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

TL;DR

This paper explores how Geometric Algebra can unify and improve various computational processes in XR and CG, such as animation, rendering, and scene editing, by offering a more precise and efficient mathematical framework.

Contribution

It introduces the use of Geometric Algebra as a unifying mathematical approach for neurosymbolic XR scene authoring, animation, and neural rendering, highlighting its advantages over traditional methods.

Findings

GA enhances the fidelity of character animations.

GA streamlines real-time rendering and scene editing.

GA improves computational efficiency in XR applications.

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.