cs.CVApr 13, 2026
SaveGeoparsing: Diagram Parsing for Plane and Solid Geometry with a Unified Formal Language
Organizations: MAIS, Institute of Automation of Chinese Academy of Sciences · School of Artificial Intelligence, University of Chinese Academy of Sciences · Future Living Lab of Alibaba
Abstract
Multimodal Large Language Models (MLLMs) have achieved remarkable progress but continue to struggle with geometric reasoning, primarily due to the perception bottleneck regarding fine-grained visual elements. While formal languages have aided plane geometry understanding, solid geometry which requires spatial understanding remains largely unexplored. In this paper, we address this challenge by designing a unified formal language that integrates plane and solid geometry, comprehensively covering geometric structures and semantic relations. We construct GDP-29K, a large-scale dataset comprising 20k plane and 9k solid geometry samples collected from diverse real-world sources, each paired with its ground-truth formal description. To ensure syntactic correctness and geometric consistency, we propose a training paradigm that combines Supervised Fine-Tuning with Reinforcement Learning via Verifiable Rewards. Experiments show that our approach achieves state-of-the-art parsing performance. Furthermore, we demonstrate that our parsed formal descriptions serve as a critical cognitive scaffold, significantly boosting MLLMs' capabilities for downstream geometry reasoning tasks. Our data and code are available at Geoparsing.
Figures & tables
Figure 1: Hallucinations in geometric parsing by SOTA MLLMs. Gemini-3-Pro and GPT-5.1 struggle to correctly parse slightly complex plane geometry and simple solid geometry. Red text indicates parsing errors.
Figure 2: Overview of the GDP-29K dataset for geometry diagram parsing. The dataset spans plane geometry (printed and handwritten) and solid geometry, with each diagram annotated by formal language that captures primitives and semantic constraints.
| Statistic | Number |
|---|---|
| Dataset Scale & Style | |
| Total Images | 28,882 |
| - Plane Geometry (PG) | 19,965 |
| - Solid Geometry (SG) | 8,917 |
| Style | |
| - Printed | 23,366 |
Table 1: Detailed statistics of GDP-29K.
Table 2: Comparison with existing multimodal math benchmarks. SG: Solid Geometry, PG: Plane Geometry. GQA: Geometry QA, MQA: Math QA. Style: P = P rinted, H = H andwritten. Source: S = S elf-sourced, O =Collected from O pen-source Dataset. PGFL/SGFL: Plane/Solid Geometry Formal Language.
Figure 3: Overview of our geometry diagram parsing framework. We first construct SFT training pairs from GDP-29K and obtain initial parser GDP-SFT via instruction fine-tuning. We then further optimize the parser with reinforcement learning via verifiable rewards that enforce both format correctness and geometric validity. The final model GDP-RL generates unified formal language descriptions for both plane and solid geometry diagrams.
| Model | Points | Lines | Circles | Semantics | Overall | ||||||||
| P | R | F1 | P | R | F1 | P | R | F1 | P | R | F1 | Score | |
| Traditional Methods | |||||||||||||
| InterGPS | 47.3 | 90.0 | 62.2 | 3.6 | 52.8 | 6.7 | 1.0 | 7.7 | 1.8 | 11.4 | 11.8 | 11.6 | 20.6 |
| PGDPNet | 88.8 | 94.1 | 91.4 | 66.8 | 73.4 | 70.1 | 61.6 | 61.2 | 61.4 | 64.1 | 51.5 | 57.1 | 70.6 |
| Open-source MLLMs | |||||||||||||
| LLaVA-OneVision-1.5-7B | 94.6 | 95.2 | 94.3 | 51.9 | 52.1 | 52.0 | 54.4 | 57.1 | 55.7 | 57.5 | 58.4 | 57.9 | 65.0 |
Table 3: Performance comparison on the PGDP-2K test benchmark. Models are categorized by type. P : Precision, R : Recall, F1 : F1-Score. Semantics : Semantic Relations. Overall Score represents the aggregate parsing accuracy. Bold denotes the best performance.
| Model | Points | Lines | Circles | Planes | Solids | Overall | ||||||||
| P | R | F1 | P | R | F1 | P | R | F1 | P | R | F1 | Acc | Score | |
| Open-source MLLMs | ||||||||||||||
| LLaVA-OneVision-1.5-7B | 91.6 | 96.9 | 94.2 | 54.8 | 66.5 | 60.8 | 34.5 | 57.4 | 43.1 | 31.2 | 62.3 | 41.5 | 69.6 | 61.9 |
| Qwen3-VL-4B-Instruct | 99.0 | 98.5 | 98.8 | 48.7 | 69.5 | 57.3 | 44.7 | 47.2 | 46.0 | 35.0 | 73.9 | 47.5 | 70.8 | 64.0 |
| InternVL3.5-8B-Instruct | 98.5 | 94.8 | 96.6 | 57.8 | 50.4 | 53.8 | 23.3 | 86.1 | 36.7 | 62.4 | 72.6 | 67.1 | 69.7 | 64.8 |
| Qwen3-VL-8B-Instruct | 98.9 | 99.1 | 99.0 | 42.3 | 65.4 | 51.4 | 48.9 | 57.6 | 52.9 | 32.0 | 75.1 | 44.8 | 77.5 | 65.1 |
Table 4: Performance comparison on the SGDP-1K test benchmark. We report Precision ( P ), Recall ( R ), and F1-Score ( F1 ) for multi-component primitives, and Accuracy ( Acc ) for the overall solid type. Overall Score represents the aggregate parsing accuracy. Bold denotes the best performance.
