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The last two decades, since the seminal work of Selig, has seen projective geometric algebra (PGA) gain popularity as a modern coordinate-free framework for doing classical Euclidean geometry and other Cayley-Klein geometries. This framework is based upon a degenerate Clifford algebra, and it is the purpose of this paper to delve deeper into its internal algebraic structure and extract meaningful information for the purposes of PGA. This includes exploiting the split extension structure to realise the natural decomposition of elements of this Clifford algebra into Euclidean and ideal parts. This leads to a beautiful demonstration of how Playfair's axiom for affine geometry arises from the ambient degenerate quadratic space. The highlighted split extension property of the Clifford algebra also corresponds to a splitting of the group of units and the Lie algebra of bivectors. Central to these results is that the degenerate Clifford algebra $\mathrm{Cl}(V)$ is isomorphic to the twisted trivial extension $\mathrm{Cl}(V/\langle e_0\rangle)\ltimes_\alpha\mathrm{Cl}(V/\langle e_0\rangle)$, where $e_0$ is a degenerate vector and $\alpha$ is the grade-involution., Comment: 17 pages. Submitted to Advances in Applied Clifford Algebras

This paper introduces a novel integration of Large Language Models (LLMs) with Conformal Geometric Algebra (CGA) to revolutionize controllable 3D scene editing, particularly for object repositioning tasks, which traditionally requires intricate manual processes and specialized expertise. These conventional methods typically suffer from reliance on large training datasets or lack a formalized language for precise edits. Utilizing CGA as a robust formal language, our system, shenlong, precisely models spatial transformations necessary for accurate object repositioning. Leveraging the zero-shot learning capabilities of pre-trained LLMs, shenlong translates natural language instructions into CGA operations which are then applied to the scene, facilitating exact spatial transformations within 3D scenes without the need for specialized pre-training. Implemented in a realistic simulation environment, shenlong ensures compatibility with existing graphics pipelines. To accurately assess the impact of CGA, we benchmark against robust Euclidean Space baselines, evaluating both latency and accuracy. Comparative performance evaluations indicate that shenlong significantly reduces LLM response times by 16% and boosts success rates by 9.6% on average compared to the traditional methods. Notably, shenlong achieves a 100% perfect success rate in common practical queries, a benchmark where other systems fall short. These advancements underscore shenlong's potential to democratize 3D scene editing, enhancing accessibility and fostering innovation across sectors such as education, digital entertainment, and virtual reality., Comment: 17 pages, 8 figures

Differential forms is a highly geometric formalism for physics used from field theories to General Relativity (GR) which has been a great upgrade over vector calculus with the advantages of being coordinate-free and carrying a high degree of geometrical content. In recent years, Geometric Algebra appeared claiming to be a unifying language for physics and mathematics with a high level of geometrical content. Its strength is based on the unification of the inner and outer product into a single geometric operation, and its easy interpretation. Given their similarities, in this article we compare both formalisms side-by-side to narrow the gap between them in literature. We present a direct translation including differential identities, integration theorems and various algebraic identities. As an illustrative example, we present the case of classical electrodynamics in both formalism and finish with their description of GR., Comment: 26 pages, 6 figures

We present, for the first time, a novel theoretical approach to address the problem of correspondence free multivector cloud registration in conformal geometric algebra. Such formalism achieves several favorable properties. Primarily, it forms an orthogonal automorphism that extends beyond the typical vector space to the entire conformal geometric algebra while respecting the multivector grading. Concretely, the registration can be viewed as an orthogonal transformation (\it i.e., scale, translation, rotation) belonging to $SO(4,1)$ - group of special orthogonal transformations in conformal geometric algebra. We will show that such formalism is able to: $(i)$ perform the registration without directly accessing the input multivectors. Instead, we use primitives or geometric objects provided by the conformal model - the multivectors, $(ii)$ the geometric objects are obtained by solving a multilinear eigenvalue problem to find sets of eigenmultivectors. In this way, we can explicitly avoid solving the correspondences in the registration process. Most importantly, this offers rotation and translation equivariant properties between the input multivectors and the eigenmultivectors. Experimental evaluation is conducted in datasets commonly used in point cloud registration, to testify the usefulness of the approach with emphasis to ambiguities arising from high levels of noise. The code is available at https://github.com/Numerical-Geometric-Algebra/RegistrationGA . This work was submitted to the International Journal of Computer Vision and is currently under review.

