and efficient framework for object acquisition. Extensive evaluations and comparisons demonstrate the superiority of our method in capturing high-quality geometry and appearance properties for rendering applications.