| Model | PGDP-2K | SGDP-1K | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Points | Lines | Circles | Semantics | PPR | Points | Lines | Circles | Planes | Solids | PPR | |
| Qwen3-VL-4B | 81.5 | 33.4 | 73.8 | 37.3 | 27.4 | 93.1 | 38.6 | 42.6 | 32.8 | 70.8 | 26.2 |
| Qwen3-VL-8B | 81.5 | 35.7 | 77.2 | 45.7 | 30.4 | 94.1 | 36.1 | 49.5 | 32.3 | 77.5 | 26.4 |
| Qwen3-VL-32B | 83.3 | 54.6 | 80.3 | 50.7 | 44.8 | 94.7 | 51.5 | 55.4 | 31.7 | 83.8 | 28.8 |
| GPT-5.2-1211 | 82.9 | 64.4 | 83.7 | 60.9 | 55.8 | 91.5 | 54.9 | 61.7 | 58.6 | 72.7 | 50.2 |
| Gemini-3-Flash | 91.4 | 87.0 | 91.7 | 69.8 | 63.9 | 97.7 | 80.4 | 76.3 | 71.7 | 88.9 | 64.7 |
Table 5: Sample Accuracy (SA) and Perfect Parsing Rate (PPR) on PGDP-2K and SGDP-1K.
| Model | Plane Geometry | Solid Geometry | |||||
|---|---|---|---|---|---|---|---|
| GeoQA | PGPS9K | Geometry3K | MathVerse | SolidGeo | MathVerse | ||
| Ministral-3-8B | 39.6 | 41.2 | 44.8 | 51.2 | 9.6 | 26.0 | |
| + Ours | 41.5 (+1.9) | 47.3 (+6.1) | 53.3 (+8.5) | 52.4 (+1.2) | 8.8 (-0.6) | 26.8 (+0.8) | |
| Qwen3-VL-8B | 48.9 | 44.9 | 50.1 | 66.8 | 59.0 | 42.0 | |
| + Ours | 48.7 (-0.2) | 53.9 (+9.0) | 60.2 (+10.1) | 68.5 (+1.7) | 62.1 (+3.1) | 44.5 (+2.5) | |
| Qwen2.5-VL-32B | 59.7 | 38.1 | 46.3 | 54.9 | 52.5 | 36.1 | |
Table 6: Downstream reasoning accuracy (%) on Plane and Solid geometry benchmarks. We compare vanilla MLLMs with those augmented by our formal parsing ( + Ours ). (in red ) denotes the absolute improvement. MathVerse results are reported on its plane and solid geometry subsets, respectively.
Figure 4: Effect of Representation Forms on PGPS9K Reasoning Accuracy.
Appendix figures & tables18 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 5: Distribution of 3D structures in the SGDP subset ( ). The dataset covers a wide range of polyhedral forms and rotational solids to facilitate robust spatial perception.
Figure 6: Distribution of semantic predicates across the PGDP subset ( ). Metric constraints and orthogonal relations form the core of the geometric reasoning tasks.
| Category | Formal Syntax (Example) | Geometric Description |
|---|---|---|
| Primitives | point A | Defines a point vertex named A. |
| line A B C | A line segment passing through points A, B, and C. | |
| line k lineson A B C | A line lieson point A B C | |
| plane A B C D | A plane defined by vertices A, B, C, and D. | |
| \odot O lieson A B C | A circle with center O passing through points A, B, and C. | |
| Semantics | AB = 57 | The length of segment is . |
Table 7: Detailed syntax and examples of the formal language in GDP-29K, covering 2D primitives and 3D solid structures.
Figure 7: Prompts for geometric structural analysis. These prompts guide the model to extract the topological skeleton of the diagram.
Figure 8: Prompts for geometric structural analysis. These prompts guide the model to extract the topological skeleton of the diagram.
Figure 9: Prompts for geometric structural analysis. These prompts guide the model to extract the topological skeleton of the diagram.
Figure 10: Prompts for 3D geometric structural analysis. These prompts guide the model to classify the solid type and identify its defining vertices.
Figure 11: Prompts for solid geometric plane extraction. These prompts enforce the Maximal Point Set Principle to ensure all coplanar labeled points are grouped together.
Figure 12: Representative plane geometry samples from the GDP-29K dataset.
Figure 13: Representative plane geometry samples from the GDP-29K dataset.
Figure 14: Representative plane geometry samples from the GDP-29K dataset.
Figure 15: Representative solid geometry samples from the GDP-29K dataset.
Figure 16: Representative solid geometry samples from the GDP-29K dataset.
Figure 17: Representative solid geometry samples from the GDP-29K dataset.
Figure 18: Representative solid geometry samples from the GDP-29K dataset.
Figure 19: Qualitative comparison between Direct Inference and our method (+ Ours) on PGPS9K. Our formal parsing provides precise symbolic grounding that rectifies reasoning errors.
Figure 20: Qualitative comparison between Direct Inference and our method (+ Ours) on PGPS9K. Our formal parsing provides precise symbolic grounding that rectifies reasoning errors.
Figure 21: Qualitative comparison between Direct Inference and our method (+ Ours) on PGPS9K. Our formal parsing provides precise symbolic grounding that rectifies reasoning errors.
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