This paper introduces GA-Unity, the first Unity package specifically designed for seamless integration of Geometric Algebra (GA) into collaborative networked applications. Indeed, in such contexts, it has been demonstrated that using multivectors for interpolation between transmitted poses reduces runtime by 16% and bandwidth usage by an average of 50% compared to traditional representation forms (vectors and quaternions); we demonstrate that GA-Unity further enhances runtime performance. Tailored for 3D Conformal Geometric Algebra, GA-Unity also offers an intuitive interface within the Unity game engine, simplifying GA integration for researchers and programmers. By eliminating the need for users to develop GA functionalities from scratch, GA-Unity expedites GA experimentation and implementation processes. Its seamless integration enables easy representation of transformation properties using multivectors, facilitating deformations and interpolations without necessitating modifications to the rendering pipeline. Furthermore, its graphical interface establishes a GA playground for developers within the familiar confines of a modern game engine. In summary, GA-Unity represents a significant advancement in GA accessibility and usability, particularly in collaborative networked environments, empowering innovation and facilitating widespread adoption across various research and programming domains while upholding high-performance standards., Comment: 15 pages, 2 figures, paper refined and shortened

We introduce randomized algorithms to Clifford's Geometric Algebra, generalizing randomized linear algebra to hypercomplex vector spaces. This novel approach has many implications in machine learning, including training neural networks to global optimality via convex optimization. Additionally, we consider fine-tuning large language model (LLM) embeddings as a key application area, exploring the intersection of geometric algebra and modern AI techniques. In particular, we conduct a comparative analysis of the robustness of transfer learning via embeddings, such as OpenAI GPT models and BERT, using traditional methods versus our novel approach based on convex optimization. We test our convex optimization transfer learning method across a variety of case studies, employing different embeddings (GPT-4 and BERT embeddings) and different text classification datasets (IMDb, Amazon Polarity Dataset, and GLUE) with a range of hyperparameter settings. Our results demonstrate that convex optimization and geometric algebra not only enhances the performance of LLMs but also offers a more stable and reliable method of transfer learning via embeddings.

Extracting scientific understanding from particle-physics experiments requires solving diverse learning problems with high precision and good data efficiency. We propose the Lorentz Geometric Algebra Transformer (L-GATr), a new multi-purpose architecture for high-energy physics. L-GATr represents high-energy data in a geometric algebra over four-dimensional space-time and is equivariant under Lorentz transformations, the symmetry group of relativistic kinematics. At the same time, the architecture is a Transformer, which makes it versatile and scalable to large systems. L-GATr is first demonstrated on regression and classification tasks from particle physics. We then construct the first Lorentz-equivariant generative model: a continuous normalizing flow based on an L-GATr network, trained with Riemannian flow matching. Across our experiments, L-GATr is on par with or outperforms strong domain-specific baselines., Comment: 10+12 pages, 5+2 figures, 2 tables, v2: Extend acknowledgements, add link to github repo

In this paper, we present a series of techniques to describe General Relativity using Geometric Algebra (GA). We emphasize the physical interpretation of quantities and provide a step-by-step guide for performing calculations. In doing so, we show how GA offers insightful information on the physical meaning of the connection coefficients, the Riemann tensor, and other geometrical quantities., Comment: 9 pages, 2 figures

Geometric algebra (GA) is a mathematical tool for geometric computing, providing a framework that allows a unified and compact approach to geometric relations which in other mathematical systems are typically described using different more complicated elements. This fact has led to an increasing adoption of GA in applied mathematics and engineering problems. However, the scarcity of symbolic implementations of GA and its inherent complexity, requiring a specific mathematical background, make it challenging and less intuitive for engineers to work with. This prevents wider adoption among more applied professionals. To address this challenge, this paper introduces SUGAR (Symbolic and User-friendly Geometric Algebra Routines), an open-source toolbox designed for Matlab and licensed under the MIT License. SUGAR facilitates the translation of GA concepts into Matlab and provides a collection of user-friendly functions tailored for GA computations, including support for symbolic operations. It supports both numeric and symbolic computations in high-dimensional GAs. Specifically tailored for applied mathematics and engineering applications, SUGAR has been meticulously engineered to represent geometric elements and transformations within two and three-dimensional projective and conformal geometric algebras, aligning with established computational methodologies in the literature. Furthermore, SUGAR efficiently handles functions of multivectors, such as exponential, logarithmic, sinusoidal, and cosine functions, enhancing its applicability across various engineering domains, including robotics, control systems, and power electronics. Finally, this work includes four distinct validation examples, demonstrating SUGAR's capabilities across the above-mentioned fields and its practical utility in addressing real-world applied mathematics and engineering problems., Comment: 33 pages, 6 figures, journal paper submitted to ACM TOMS

