Natural-Language to Geometry Diagrams: A Constraint-Based Pipeline for Precise Visual Reasoning

Abstract

We present GenGX, a system that generates precise geometric diagrams from natural-language descriptions by combining large language model (LLM) interpretation with symbolic constraint solving. User prompts are translated by an LLM autoformalizer into a structured intermediate representation (IR) encoding geometric entities, relationships, and construction semantics. The IR is passed to CoreGX, a constraint solver that synthesizes a deterministic construction sequence — operating above classical Euclidean primitives — that provably realizes the specified figure without numerical optimization. The system handles classical constructions, conics, curves, and transformations, and resolves both discrete and continuous ambiguity through explicit IR specifiers and a numeric clarity optimizer that selects a visually canonical representative from any underdetermined family of valid diagrams. This hybrid architecture avoids the spatial inaccuracies endemic to purely generative text-to-image approaches, produces reproducible results, and allows users to inspect and correct the IR directly.