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.

翻译：本立场论文深入探讨了几何代数在推动计算机图形学和扩展现实特定领域（尤其是角色动画、渲染、绑定、神经渲染以及生成式AI驱动的场景编辑）中的变革性作用。常见的计算机图形学算法需要在物体渲染、绑定模型动画、软体变形和XR模拟等操作中处理旋转、平移和均匀缩放。传统的表示形式（如矩阵、四元数和向量）往往在精度和性能上存在局限。几何代数应用的最新突破表明，通过将几何形态与变换封装为统一的代数表达式，并在多步变换中保持关键的几何属性，几何代数能显著优化这些过程。此外，本文探讨了几何代数如何作为神经符号XR场景创作的统一数学基础，桥接学习得到的神经表示与显式几何推理。论文概述了基于几何代数的方法如何提升绑定角色动画的保真度、增强软体模拟效果、简化实时渲染流程，并优化神经与生成式AI场景编辑。几何代数为这些过程提供了一个连贯且高效的框架，从而在XR环境中实现更优的视觉呈现与计算效率。