FormalAnalyticGeo: A Neural-Symbolic Based Framework for Multimodal Analytic Geometry Problem Generation
2026-07-14 • Artificial Intelligence
Artificial IntelligenceMultiagent SystemsSymbolic Computation
AI summaryⓘ
The authors developed a new system called FormalAnalyticGeo to automatically create and verify math problems involving analytic geometry, which has been hard to explore due to lack of good example problems. Their method uses a special language (CDL) to turn text problems into precise geometry diagrams using a technique called Signed Distance Fields. Four AI components work together to generate problems, convert them into diagrams, measure answers accurately, and check quality without needing human help. Using this, they created a large dataset of over 7,000 verified problems with accurate answers. Their experiments show these problems are quite close to exact symbolic solutions.
Analytic GeometryMultimodal Large Language ModelsCondition Description LanguageSigned Distance FieldFormal LanguageDiagram RenderingAI VerificationDataset Generation
Authors
Ruoran Xu, Wending Gao, Qiufeng Wang
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.