FormalAnalyticGeo: A Neural-Symbolic Based Framewor...

FormalAnalyticGeo: A Neural-Symbolic Based Framewor... | AI Research

Key Takeaways

FormalAnalyticGeo is a new framework designed to solve the scarcity of high-quality, annotated data for analytic geometry—a field that combines algebraic equ...
We present FormalAnalyticGeo, a scalable framework for fully automatic generation of multimodal analytic geometry problems.
Structured feedback from the Quality Verifier drives automatic retry, forming a closed loop that eliminates any need for human annotation.
Our framework and dataset will be publicly released.
FormalAnalyticGeo is a new framework designed to solve the scarcity of high-quality, annotated data for analytic geometry—a field that combines algebraic equations with visual diagrams.

Paper AbstractExpand

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 this http URL 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.

FormalAnalyticGeo is a new framework designed to solve the scarcity of high-quality, annotated data for analytic geometry—a field that combines algebraic equations with visual diagrams. While Multimodal Large Language Models (MLLMs) have improved at general math, they often struggle with the precise geometric requirements of conic sections (like ellipses and hyperbolas). This framework automates the creation of complex, multimodal geometry problems, ensuring that the text and the visual diagrams are perfectly aligned without requiring any human intervention.

A New Language for Geometry

The core of the framework is the Condition Description Language (CDL). Existing formal languages for geometry were built for simple shapes like triangles and circles, making them unable to handle the coordinate systems and complex curves found in analytic geometry. CDL acts as a bridge: it translates natural language problems into a structured format that a computer can understand. This allows the system to perform automatic consistency checks, ensuring that every geometric object mentioned in a problem is properly defined and renderable.

From Code to Precise Diagrams

To turn these formal descriptions into accurate images, the framework uses a specialized rendering engine based on Signed Distance Fields (SDFs). Traditional plotting tools often fail to handle constraint-driven layouts, such as placing a point on a curve at a specific distance from a focus. The SDF engine treats geometric shapes as mathematical functions, allowing the system to use gradient descent to solve for unknown positions. This ensures that the resulting diagrams are geometrically exact, with a known mapping between coordinates and pixels, which is essential for accurate visual measurement.

A Closed-Loop Verification System

FormalAnalyticGeo operates as a multi-stage pipeline involving a Generator, a Formalizer, an SDF engine, and a Measurer. To maintain high standards, a Quality Verifier monitors the process at three different stages. If any component produces an error—such as a diagram that doesn't match the text or a measurement that is logically inconsistent—the system provides structured feedback and triggers an automatic retry. This closed-loop process eliminates the need for human annotators, as the system only keeps problems that pass all verification gates.

Results and Impact

The framework successfully produced AnalyticGeo7K, a dataset containing over 7,000 verified multimodal problems. Testing shows that the generated problems are highly accurate, with a median ground-truth relative error of just 0.70%. Furthermore, 82.3% of the answers derived from these diagrams fall within 5% of the exact symbolic solution. By providing this large-scale, verified dataset, the researchers aim to help advance the ability of AI models to reason about complex spatial and mathematical relationships.