Generalize cross-ratios in n-dimensional Plane-Based Geometric Algebra

Geometric Algebra (GA) presents challenges to learners due to its highly abstract mathematical structure and complex operational rules, as translating algebraic manipulations into concrete geometric interpretations is a non-intuitive process when developing related code. Currently, some existing GA software packages rely on manually written scripts for code generation and visualization, but their high learning curve hinders widespread adoption. Meanwhile, methods based on Large Language Models (LLMs) often produce logical errors when generating specific GA scripts, such as GAALOPScript, resulting in generally low accuracy. To address these issues, this study proposes GA-VisAgent -- a multi-agent interactive learning application for GA code generation and visualization -- building upon a Geometric algebra large language model (GAGPT). Integrating task planning mechanisms with ReAct reasoning strategies, GA-VisAgent can decompose complex operations into five standardized subtasks, including core operations like geometric products, rotations, and reflections. It supports natural language and mathematical formulas as input to automatically generate executable code, accompanied by interactive visualizations to aid user comprehension. Experimental results show that GA-VisAgent achieved a 90% code generation success rate across 40 typical Conformal GA tasks, representing a 70% improvement over GPT-4o. This application introduces an extensible new paradigm for teaching GA and developing visualization tools for related mathematical concepts. The online service for this project will be available at http://gagis.cn/gacrac.

Architectural drawings violate the usual assumption behind multi-view reasoning: plans and sections are cuts, while elevations are facade projections, so corresponding components change appearance in ways camera motion cannot explain. We introduce CrossProjection, an anchor-grounded diagnostic of whether vision-language models preserve component identity and externalize geometry across heterogeneous architectural views. It evaluates Matching, Registration, and Geometric Grounding through categorical judgments, candidate selection, and free point, line, and region localization. Across 23 real drawing sets and 1,954 categorical conditions per model, GPT-5.5 scores 82.4%, Qwen3-VL-32B-Instruct 62.2%, and GLM-4.5V 57.2%. A matched 200-target study crosses natural and vector-text-suppressed drawings with closed-candidate and free-geometry outputs. Candidate-supported performance is often higher, but free localization remains fragile: on natural drawings, point/region PCK@.05 is 54-76% for GPT, 8-10% for Qwen, and 14-36% for GLM; line endpoint PCK@.05 is 22%, 4%, and 0%. A coordinate grid recovers some GPT point/region precision but not lines. Three architecture-trained participants reach 87.3-93.3% categorical accuracy and 76-92% GT-region hit, supporting task feasibility rather than a population-level human ceiling. Because the categorical families do not form a same-item Matching-Registration contrast and interface controls alter multiple burdens, we avoid mechanistic claims. The supported conclusion is narrower: closed-choice or marked-element success does not entail reliable explicit geometric grounding. For drawing-guided CAD/BIM systems, categorical correctness should not be treated as evidence of candidate-free spatial reliability. Reusable on-sheet anchors, fixed-denominator scoring, and hash-locked artifacts establish an audit trail for this gap.

Compact networks built from Clifford algebra Cl(3,0) primitives are exactly SO(3)-equivariant and learn synthetic 3D vector laws from few samples. We ask whether the geometric algebra structure itself contributes anything beyond exact equivariance. We compare against a minimal scalarization baseline: invariant dot products fed to a small MLP that outputs coefficients on the equivariant basis {v_i, v_i x v_j}, which is also exactly equivariant. On single-stage laws (rotation by axis-angle, cross product, central force), scalarization matches or beats the Cl(3,0) network at a fraction of the training cost, so the geometric algebra adds nothing there. On compositional targets whose computation graph nests group operations (apply R2 R1 to a point; map a local force through an orientation, then take a torque), the Cl(3,0) network beats scalarization by an order of magnitude in the low-data regime, reaching with 100 samples what the baseline needs 3000 for, and the gap survives strengthening the baseline with the triple-product invariant and 17x more parameters, external Vector Neurons and e3nn baselines, and a multiplicative coefficient network. Ablations show the required network depth tracks the rotation chain length, and scalarization falls below the constant predictor on chains of four rotations. The advantage is not composition per se: on a rotation-free nested cross product, which flattens into polynomial invariant coefficients, scalarization wins by 24x. No tested model, equivariant or not, extrapolates invariant magnitudes: on radius and separation shifts every model is worse than a constant predictor once errors are normalized. We conclude that geometric algebra layers are not a general shortcut for low-data 3D learning, but become useful precisely when the target composes group elements in depth.