GeoSDF: Plane Geometry Diagram Synthesis via Signed Distance Field

2025-06-16

AI Summary

Automated synthesis of planar geometric diagrams faces dual challenges: labor-intensive manual modeling and insufficient accuracy/realism in learning-based approaches. This paper introduces the first end-to-end framework based on symbolic signed distance fields (SDFs), which encodes geometric primitives and their relational constraints as differentiable symbolic functions, enabling high-fidelity diagram generation via gradient-based SDF optimization. We pioneer a unified symbolic SDF modeling paradigm coupled with a self-verification mechanism that jointly ensures mathematical correctness and visual plausibility. The method supports fully automatic synthesis of diagrams ranging from middle-school to International Mathematical Olympiad (IMO)-level complexity. On a standard benchmark, it achieves 95.2% geometric accuracy—surpassing the state-of-the-art by over 20 percentage points—thereby significantly enhancing the quality and reliability of diagram generation in AI-powered mathematical reasoning.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Visual Reasoning & Symbolic Representations
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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 computer tools (e.g., Matplotlib and GeoGebra) to manually generate precise diagrams, but it usually requires huge, complicated calculations cost. Recently, researchers start to work on learning-based methods (e.g., Stable Diffusion and GPT4) to automatically generate diagrams, saving operational cost but usually 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 in the SDF, then construct a series of constraint functions to represent geometric relationships, next we optimize such 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 easily represent geometric elements and those constraints, and our synthesized geometry diagrams can be self-verified in the SDF, ensuring both mathematical accuracy and visual plausibility. In experiments, our GeoSDF synthesized both normal high-school level and IMO-level geometry diagrams. Through both qualitative and quantitative analysis, we can see that synthesized diagrams are realistic and accurate, and our synthesizing process is simple and efficient. Furthermore, we obtain a very high accuracy of solving geometry problems (over 95% while the current SOTA accuracy is around 75%) by leveraging our self-verification property. All of these demonstrate the advantage of GeoSDF, paving the way for more sophisticated, accurate, and flexible generation of geometric diagrams for a wide array of applications.

Problem

• Automate plane geometry diagram synthesis efficiently
• Improve accuracy and realism in generated diagrams
• Enable self-verification for mathematical correctness

Innovation

• Uses Signed Distance Field for diagram synthesis
• Defines symbolic language for geometric constraints
• Self-verification ensures accuracy and plausibility