GeoUni: A Unified Model for Generating Geometry Diagrams, Problems and Problem Solutions

TL;DR

GeoUni introduces a unified geometry expert model that jointly handles diagram generation, problem solving, and problem creation in both English and Chinese. It combines a geometry-focused diagram tokenizer (Geo-MAGVIT), multi-task instruction tuning with Diagram Formalization Unified Prompting, and a reasoning-enhancement branch using LoRA and GRPO, yielding strong performance across reasoning, text-to-diagram generation, and novel problem creation. Key contributions include the Geo-MAGVIT diagram tokenizer with topo-structural and text reconstruction losses, a unified prompting scheme with dedicated task tokens, and a Geo-Reasoning-Adapter that boosts geometric reasoning without harming diagram generation. Extensive experiments on Formalgeo7K and SynthGeo228K demonstrate state-of-the-art reasoning in English and competitive results in Chinese, as well as superior diagram fidelity measured by GSMS/GPMS and BLEU-based structural fidelity. The work has practical impact by enabling end-to-end geometry tutoring, personalized problem generation, and high-fidelity diagram creation within a single model.

Abstract

We propose GeoUni, the first unified geometry expert model capable of generating problem solutions and diagrams within a single framework in a way that enables the creation of unique and individualized geometry problems. Traditionally, solving geometry problems and generating diagrams have been treated as separate tasks in machine learning, with no models successfully integrating both to support problem creation. However, we believe that mastery in geometry requires frictionless integration of all of these skills, from solving problems to visualizing geometric relationships, and finally, crafting tailored problems. Our extensive experiments demonstrate that GeoUni, with only 1.5B parameters, achieves performance comparable to larger models such as DeepSeek-R1 with 671B parameters in geometric reasoning tasks. GeoUni also excels in generating precise geometric diagrams, surpassing both text-to-image models and unified models, including the GPT-4o image generation. Most importantly, GeoUni is the only model capable of successfully generating textual problems with matching diagrams based on specific knowledge points, thus offering a wider range of capabilities that extend beyond current models.

Figures (20)

• Figure 1: Overview of GeoUni
• Figure 2: Overview of Geo-MAGVIT
• Figure 3: Diagram Formalization Unified Prompting
• Figure 4: Text-to-Diagram
• Figure 5: Problem Creation