GeoFocus: Global-to-Local Geometry Framework

Updated 4 July 2026

GeoFocus is a training framework for geometry problem-solving that employs a global-to-local approach, combining global topology reconstruction with focused local feature detection.
It uses the VertexLang module to represent diagram topology compactly, reducing training overhead by 20% compared to traditional code-based methods.
The Critical Local Perceptor enhances accuracy by 4.7% on benchmarks by reliably detecting theory-critical local structures such as midpoints, angles, and parallel lines.

GeoFocus is a training framework for multimodal geometry problem-solving that targets a specific failure mode of large multimodal models: they may recognize the rough overall diagram or rely on text, yet still miss the small local visual facts that trigger the correct theorem. It is organized as a global-to-local perception pipeline with two modules: the VertexLang Topology Percepter, which teaches global topology understanding through a compact formal representation, and the Critical Local Perceptor, which teaches attention to theory-relevant local structures such as angles, parallel lines, midpoint relations, and comparative distances. On Geo3K, GeoQA, and FormalGeo7K, GeoFocus reports a 4.7% accuracy improvement over leading specialized models; it also reports a 61% increase in critical local feature coverage relative to previous methods and a 20% reduction in global perception training time through VertexLang-based topology reconstruction (Deng et al., 9 Feb 2026).

1. Problem formulation and scope

GeoFocus is framed around the claim that geometry problem-solving requires two distinct perception skills that standard large multimodal model training does not align well: global topology understanding of the whole figure and fine-grained local relation perception of small theory-bearing structures. In the paper’s formulation, geometry diagrams do not primarily encode meaning through object categories, but through structural relations such as incidence, connectivity, equality, comparison, and configuration. A model may recognize that a diagram contains a triangle or a circle, yet still fail on whether a point is a midpoint, whether two lines are parallel, whether a segment is an angle bisector, or whether three points are collinear (Deng et al., 9 Feb 2026).

The framework is therefore designed as a sequential and asymmetric global-to-local curriculum. Global perception comes first through topology reconstruction; local perception is then layered on top through focused supervision on critical geometric structures. The authors explicitly report that the training order matters: global-first, then local performs better than the reverse. This supports the paper’s central intuition that one first understands the whole figure and only then inspects the decisive local substructures (Deng et al., 9 Feb 2026).

A common misunderstanding is to read “GeoFocus” as a geographic or geospatial system. In this work, the “Geo” denotes geometry problem-solving, not geolocation or GIS. The relevant spatial units are diagram entities and their mathematical relations rather than geographic coordinates. This distinguishes GeoFocus from other region-focused multimodal systems that operate on natural images, remote sensing, or visual search tasks (Deng et al., 9 Feb 2026).

2. Global-to-local architecture

The overall framework is a two-stage perception pipeline. First, the VertexLang Topology Percepter teaches the model to reconstruct the global topology of geometric figures using a compact formal language. Second, the Critical Local Perceptor teaches the model to answer focused local questions about theory-relevant structures generated from synthetic geometry metadata. The final model takes the original geometry image and question and answers directly; GeoFocus adds no extra inference-time modules or overhead (Deng et al., 9 Feb 2026).

This design is explicitly training-time rather than inference-time. GeoFocus is not a test-time crop-search method, nor does it insert an external theorem prover or symbolic parser into the reasoning chain during evaluation. Instead, it uses auxiliary training tasks to internalize both topology and local-structure perception. The larger methodological claim is that geometry failures are often perception failures before they are reasoning failures: if the model does not reliably perceive the right global scaffold and the right local cues, later symbolic or textual reasoning remains brittle (Deng et al., 9 Feb 2026).

The two modules are intended to be complementary rather than redundant. The topology component emphasizes the whole figure—vertices, edges, circles, and adjacency—while the local component emphasizes the micro-structures where theorems attach. In the ablations, these gains are additive. For a general Qwen2.5-VL-3B model, the baseline total score is 74.2, rising to 76.5 with topology only, 77.1 with local only, and 79.6 with both. For Qwen2.5-VL-7B, the corresponding totals are 119.4, 122.4, 121.9, and 123.5. The same pattern also appears for GRPO-based geometry-specific LMMs, where GeoFocus improves 164.3 → 170.1 at 3B and 185.3 → 190.9 at 7B (Deng et al., 9 Feb 2026).

