GeoLoom: High-quality Geometric Diagram Generation from Textual Input

TL;DR

GeoLoom is a novel framework that converts natural language geometric descriptions into high-quality, accurate diagrams using formal language translation and Monte Carlo optimization, outperforming existing methods.

Contribution

It introduces GeoLoom, combining autoformalization and coordinate solving, along with a new dataset GeoNF and a constraint-based evaluation metric for improved diagram generation.

Findings

Outperforms state-of-the-art in structural fidelity

Uses formal language and Monte Carlo optimization for accuracy

Provides a new dataset and evaluation metric for geometric diagram generation

Abstract

High-quality geometric diagram generation presents both a challenge and an opportunity: it demands strict spatial accuracy while offering well-defined constraints to guide generation. Inspired by recent advances in geometry problem solving that employ formal languages and symbolic solvers for enhanced correctness and interpretability, we propose GeoLoom, a novel framework for text-to-diagram generation in geometric domains. GeoLoom comprises two core components: an autoformalization module that translates natural language into a specifically designed generation-oriented formal language GeoLingua, and a coordinate solver that maps formal constraints to precise coordinates using the efficient Monte Carlo optimization. To support this framework, we introduce GeoNF, a dataset aligning natural language geometric descriptions with formal GeoLingua descriptions. We further propose a constraint-based evaluation metric that quantifies structural deviation, offering mathematically grounded supervision for iterative refinement. Empirical results demonstrate that GeoLoom significantly outperforms state-of-the-art baselines in structural fidelity, providing a principled foundation for interpretable and scalable diagram generation.