G eo DANO : Geometric VLM with Domain Agnostic Vision Encoder (2025)

Abstract: We introduce GeoDANO, a geometric vision-language model (VLM) with a domain-agnostic vision encoder, for solving plane geometry problems. Although VLMs have been employed for solving geometry problems, their ability to recognize geometric features remains insufficiently analyzed. To address this gap, we propose a benchmark that evaluates the recognition of visual geometric features, including primitives such as dots and lines, and relations such as orthogonality. Our preliminary study shows that vision encoders often used in general-purpose VLMs, e.g., OpenCLIP, fail to detect these features and struggle to generalize across domains. To overcome the limitation, we develop GeoCLIP, a CLIP-based model trained on synthetic geometric diagram–caption pairs. Benchmark results show that GeoCLIP outperforms existing vision encoders in recognizing geometric features. We then propose our VLM, GeoDANO, which augments GeoCLIP with a domain adaptation strategy for unseen diagram styles. GeoDANO outperforms specialized methods for plane geometry problems and GPT-4o on MathVerse. The implementation is available at https://github.com/ml-postech/GeoDANO.

## introduction large language models (llms) have achieved remarkable success in automated math problem solving, particularly through code-generation capabilities integrated with proof assistants @xcite @xcite @xcite . although llms excel at generating solution steps and correct answers in algebra and calculus @xcite , their unimodal nature limits performance in plane geometry, where the solution depends on both diagram and text @xcite . specialized vision-language models (vlms) have accordingly been developed for plane geometry problem solving (pgps) @xcite figure 1 : examples of diagram-caption pairs and their solution steps written in formal languages from the geoqa and pgps9k datasets. the problem description highlights the visual geometric premises and numerical variables in green and red, respectively. a significant difference in the style of the diagram and formal language can be observed. 2022; @xcite @xcite @xcite . yet, whether these models genuinely leverage diagrams or rely almost exclusively on textual features remains unclear. this ambiguity arises because existing pgps datasets typically embed sufficient geometric details within problem statements, potentially making the vision encoder unnecessary @xcite . fig. 1 illustrates example questions from geoqa and pgps9k, where solutions can be derived without referencing the diagrams. we propose a new benchmark created via a synthetic data engine, which systematically evaluates the ability of vlm vision encoders to recognize geometric premises. our empirical findings reveal that previously suggested self-supervised learning (ssl) approaches, e.g., vector quantized variational auto-encoder (vq-vae) @xcite and masked auto-encoder (mae) @xcite , and widely adopted encoders, e.g., openclip @xcite and di-nov2 @xcite , struggle to detect geometric features such as perpendicularity and degrees. to this end, we propose geoclip, a model pre-trained on a large corpus of synthetic diagram-caption pairs. by varying diagram styles (e.g., color, font size, resolution, line width), geoclip learns robust geometric representations and outperforms prior ssl-based methods on our benchmark. building on geoclip, we introduce a few-shot domain adaptation technique that efficiently transfers the recognition ability to realworld diagrams. we finally propose geodano by combining this domain-adapted geoclip with an llm, forming a domain-agnostic vlm for solving pgps tasks. in our experiments on mathverse @xcite , which encompasses diverse plane geometry tasks and diagram styles, geodano consistently outperforms both task-specific pgps models and generalist vlms. ablation studies confirm the effectiveness of our domain adaptation strategy, showing improvements in optical character recognition (ocr)-based tasks and robust diagram embeddings across different styles. we summarize the contributions as follows: we propose a novel benchmark for systematically assessing how well vision encoders recognize geometric premises in plane geometry diagrams ( §3); we introduce geoclip, a vision encoder capable of accurately detecting visual geometric premises ( §4.1), and a few-shot domain adaptation technique that efficiently transfers this capability across different diagram styles ( §4.2); we show that our vlm, named geodano, incorporating domain-adapted geoclip, surpasses existing specialized pgps vlms and generalist vlms on the mathverse benchmark ( §5.2) and effectively interprets diverse diagram styles ( §5.3). ## related work in this section, we summarize the known pgps benchmarks and models. detailed comparison with previous work is reported in appendix a. ## benchmark for geometric premises in this section, we first develop a benchmark for evaluating a vision encoder's performance in recognizing geometric features from a diagram. we then report the performance of well-known vision encoders on this benchmark. ## geoclip: enhanced vision encoder in this section, we first propose geoclip, a new vision encoder designed to recognize geometric premises from diverse styles of diagrams. to transfer the recognition to real-world pgps benchmarks, we then propose a domain adaptation technique for geoclip that leverages a small set of diagram-caption pairs from target domains. ## experiments in this section, we evaluate the pgps performance of our vlm equipped with the domain-adapted geoclip on mathverse @xcite . we compare its performance against established pgps baselines. we also present ablation studies highlighting our vlm's strong visual feature recognition and resilience to domain shifts, both of which are facilitated by the adapted vision encoder. ## conclusion in this work, we propose a domain-agnostic pgps method, geodano, by implementing a synthetic data engine and proposing a contrastive learning framework with domain adaptation. we demonstrate the effectiveness of geodano in visual feature perception at both vlm and vision encoder levels by evaluating on the mathverse and through a newly proposed geometric feature recognition benchmark for vision encoders. eventually, the reasoning ability in plane geometry problems is enhanced with the improved perceptual capabilities.