A plausible implication is that GeoFocus is best understood as a perception-enhancement curriculum for multimodal geometry, not as a standalone symbolic solver. The model still answers directly at inference time, but it has been trained to see the figure in a more geometry-specific way (Deng et al., 9 Feb 2026).

3. Critical Local Perceptor

The Critical Local Perceptor is the local-structure supervision mechanism. Its purpose is to teach the model to attend to small visual patterns that carry direct theoretical meaning, such as midpoint cues, bisectors, perpendicular feet, collinearity, or angle and length comparisons. The paper states that it collected 5,000 middle-school geometry QA pairs and distilled 13 categories of critical local structures, divided into Basic Measurement and Relational Reasoning (Deng et al., 9 Feb 2026).

For numerical properties, the paper defines measurable properties $K$ such as angles, lengths, and areas. The quantification template pool is

$W_Q(K_i) = DT_{K_i} \mid K_i \in [\text{Angle}, \text{Length}, \text{Area}],$

and comparison templates are

$W_C(K_i, K_j) = \begin{cases} DT_{K_i > K_j}, & \text{if } K_i \text{ is greater}, \ DT_{K_i < K_j}, & \text{if } K_i \text{ is smaller}, \ DT_{K_i = K_j}, & \text{if } K_i \text{ is equal}. \end{cases}$

These yield the five concrete local template categories in the Basic Measurement group: Angle Value, Angle Compare, Length Value, Length Compare, and Area Compare (Deng et al., 9 Feb 2026).

For relations, the paper defines line-line and point-line template families. With $\mathcal{E}$ the set of geometric lines,

$W_L(\mathcal{E}) = \begin{cases} DT_{\text{Shape}(\mathcal{E}_i, \dots, \mathcal{E}_n)}, & \text{if } \mathcal{E}_i, \dots, \mathcal{E}_n \text{ satisfy shape properties,} \ DT_{\text{Bisector}(\mathcal{E}_i, \mathcal{E}_j, \mathcal{E}_k)}, & \text{if } \mathcal{E}_i \text{ bisects the angle between } \mathcal{E}_j \text{ and } \mathcal{E}_k, \ DT_{\text{Parallel}(\mathcal{E}_i, \mathcal{E}_j)}, & \text{if } \mathcal{E}_i \parallel \mathcal{E}_j, \ DT_{\text{Perpendicular}(\mathcal{E}_i, \mathcal{E}_j)}, & \text{if } \mathcal{E}_i \perp \mathcal{E}_j. \end{cases}$

With $V$ the set of points,

$W_P(V_i, \mathcal{E}_j) = \begin{cases} DT_{\text{Midpoint}(V_i, \mathcal{E}_j)}, & \text{if } V_i \text{ divides } \mathcal{E}_j \text{ into equal segments,} \ DT_{\text{On-Circle}(V_i, \mathcal{E}_j)}, & \text{if } V_i \text{ lies on the circumference of } \mathcal{E}_j, \ DT_{\text{Perp.-Foot}(V_i, \mathcal{E}_j)}, & \text{if } V_i \text{ is the perp. foot of } \mathcal{E}_j, \ DT_{\text{Col.}(V_i, V_j, V_k)}, & \text{if } V_i, V_j, V_k \text{ lie on the same line.} \end{cases}$

These instantiate the remaining eight template categories: Shape Check, Bisector Check, Perp. Check, Parallel Check, Midpoint Check, On Circle, Perp. Foot Check, and Collinearity (Deng et al., 9 Feb 2026).

The local questions are generated automatically from synthetic geometry metadata through two transformation operators. For numerical properties,

$T_P(K) = H_P\big(W_Q(K) \cup W_C(K_i, K_j)\big),$

and for relations,

$T_R(\mathcal{E}, V) = H_R\big(W_L(\mathcal{E}_i, \mathcal{E}_j) \cup W_P(V_i, \mathcal{E}_j)\big).$

A concrete example given in the paper is that from the description “Point F is the midpoint of line segment AC,” the system matches the Point-Line category, finds the keyword “midpoint,” and instantiates the midpoint template (Deng et al., 9 Feb 2026).

