GeoSDF: Plane Geometry Diagram Synthesis via Signed Distance Field

TMLR Paper7552 Authors

Abstract: Plane Geometry Diagram Synthesis has been a crucial task in computer graphics, with applications ranging from educational tools to AI-driven mathematical reasoning. Traditionally, we rely on manual tools (e.g., Matplotlib and GeoGebra) to generate precise diagrams, but this usually requires huge, complicated calculations. Recently, researchers start to work on model-based methods (e.g., Stable Diffusion and GPT5) to automatically generate diagrams, saving operational cost but typically suffering from limited realism and insufficient accuracy. In this paper, we propose a novel framework, GeoSDF, to automatically generate diagrams efficiently and accurately with Signed Distance Field (SDF). Specifically, we first represent geometric elements (e.g., points, segments, and circles) in SDF, then construct a series of constraint functions to represent geometric relationships. Next, we optimize those constructed constraint functions to get an optimized field of both elements and constraints. Finally, by rendering the optimized field, we can obtain the synthesized diagram. In our GeoSDF, we define a symbolic language to represent geometric elements and constraints, and our synthesized geometry diagrams can be self-verified in SDF, ensuring both mathematical accuracy and visual plausibility. Through extensive experiments, we demonstrate GeoSDF’s ability to synthesize high-quality geometry diagrams across various levels of complexity, including standard high-school problems and IMO-level (International Mathematical Olympiad) challenges.We achieve an impressive 88.67\% synthesis accuracy as evaluated by human experts on the IMO problem set. Furthermore, leveraging the self-verification property, we attain a geometry problem-solving accuracy exceeding 95\%, outperforming the current state-of-the-art (approximately 75\%) by a significant margin of ~20\%. These results highlight the advantages of GeoSDF, paving the way for more sophisticated, accurate, and flexible geometric diagram generation across a wide range of applications. The accompanying code, datasets, and all synthesized outputs will be released to benefit the research community upon acceptance of the paper.

Submission Type: Regular submission (no more than 12 pages of main content)

Assigned Action Editor: ~ERIC_EATON1

Submission Number: 7552