CrossProjection: Geometric Grounding Beyond Viewpoint Change in Architectural Drawings

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/

Spatial reasoning is fundamental to robotics, autonomy, and embodied AI, yet modern vision-language models (VLMs) remain unreliable on metric distance queries. A common assumption is that consistent predictions across viewpoints reflect geometric grounding. We test this assumption and find the opposite: leading VLMs often produce view-invariant and consistent answers even when those answers are incorrect, indicating weak coupling between predictions and viewpoint-specific visual evidence. We introduce \textbf{ViewDiag}, a controlled multi-view evaluation protocol built from Hypersim, ScanNet, and KITTI360, comprising 176 object-pair tracks across 80 scenes with 2--10 views per track. The protocol evaluates models along three axes: metric accuracy, distributional concentration, and internal collapse, the last of which is assessed using a latent feature probe. Across diverse models, we observe a consistent pattern of high prediction stability paired with substantial error, clustering in a regime characterized by strong consistency but low accuracy. \noindent These results challenge the common use of cross-view consistency as a proxy for geometric understanding. Instead, we show that stable predictions may reflect prior-driven collapse rather than evidence-sensitive reasoning. ViewDiag provides a controlled benchmark and diagnostic framework for evaluating whether spatial VLMs are not only accurate, but also meaningfully coupled to visual evidence.

Consistent cross-view understanding under extreme viewpoint changes is essential for spatial intelligence, as it enables models to recognize the same scene across extreme viewpoint gaps. Cross-view localization naturally provides a promising pathway toward this ability, as it requires a model to align ground-view imagery with geo-referenced satellite-view imagery despite drastic appearance changes to estimate camera poses. Recent visual foundation models have made this long-standing localization problem increasingly feasible by providing rich 2D representations for cross-view matching. However, we argue that cross-view localization should not be viewed merely as 2D matching or pose estimation. In this work, we revisit cross-view localization as more than pose estimation and investigate how it can help the model develop consistent cross-view understanding under extreme viewpoint changes, including stable semantics, reliable structure, and transferable geometry. We identify three key limitations of existing methods that prevent them from achieving this. They usually lack explicit 3D grounding, rely on strict point-wise matching that can weaken semantic consistency, and learn from an absolute objective that provides limited guidance for geometric reasoning. To address these limitations, we propose CROSS, a unified cross-view localization framework built upon 3D-grounded alignment, structure-aware matching, and hypothesis ranking. This formulation makes structure learning an intrinsic requirement, encourages semantic representations to remain stable, and enables the model to acquire transferable geometry. Extensive experiments on the KITTI and VIGOR datasets show that CROSS achieves state-of-the-art performance in cross-view localization. More importantly, CROSS effectively learns stable semantics, reliable structure, and transferable geometry across extremely different viewpoints.