Each local perception sample contains an image, a question, a correct answer, and an incorrect answer. Training uses Direct Preference Optimization (DPO) rather than plain supervised fine-tuning. The stated objective is

$\mathcal{L}_{\text{DPO}} = -\mathbb{E}_{\mathbf{Q}, \mathbf{I}, \mathbf{R}^+}[\log P(\mathbf{R}^+ | \mathbf{Q, I})] + \mathbb{E}_{\mathbf{Q}, \mathbf{I}, \mathbf{R}^-}[\log P(\mathbf{R}^- | \mathbf{Q, I})].$

The paper’s interpretation is that DPO avoids over-constraining exact token sequences and instead teaches the model to prefer correct local interpretations over incorrect ones. Empirically, on the CogAlign-data perception benchmark at equal 65k scale, the local perceptor reaches 72.5 for 3B and 79.4 for 7B, compared with 64.7 and 75.9 for CogAlign. In the DPO-versus-supervised comparison, total score rises from 75.9 → 77.1 at 3B and 114.6 → 121.9 at 7B (Deng et al., 9 Feb 2026).

The paper’s 61% coverage claim is based on category coverage over a 17-type universe of local task types. The reported coverage values are 0.76 for the Critical Local Perceptor, 0.47 for CogAlign, 0.18 for GeoPep, and 0.12 for MAVIS. The relative gain over CogAlign is approximately 61.7%, which the paper reports as a 61% increase in critical local feature coverage (Deng et al., 9 Feb 2026).

4. VertexLang and global topology perception

If the Critical Local Perceptor addresses theorem-bearing micro-structures, VertexLang addresses the global topology of the diagram. It is introduced as a compact topology formal language intended to replace bulkier code-based encodings of geometry figures. The paper argues that code-based reconstruction preserves topology better than pixel reconstruction, but remains verbose and includes drawing-tool syntax unrelated to the actual geometric content. In the cited comparison, code-based reconstruction requires about 2.5k characters per image, whereas VertexLang represents the same topology in about 0.3k characters on average (Deng et al., 9 Feb 2026).

VertexLang consists of three dictionaries. The circle-radius dictionary is

$W_Q(K_i) = DT_{K_i} \mid K_i \in [\text{Angle}, \text{Length}, \text{Area}],$ 0

where $W_Q(K_i) = DT_{K_i} \mid K_i \in [\text{Angle}, \text{Length}, \text{Area}],$ 1 denotes a circle center and $W_Q(K_i) = DT_{K_i} \mid K_i \in [\text{Angle}, \text{Length}, \text{Area}],$ 2 its radius. The vertex-coordinate dictionary is

$W_Q(K_i) = DT_{K_i} \mid K_i \in [\text{Angle}, \text{Length}, \text{Area}],$ 3

where $W_Q(K_i) = DT_{K_i} \mid K_i \in [\text{Angle}, \text{Length}, \text{Area}],$ 4 is a vertex label and $W_Q(K_i) = DT_{K_i} \mid K_i \in [\text{Angle}, \text{Length}, \text{Area}],$ 5 are normalized coordinates. The connectivity dictionary is

$W_Q(K_i) = DT_{K_i} \mid K_i \in [\text{Angle}, \text{Length}, \text{Area}],$ 6

where $W_Q(K_i) = DT_{K_i} \mid K_i \in [\text{Angle}, \text{Length}, \text{Area}],$ 7 indicates straight-line connectivity. The figure is then rendered by

$W_Q(K_i) = DT_{K_i} \mid K_i \in [\text{Angle}, \text{Length}, \text{Area}],$ 8

Lines and polygons are not stored as separate primitives; they are implicit in connectivity and vertex placement. This keeps the language deliberately minimal and topology-centric (Deng et al., 9 Feb 2026).

