Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences最新文献_第8页

One algebra for all: geometric algebra methods for neurosymbolic XR scene authoring, animation and neural rendering. 一个代数为所有：几何代数方法的神经符号XR场景创作，动画和神经渲染。

IF 3.7 3区综合性期刊

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2026-08-13 DOI: 10.1098/rsta.2025.0109

Manos Kamarianakis, Antonis Protopsaltis, George Papagiannakis

{"title":"One algebra for all: geometric algebra methods for neurosymbolic XR scene authoring, animation and neural rendering.","authors":"Manos Kamarianakis, Antonis Protopsaltis, George Papagiannakis","doi":"10.1098/rsta.2025.0109","DOIUrl":"10.1098/rsta.2025.0109","url":null,"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 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'.","PeriodicalId":19879,"journal":{"name":"Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"384 2326","pages":""},"PeriodicalIF":3.7,"publicationDate":"2026-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148723645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Splossoms: blossoming polynomials on the sphere via geometric algebra. 开花：通过几何代数在球体上开花的多项式。

IF 3.7 3区综合性期刊

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2026-08-13 DOI: 10.1098/rsta.2025.0106

Alyn Rockwood

{"title":"Splossoms: blossoming polynomials on the sphere via geometric algebra.","authors":"Alyn Rockwood","doi":"10.1098/rsta.2025.0106","DOIUrl":"10.1098/rsta.2025.0106","url":null,"abstract":"We introduce splossoms, the spherical analogue of polynomial blossoms, extending Ramshaw's theory of blossoming into spherical geometry. We note that the classical blossom axioms require spherical reinterpretation: strict symmetry does not hold on S2, and we identify the appropriate spherical analogues that the splossom satisfies. Splossoms provide a structured approach to designing spherical curves, generalizing Shoemake's SLERP interpolation. In addition, we define spolynomials, iterated spherical interpolants that mirror polynomial structures such as Bézier and B-spline forms, and we study their continuity. Potential applications include robotics, CNC milling, virtual/augmented reality and computer animation, where orientation curves on spheres are central. The splossom is naturally expressed in the framework of geometric algebra (GA), where rotors and spinors generalize quaternionic rotation and make spherical curve construction both simpler and more powerful. We argue that while splines on Lie groups and algebras are of general mathematical interest, GA provides a direct and computationally efficient setting for implementation. This article is part of the theme issue 'Modern applications of geometric algebra'.","PeriodicalId":19879,"journal":{"name":"Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"384 2326","pages":""},"PeriodicalIF":3.7,"publicationDate":"2026-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148725433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Geometric algebra integral transforms and their applications. 几何代数积分变换及其应用。

IF 3.7 3区综合性期刊

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2026-08-13 DOI: 10.1098/rsta.2025.0108

Eckhard Hitzer

引用次数: 0

The role of quantum entanglement in three players cooperative game. 量子纠缠在三人合作博弈中的作用。

IF 3.7 3区综合性期刊

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2026-08-13 DOI: 10.1098/rsta.2025.0111

Jaroslav Hrdina

引用次数: 0

Null geometric algebra with geometric interpretation. 零几何代数与几何解释。

IF 3.7 3区综合性期刊

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2026-08-13 DOI: 10.1098/rsta.2025.0096

Hongbo Li

引用次数: 0

Generative geospatial modelling with geometric algebra. 基于几何代数的生成地理空间建模。

IF 3.7 3区综合性期刊

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2026-08-13 DOI: 10.1098/rsta.2025.0103

Zhaoyuan Yu, Jian Wang, Zengjie Wang, Yi Liu, Dmitry Shirokov, Dietmar Hildenbrand, Linwang Yuan

引用次数: 0

A Clifford algebra approach to bosonic strings with fermionic boundaries. 具有费米子边界的玻色子弦的Clifford代数方法。

IF 3.7 3区综合性期刊

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2026-08-13 DOI: 10.1098/rsta.2025.0095

Andrew Hamilton, Tyler McMaken

引用次数: 0

A geometric algebra approach to coincidence site lattices. 符合点格的几何代数方法。

IF 3.7 3区综合性期刊

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2026-08-13 DOI: 10.1098/rsta.2025.0097

Marco A Rodríguez-Andrade, José L Aragón

{"title":"A geometric algebra approach to coincidence site lattices.","authors":"Marco A Rodríguez-Andrade, José L Aragón","doi":"10.1098/rsta.2025.0097","DOIUrl":"10.1098/rsta.2025.0097","url":null,"abstract":"When a lattice in Rn is superimposed on a rotated copy with coincident origins, coincidence points may arise for specific rotations, forming a coincidence site lattice (CSL). In crystallography, particularly in two and three dimensions, CSL theory is fundamental for describing low-energy grain boundaries. Geometric (Clifford) algebra provides a natural framework for this problem, as orthogonal transformations can be expressed as products of reflections. This article presents a unified account of the geometric algebra approach to coincidence isometries, including a Cartan-type decomposition of coincidence rotations for hypercubic lattices Zn. We then summarize the analytical characterization of coincidence rotations, indices and bases for planar hexagonal lattices. The main new result is a rigorous extension of these results to honeycomb structures, such as graphene, which are periodic but not Bravais lattices. We prove that the coincidence structure of two rotated honeycombs is completely determined by the coincidence lattice of their associated hexagonal Bravais lattices. As an application, structures near the first magic angle in twisted bilayer graphene are analysed, yielding quantitative agreement with experimentally observed Moiré lattice parameters. This article is part of the theme issue 'Modern applications of geometric algebra'.","PeriodicalId":19879,"journal":{"name":"Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"384 2326","pages":""},"PeriodicalIF":3.7,"publicationDate":"2026-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148722263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Closed-form solutions for rigid transformations between heterogeneous groups of noisy geometric primitives. 噪声几何基元异质群间刚性变换的闭型解。

IF 3.7 3区综合性期刊

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2026-08-13 DOI: 10.1098/rsta.2025.0110

Hugo Hadfield

引用次数: 0

Geometric algebra: an overview about different implementation representations and memory requirements. 几何代数：关于不同实现表示和内存需求的概述。

IF 3.7 3区综合性期刊

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2026-08-13 DOI: 10.1098/rsta.2025.0104

Oliver Rettig, Jaroslav Hrdina, Dietmar Hildenbrand

{"title":"Geometric algebra: an overview about different implementation representations and memory requirements.","authors":"Oliver Rettig, Jaroslav Hrdina, Dietmar Hildenbrand","doi":"10.1098/rsta.2025.0104","DOIUrl":"10.1098/rsta.2025.0104","url":null,"abstract":"Several implementations of geometric algebra are available, and many have been approved in the past and can be found in various software codes. For low-dimensional algebras, representations as a sum of basis blades are typically the most suitable. However, isomorphic or irreducible matrix representations are also options. Techniques such as compression on a per-coordinate, per-grade or per-group basis, as well as the use of sparse matrices, can help reduce memory consumption. In the case of high-dimensional algebras, factorized representations can significantly decrease the memory needed for blades and versors, reducing it to O(n2), as opposed to the O(2n) required to represent a complete basis of blades. This is particularly relevant in the context of quantum computing, where high dimensionality and memory usage are increasingly important considerations. This paper reviews various implementation strategies, classifies the corresponding software tools and libraries and highlights recent advancements aimed at reducing memory consumption within the tool Gaalop. This article is part of the theme issue 'Modern applications of geometric algebra'.","PeriodicalId":19879,"journal":{"name":"Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"384 2326","pages":""},"PeriodicalIF":3.7,"publicationDate":"2026-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148724952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0