FormalAnalyticGeo: A Neural-Symbolic Based Framework for Multimodal Analytic Geometry Problem Generation

2026-07-14

AI Summary

This work addresses the challenges of data scarcity and the difficulty of simultaneously satisfying layout constraints and geometric precision in multimodal analytic geometry problems. We propose the first neuro-symbolic framework that bridges free-form text and precise signed distance field (SDF)-based diagram generation through a formal intermediate language, Coordinate Description Language (CDL). A closed-loop pipeline orchestrated by four large language models enables fully automated problem generation, formalization, measurement, and verification—without requiring human annotations—significantly enhancing the accuracy and scalability of geometric representations. Using this approach, we construct AnalyticGeo7K, a dataset comprising over 7,000 samples, achieving a median relative error of only 0.70% and ensuring that 82.3% of answers exhibit errors within 5%.

Abstract

Math reasoning has achieved significant progress with the rapid advancement of Multimodal Large Language Models (MLLMs), however analytic geometry remains largely underexplored, primarily due to the scarcity of annotated samples. Existing diagram generation approaches struggle with analytic geometry: template methods cannot handle constraint-driven layouts, and generative models lack the geometric precision to render annotated conic curves correctly. We present FormalAnalyticGeo, a scalable framework for fully automatic generation of multimodal analytic geometry problems. Leveraging the rigor of formal languages, we design the framework around CDL (Condition Description Language), a formal intermediate representation that bridges free-form problem text with precise diagram rendering via a Signed Distance Field (SDF) engine. The framework employs four specialized LLM components in sequence: a Generator that produces diverse analytic geometry problems, a Formalizer that converts each problem into CDL for SDF-based rendering, a Measurer that extracts ground-truth answers through vision-based measurement on the rendered diagrams, and a Quality Verifier that checks outputs at three stages. Structured feedback from the Quality Verifier drives automatic retry, forming a closed loop that eliminates any need for human annotation. Applying FormalAnalyticGeo at scale yields AnalyticGeo7K, a dataset of over 7K verified multimodal problems, each with aligned text, diagram, formal annotation, and ground truth.Experiments show that the generated problems achieve a median ground-truth relative error of 0.70\%, with 82.3\% of answers falling within 5\% of the exact symbolic solution. Our framework and dataset will be publicly released.

Problem

• analytic geometry
• multimodal problem generation
• diagram generation
• annotated conic curves
• constraint-driven layouts

Innovation

• Neural-Symbolic Framework
• Analytic Geometry
• Formal Language
• Signed Distance Field
• Multimodal Problem Generation