The synthetic generation pipeline called the VertexLang Reconstructor builds both the diagram and its corresponding VertexLang text. It initializes $W_Q(K_i) = DT_{K_i} \mid K_i \in [\text{Angle}, \text{Length}, \text{Area}],$ 9, $W_C(K_i, K_j) = \begin{cases} DT_{K_i > K_j}, & \text{if } K_i \text{ is greater}, \ DT_{K_i < K_j}, & \text{if } K_i \text{ is smaller}, \ DT_{K_i = K_j}, & \text{if } K_i \text{ is equal}. \end{cases}$ 0, and $W_C(K_i, K_j) = \begin{cases} DT_{K_i > K_j}, & \text{if } K_i \text{ is greater}, \ DT_{K_i < K_j}, & \text{if } K_i \text{ is smaller}, \ DT_{K_i = K_j}, & \text{if } K_i \text{ is equal}. \end{cases}$ 1 as empty; optionally adds circle entries; repeatedly generates normalized coordinates and unique vertex labels; determines adjacent vertices according to connectivity constraints; then renders the image through $W_C(K_i, K_j) = \begin{cases} DT_{K_i > K_j}, & \text{if } K_i \text{ is greater}, \ DT_{K_i < K_j}, & \text{if } K_i \text{ is smaller}, \ DT_{K_i = K_j}, & \text{if } K_i \text{ is equal}. \end{cases}$ 2. The result is paired image–topology data for reconstruction training (Deng et al., 9 Feb 2026).

The topology training stage does not rely on plain supervised fine-tuning alone. The paper argues that supervised fine-tuning can inject task-specific knowledge but hurts transfer from reconstruction to reasoning, while pure reinforcement learning is unstable early because the model is too weak at image reconstruction to obtain useful reward signals. GeoFocus therefore introduces DynamicGT-RL, which probabilistically mixes model-generated outputs with occasional ground-truth guidance. The policy objective is

with

The modified advantage term is

The reconstruction reward combines coordinate quality and connectivity quality. Vertex scores depend on Euclidean distance $W_C(K_i, K_j) = \begin{cases} DT_{K_i > K_j}, & \text{if } K_i \text{ is greater}, \ DT_{K_i < K_j}, & \text{if } K_i \text{ is smaller}, \ DT_{K_i = K_j}, & \text{if } K_i \text{ is equal}. \end{cases}$ 6 to the ground truth:

with average coordinate reward

and total reward

In the DynamicGT-RL ablation, total score for the 3B model changes from 74.2 baseline to 71.9 with supervised learning, 74.4 with RL, and 76.5 with DynamicGT-RL; for 7B the corresponding values are 119.4, 116.8, 120.9, and 122.4. The best GT ratio is reported as 50% (Deng et al., 9 Feb 2026).

The efficiency claims around VertexLang are tied to this compression of topology targets. On 2,500 images, using Gemini-2.0-Flash to generate code baselines, the authors report an 88% reduction in character count relative to code-based methods and state that VertexLang yields a 20% reduction in training time compared to CodeLang under identical settings. On in-domain reconstruction, Qwen2.5-VL-3B improves Connect. Acc. 41.9 → 62.5 and Coord. Acc. 8.9 → 30.9; Qwen2.5-VL-7B improves 48.9 → 68.9 and 38.7 → 51.6. On out-of-domain reconstruction over FormalGeo7K with FGeo-Parser, VertexLang also outperforms CodeLang in both BLEU-4 and Cons. CdlAcc (Deng et al., 9 Feb 2026).

5. Training protocol, datasets, and empirical performance

GeoFocus is evaluated on three primary geometry benchmarks: GeoQA, Geo3K, and FormalGeo7K. GeoQA contains 754 geometry questions from Chinese secondary school exams; Geo3K contains 601 textbook-derived geometry questions; FormalGeo7K contributes 1,050 test problems from manually curated online geometry resources. Perception evaluation uses CogAlign-data, and out-of-domain evaluation uses We-Math, MathVerse, MathVision, MathVista, HallusionBench, and ChartQA (Deng et al., 9 Feb 2026).

The reported backbone models are Qwen2.5-VL-3B, Qwen2.5-VL-7B, and Gemma3-4B. The learning rate is $\mathcal{E}$ 0. The topology stage runs for 15 epochs using EasyR1 defaults, and the local-perception stage runs for 1 epoch using standard LlamaFactory setup (Deng et al., 9 Feb 2026).

On the main geometry benchmarks, GeoFocus-3B reports 50.4 on Geo3K, 64.3 on GeoQA, and 55.4 on FormalGeo7K. GeoFocus-7B reports 55.3, 71.9, and 63.5. Compared with GeoUni at 50.0 / 66.7 / 59.8, the per-dataset gains are 5.3, 5.2, and 3.7, averaging approximately 4.73, which the paper states as a 4.7% average improvement over leading specialized models (Deng et al., 9 Feb 2026).

