One algebra for all : Geometric Algebra methods for neurosymbolic XR scene authoring, animation and neural rendering

TL;DR

The paper argues that Geometric Algebra provides a single, geometry-preserving algebraic framework to unify computer graphics and XR tasks that are traditionally handled by disparate formalisms such as vectors, matrices and quaternions. Key contributions include CGA-based mesh processing, GA-centric rigged animation with a compact multivector skinning equation, and a production-oriented GA-Unity integration that reduces bandwidth for networked XR. It also introduces neural and generative graphics pipelines like Shenlong and NeuralGASh that use CGA representations to enable real-time, geometry-aware AI-driven scene editing and shading without precomputation. The authors outline practical adoption paths, including matrix-free hardware acceleration ideas and standardized GA libraries, to realize a grand unification of geometry, simulation and intelligence in real-time graphics.

Abstract

This position paper delves into the transformative role of Geometric Algebra (GA) in advancing specific areas of Computer Graphics (CG) and Extended Reality (XR), particularly in character animation, rendering, rigging, neural rendering, and generative AI-driven scene editing. Common CG algorithms require handling rotations, translations, and dilations (uniform scalings) in operations such as object rendering, rigged model animation, soft-body deformation, and XR simulations. Traditional representation forms - such as matrices, quaternions, and vectors - often introduce limitations in precision and performance. Recent breakthroughs in the use of GA suggest it can significantly enhance these processes by encapsulating geometric forms and transformations into uniform algebraic expressions, which maintain critical geometric properties throughout multi-step transformations. Furthermore, we explore how GA can serve as a unifying mathematical substrate for neurosymbolic XR scene authoring, bridging learned neural representations and explicit geometric reasoning. This paper outlines how GA-based approaches can improve the fidelity of rigged character animations, enhance soft-body simulations, streamline real-time rendering, and optimize neural and generative AI scene editing. GA offers a coherent and efficient framework for these processes, resulting in superior visual outcomes and computational efficiency, particularly in XR environments.

Figures (8)

• Figure 1: The "original" euclidean animation equation involves vectors and matrices can be transformed to an equivalent one that only involves multivectors C4-B1J1. Results for keyframes are identical, with interpolated frames showing minor deviations (see Figure \ref{['fig:vr_recorder']}). Notation:$V_{k}[m]$ is the position vector of the position of the $m$-th vertex at animation time $k$ (in homogeneous coordinates), $T_{n,k}$ is the 4x4 matrix storing the transformation of the $n$-th bone at animation time $k$ and $O_{n}$ is the offset matrix corresponding to the $n$-th bone. $w_{m,n}$ denotes the weight of the $n$-th bone on the $m$-th vertex, $I_m$ is the set of indices of bones that affect the $m$-th vertex and $v[m]$ is the original position of the $m$-th vertex. $C_k[m], M_{n,k}, B_n$ and $c[m]$ are the equivalent multivector forms of $V_k[m], T_{n,k}, B_n$ and $v[m]$.
• Figure 2: GA for Accurate Psychomotor Action Replay: Our VR Record and Replay system (left, center) enables high-fidelity playback of user actions for analysis or training C9. (Right) The system's accuracy is built on GA, which provides a more robust transformation representation with significantly lower relative error compared to traditional quaternions.
• Figure 3: Robust Real-Time Surgery Training in XR: Our GA framework powers high-fidelity collaborative medical training, enabling dynamic, artifact-free mesh cutting and deformation on deformable patient models C3J7.
• Figure 4: A Geometrically-Aware Neural Architecture: The NeuralGASh pipeline consumes CGA multivectors—which compactly encode vertex position and normal data—to directly predict Spherical Harmonic (SH) coefficients for light and shadow, eliminating traditional pre-computation steps geronikolakis2025neural.
• Figure 5: Enhanced Realism with GA-Powered Neural Rendering: (Left) Standard Unity engine lighting on a dynamic character. (Right) The same scene rendered with our "NeuralGASh" pipeline, which achieves superior, dynamic lighting and shadow fidelity without pre-computation geronikolakis2025neural.