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

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 artificial intelligence (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. This article is part of the theme issue 'Modern applications of geometric algebra'.