On the CogAlign-data perception benchmark, the Critical Local Perceptor alone yields large gains over both baseline and CogAlign. For 3B, baseline average is 63.3, CogAlign reaches 64.7, and the local perceptor reaches 72.5. For 7B, the corresponding values are 75.0, 75.9, and 79.4. On real-world geometry problem-solving datasets, the combination of CogAlign and the local perceptor further improves totals, though the local perceptor by itself already transfers better than CogAlign-only training (Deng et al., 9 Feb 2026).

The curriculum order is explicitly important. For the 3B model, Local $\mathcal{E}$ 1 Topo yields 75.2, whereas Topo $\mathcal{E}$ 2 Local yields 79.6. For the 7B model, the corresponding totals are 122.2 and 123.5. This is one of the clearest experimental validations of the framework’s global-to-local thesis (Deng et al., 9 Feb 2026).

On out-of-domain benchmarks, GeoFocus also improves over baseline and GRPO. For the 3B model, total score across HallusionBench, WeMath, MathVerse, MathVision, MathVista, and ChartQA rises from 297.4 baseline and 327.0 GRPO to 331.6 with GeoFocus. For 7B, the corresponding totals are 346.5, 364.4, and 370.0. The paper states that GeoFocus achieves an average 4.8% improvement over baseline across these out-of-domain sets and shows stronger robustness in MathVerse under Vision Intensive, Vision Dominant, and Vision Only settings (Deng et al., 9 Feb 2026).

6. Interpretation, limitations, and position within broader focus-based multimodal research

The paper’s strongest conceptual claim is that geometry problem-solving is constrained by perception quality as much as by downstream reasoning. GeoFocus accordingly separates perception into two hierarchically related skills: a compact global graph-like abstraction of the full figure and a theory-sensitive local recognition of decisive substructures. The qualitative analyses support this distinction: the baseline model tends to misclassify triangle types, miss circle-angle categories, fail to recognize collinearity, or recall a theorem verbally while grounding it to the wrong local configuration, whereas GeoFocus is described as better at jointly understanding global topology and critical local structures (Deng et al., 9 Feb 2026).

One limitation explicitly discussed concerns point-circle positional conflicts in VertexLang rendering. If coordinates and topology constraints conflict—for example, if a point is intended to lie on a circle but the explicit coordinates do not satisfy the circle equation—VertexLang resolves the conflict by prioritizing vertex coordinates over positional constraints. The paper notes that this can introduce slight inaccuracies for points intended to lie exactly on the circle. The inclusion of the On Circle task in the Critical Local Perceptor is presented as a partial compensation for this limitation (Deng et al., 9 Feb 2026).

A second boundary of the method is that GeoFocus remains a training-time enhancement without explicit inference-time symbolic theorem proving. This suggests that it improves what the model sees, but does not itself constitute a formal proof engine. A plausible implication is that the framework is complementary to symbolic or theorem-based solvers rather than a substitute for them (Deng et al., 9 Feb 2026).

Within the broader research landscape, GeoFocus belongs to a family of methods that improve multimodal models by restructuring how they attend to spatially decisive evidence, but it differs from several adjacent approaches. DyFo is a training-free inference-time dynamic focus controller for large multimodal models in fine-grained visual understanding, using Monte Carlo Tree Search and an external visual expert rather than a geometry-specific training curriculum (Li et al., 21 Apr 2025). FOCUS for fine-grained VQA uses internal MLLM representations to guide crop selection at inference time and is likewise training-free, whereas GeoFocus operates at training time and adds no explicit crop search at test time (Zhong et al., 26 Jun 2025). Focusable Monocular Depth Estimation reframes depth prediction as prompt-conditioned region-focused geometry, emphasizing foreground, boundary, and global scene geometry, but it addresses depth estimation rather than symbolic geometry problem-solving (Du et al., 12 May 2026). FocusTune uses geometry-aware focus-guided sampling for scene coordinate regression in visual localization, concentrating training on regions tied to reconstructed 3D structure rather than on theorem-bearing diagram structures (Nguyen et al., 2023).

These comparisons clarify what GeoFocus is and is not. It is not an image-cropping system, not a geospatial model, and not a generic fine-grained visual search mechanism. It is specifically a global-to-local perception curriculum for multimodal geometry, centered on the idea that geometry reasoning improves when global topology and local theory-bearing structures are trained separately, compactly, and in sequence (Deng et al., 9 Feb 2026).