Conformal Geometric Algebra as a Symbolic Interface for LLM-Driven 3D Scene Editing

TL;DR

This paper evaluates Conformal Geometric Algebra (CGA) as a compact symbolic interface for LLM-driven 3D scene editing, demonstrating improved compositional fidelity and reliability over traditional methods.

Contribution

It introduces the use of CGA as a symbolic interface for 3D scene editing, showing its advantages in preserving instruction order and reducing token costs.

Findings

CGA achieves 97.5% fidelity in ordered instruction chains.

CGA outperforms Euclidean baseline by 21 percentage points in semantic suite.

Compact representations surpass Euclidean 4x4 matrices in reliability.

Abstract

What symbolic format should an LLM emit for reliable 3D scene editing from natural language, and does algebraic structure help beyond compact syntax? We evaluate Conformal Geometric Algebra (CGA) as a compact symbolic interface against a verbose Euclidean 4$\times$4 matrix baseline and a non-CGA Compact SE3 control in a natural-language 3D editing pipeline with controlled prompting and deterministic geometric execution. Our primary result is compositional fidelity under sequential instruction chains. In a sequence-stress protocol (20 templates, 6 trials each; $\texttt{n=120}$ outputs per method), Simple CGA and Compact SE3 both achieve 100% parse validity, but Simple CGA preserves exact ordered operation chains more reliably (97.5% vs 90.0%, two-proportion $\texttt{p=0.016}$) with lower completion-token cost (112.6 vs 133.6 tokens). This pattern is consistent with algebraic expression form supporting compositional faithfulness beyond compactness alone. A second result is confirmatory in the powered hard semantic suite ($\texttt{n=100}$ per method): compact representations (Simple CGA 45.0%, Compact SE3 42.0%, Shenlong 44.0%) all exceed the Euclidean 4$\times$4 baseline (24.0%). Simple CGA vs Euclidean is +21 pp ($\texttt{p=0.0028}$) and Compact SE3 vs Euclidean is +18 pp ($\texttt{p=0.0103}$), while Simple CGA vs Compact SE3 is statistically close ($\texttt{p=0.7755}$). Separating parse validity from geometric correctness reveals substantial optimization headroom invisible to syntax-only metrics. Overall, compact symbolic interfaces appear to drive reliability-cost gains, with CGA motor composition providing an additional advantage on ordered instruction chains. These findings inform real-time natural-language editing in immersive and interactive 3D environments.