Printability assessment in additive manufacturing is typically conducted at the geometry level before printing to determine whether a computer-aided design (CAD) model or stereolithography (STL) file can be successfully fabricated. Task suitability, in contrast, is usually evaluated after printing to determine whether the fabricated part satisfies the requirements of its intended use. As a result, for non-expert users to print functional parts, unsuitable material or process choices may only be identified after fabrication, leading to repeated printing, material waste, and user frustration. To address this challenge, this paper leverages the reasoning and language-understanding capabilities of large language models (LLMs), while grounding the reasoning with geometry evidence and structured material/printer knowledge to generate reliable pre-print recommendations. Given a stereolithography (STL) model and a natural-language task description, the framework generates a structured recommendation covering printability, material choice, process parameters, design guidance, risks, and explanations. We evaluate the framework on focused STL benchmark scenarios with novice-style task descriptions. The proposed method achieves 75.0% printability over 96 physical validation trials, with 88.9% task suitability among successfully printed samples. It also improves Gemini 2.5 Flash-Lite material-selection accuracy from 37.5% under pure LLM to 90.0%. Expert evaluation further shows improved report quality, while post-print feedback improves recommendations on selected problematic cases. These results suggest that user task intent, geometry evidence, and structured material knowledge are all important for reliable task-driven printability assistance.
Although visual reasoning is crucial for solving complex geometry tasks, existing vision-language models rely heavily on text-only reasoning. Some recent methods introduce intermediate visual states to facilitate reasoning, but they are often hindered by inaccurate geometric representations and low rendering fidelity, ultimately leading to unreliable outputs. To address these limitations, we propose MetaReason, a framework for multimodal reasoning in plane geometry that leverages structured meta-information to enable accurate auxiliary-line construction. The framework first parses geometric images into meta-information, performs controllable edits with predefined tools to synthesize high-fidelity visual states, and then conducts reasoning based on these augmented views. To support this framework, we construct TutorGeo, a comprehensive dataset containing 17k image-to-meta conversion samples, 60k text-only reasoning traces, and 60k interleaved multimodal reasoning traces. Using this dataset, we combine supervised fine-tuning and reinforcement learning to develop robust multimodal reasoning capabilities. We also introduce ExamGeo, a benchmark derived from real-world examination problems that enables systematic evaluation across varying difficulty levels. Experimental results demonstrate that MetaReason significantly outperforms existing open-source models and achieves competitive performance against proprietary models.
Foundation models such as GPT and Claude now solve olympiad-level mathematics with remarkable proficiency, so much so that geometry problem solving has become a standard proxy for their mathematical reasoning. Yet solving a geometry problem and drawing the figure it depends on are not the same skill: progress often hinges on a faithful diagram with the right auxiliary constructions and incidences, and it is unclear that a model which reasons its way to the answer can also produce one. A growing collection of benchmarks, including MathVista, and MathVerse, measures whether models reach the correct answer, but to our knowledge, none isolate the distinct ability to construct the diagram itself, leaving this capability unmeasured. We introduce an open-source benchmark that targets this gap: 954 self-contained olympiad geometry problems, with a 297-problem hard subset, each paired with its solution and a human-authored, high-fidelity diagram in renderable Asymptote code, together with a suite of text-, code-, image-, VLM-, and constraint-based metrics for what we term diagrammatic reasoning. Evaluating current foundation models reveals a pronounced gap between solving and drawing: their diagrams are markedly less faithful, with an average compile success rate of only 36.14\%. Strong mathematical reasoning, we find, does not imply the ability to construct accurate geometric diagrams. Our benchmark and dataset can be accessed at https://huggingface.co/datasets/max98765/hard_geometry_problems_with_diagrams.
Euclidean geometry is a compelling testbed for AI reasoning, as it demands the combination of intuitive diagram understanding, axiomatic deduction, and algebraic computation. Yet, existing approaches typically address only a subset of these abilities or struggle with competition-level problems. We introduce \textit{Euclid-Omni}, a unified neuro-symbolic framework that couples a formal geometry system with Large Language Models (LLMs) and Vision-Language Models (VLMs) to tackle both calculation- and proving-style problems, in formal and natural languages, up to Olympiad-level difficulty. At its core, we develop \textit{Euclidea}, a versatile symbolic geometry solver that automatically generates reasoning steps through deductive inference and algebraic computation. Building on this, we develop a data-generation pipeline that synthesizes symbolic problems and solutions, renders diagrams, and translates them into natural language, producing large-scale, diverse datasets for training LLMs and VLMs across a wide range of reasoning settings. Experiments show that VLMs trained on our synthetic data achieve superior performance on calculation tasks, and that LLMs combined with \textit{Euclidea} are competitive with state-of-the-art systems on Olympiad-level proving problems, despite using orders of magnitude less compute and training data. Code and scripts are publicly available at https://github.com/20171130/Euclid-Omni
Geometry problem generation is useful for AI-assisted education and multimodal mathematical reasoning, but reliable synthesis remains difficult because the problem statement, diagram, constraints, and solution should be mutually consistent. Existing methods often trade off controllability and reliability: seed-based rewriting is flexible but weakly verifiable, whereas diagram-first construction improves validity but is less suited to arbitrary user-specified constraints. We introduce VeriGeo, a controllable geometry generation framework grounded in executable reasoning traces. Given user constraints such as target concepts and difficulty, an Author agent generates a problem and diagram, and a Solver agent produces a proof-aligned solution. Both agents use a shared action sequence that connects natural language, diagrams, geometric constraints, and proof steps into a verifiable representation. A three-stage pipeline checks numerical consistency, analytical realizability, and global consistency, using verification-guided reflection to repair recoverable failures and reject unrecoverable ones. Across five LLM backbones, raw generations frequently fail these checks, while VeriGeo repairs a substantial fraction of the invalid attempts. Supervised fine-tuning on 8.7k examples generated by VeriGeo achieves the best reported GeoQA performance among end-to-end multimodal LLM-based solvers, and obtains strong results on PGPS9K and MathVista-GPS, demonstrating the effectiveness of verified synthetic data for improving multimodal geometry reasoning.
Vision-language models solve geometry problems with rising accuracy, yet their intermediate states remain latent and unverifiable: a relation expressed in textual reasoning or drawing code carries no guarantee that a constraint-satisfying configuration realizes it. We observe that existing externalization methods based on rendered pixels or one-shot scripts fail to provide exact, per-action geometric guarantees. Enforcing geometric relations by algebraic definition closes this gap: the workspace becomes a constraint-checked evolving canvas. We present Draw2Think, a framework that recasts geometric reasoning from latent spatial inference into agentic interaction with the GeoGebra constraint engine. In a Propose-Draw-Verify loop, Draw2Think externalizes hypotheses onto an executable canvas, measures exact geometric quantities, and feeds structured observations back to the model, so subsequent reasoning proceeds from checked canvas state grounded by the shared workspace. This externalization makes two properties separately auditable: model-level Construction Fidelity (whether the canvas realizes the intended configuration) and engine-level Measurement Faithfulness (exact values and relations from canvas constraints). Across construction, outcome, and rendering evaluations, Draw2Think builds canvases that pass 95.9% predicate-level and 84.0% strict problem-level construction checks on GeoGoal, improves outcome accuracy by up to 4.1%/16.4% on planar/solid benchmarks, and attains 68.2%/90.5% strict/relaxed rendering scores on GenExam-math. Project page is available at https://draw2think.github.io/