Geoparsing: Diagram Parsing for Plane and Solid Geometry with a Unified Formal Language

TL;DR

This paper introduces a unified formal language for plane and solid geometry, along with a large dataset and training paradigm, to improve geometric reasoning in multimodal large language models.

Contribution

It presents a new formal language, a large-scale dataset, and a training method that enhances geometric understanding and reasoning in multimodal models.

Findings

Achieved state-of-the-art parsing performance.

Parsed formal descriptions significantly improve geometry reasoning tasks.

Constructed GDP-29K dataset with 29,000 geometry samples.

Abstract

Multimodal Large Language Models (MLLMs) have achieved remarkable progress but continue to struggle with geometric reasoning, primarily due to the perception bottleneck regarding fine-grained visual elements. While formal languages have aided plane geometry understanding, solid geometry which requires spatial understanding remains largely unexplored. In this paper, we address this challenge by designing a unified formal language that integrates plane and solid geometry, comprehensively covering geometric structures and semantic relations. We construct GDP-29K, a large-scale dataset comprising 20k plane and 9k solid geometry samples collected from diverse real-world sources, each paired with its ground-truth formal description. To ensure syntactic correctness and geometric consistency, we propose a training paradigm that combines Supervised Fine-Tuning with Reinforcement Learning via Verifiable Rewards. Experiments show that our approach achieves state-of-the-art parsing performance. Furthermore, we demonstrate that our parsed formal descriptions serve as a critical cognitive scaffold, significantly boosting MLLMs' capabilities for downstream geometry reasoning tasks. Our data and code are available at Geoparsing.