---
title: "FormalAnalyticGeo: A Neural-Symbolic Based Framework for Multimodal Analytic Geometry Problem Generation"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2607.12982v1"
permalink: "https://feed.astralor.com/items/itm_1ad0dd798ad1a584e2e7946a"
published: "2026-07-14T17:24:57.000Z"
score: 77
channel: "ai"
tags: ["AnalyticGeo7K","Condition Description Language","FormalAnalyticGeo","Signed Distance Field","MLLM"]
---

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

## AI commentary

> This paper presents an elegant neural-symbolic approach that cleverly combines formal languages and LLMs to address data scarcity in analytic geometry. It is especially valuable for researchers in educational AI, math reasoning, and multimodal data generation; the closed-loop verification mechanism is worth exploring in depth.

## At a glance

This paper presents FormalAnalyticGeo, a framework leveraging formal languages and Condition Description Language (CDL) as an intermediate representation, combined with a Signed Distance Field (SDF) engine, to fully automatically generate multimodal analytic geometry problems. It employs four specialized LLM components (Generator, Formalizer, Measurer, Quality Verifier) in a closed loop with automatic retry, eliminating the need for human annotation. Experiments yield AnalyticGeo7K, a dataset of over 7K verified problems, with a median ground-truth relative error of 0.70% and 82.3% of answers within 5% of exact symbolic solutions. This approach addresses the scarcity of annotated data that hindered MLLMs in analytic geometry.

## Key points

- Proposes CDL formal intermediate representation bridging text and precise diagram rendering.
- Designs a closed loop of four LLM components for fully automated generation and verification.
- Constructs AnalyticGeo7K dataset with over 7K verified problems and high accuracy.
- Eliminates human annotation entirely, enabling scalable generation of multimodal analytic geometry data.

## Full text

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.

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