The intention of our paper is to provide a pedagogical application of geometric algebra to a particularly well-investigated system: We formulate the geometric and dynamical properties of Friedmann-Robertson-Walker spacetimes within the language of geometric algebra and re-derive the Friedmann-equations as the central cosmological equations. Through the geometric algebra-variant of the Raychaudhuri equations, we comment on the evolution of spacetime volumes, before illustrating conformal flatness as a central property of Friedmann-cosmologies. An important aspect of spacetime symmetries are the associated conservation laws, for which we provide a geometric algebra formulation of the Lie-derivatives, of the Killing equation and of conserved quantities in Friedmann-Robertson-Walker spacetimes. Finally, we discuss the gravitational dynamics of scalar fields, with their particular relevance in cosmology, for cosmic inflation, and for dark energy., Comment: 31 pages, 4 figures

This paper explores the use of geometric algebra to study the Galilean spacetime and its physical implications. The authors introduce the concept of geometric algebra and its advantages over tensor algebra for describing physical phenomena. They define the Galilean-spacetime algebra (GSTA) as a geometric algebra generated by a four-dimensional vector space with a degenerate metric. They show how the GSTA can be used to represent Galilean transformations, rotations, translations, and boosts. The authors also derive the general form of Galilean transformations in the GSTA and show how they preserve the scalar product and the pseudoscalar. They develop a tensor formulation of Galilean electromagnetism using the GSTA and show how it reduces to the usual Maxwell equations in the non-relativistic limit. They introduce the concept of Galilean spinors as elements of the minimal left ideals of the GSTA and show how the Galilean spinors can be used to construct the Levy-Leblond equation for a free electron and its matrix representation. They provide a suitable matrix representation for the Galilean gamma matrices and the Galilean pseudoscalar. They relate the GSTA to the four component dual numbers introduced by Majernik to express Galilean transformations and show how the dual numbers can be used to develop a Newton-Cartan theory of gravity. The paper concludes by summarizing the main results and suggesting some possible applications and extensions of the GSTA., Comment: 8 pages, no figures

Many anatomical structures can be described by surface or volume meshes. Machine learning is a promising tool to extract information from these 3D models. However, high-fidelity meshes often contain hundreds of thousands of vertices, which creates unique challenges in building deep neural network architectures. Furthermore, patient-specific meshes may not be canonically aligned which limits the generalisation of machine learning algorithms. We propose LaB-GATr, a transfomer neural network with geometric tokenisation that can effectively learn with large-scale (bio-)medical surface and volume meshes through sequence compression and interpolation. Our method extends the recently proposed geometric algebra transformer (GATr) and thus respects all Euclidean symmetries, i.e. rotation, translation and reflection, effectively mitigating the problem of canonical alignment between patients. LaB-GATr achieves state-of-the-art results on three tasks in cardiovascular hemodynamics modelling and neurodevelopmental phenotype prediction, featuring meshes of up to 200,000 vertices. Our results demonstrate that LaB-GATr is a powerful architecture for learning with high-fidelity meshes which has the potential to enable interesting downstream applications. Our implementation is publicly available.

Continuous physical interaction between robots and their environment is a requirement in many industrial and household tasks, such as sanding and cleaning. Due to the complex tactile information, these tasks are notoriously difficult to model and to sense. In this article, we introduce a closed-loop control method that is constrained to surfaces. The applications that we target have in common that they can be represented by probability distributions on the surface that correlate to the time the robot should spend in a region. These surfaces can easily be captured jointly with the target distributions using coloured point clouds. We present the extension of an ergodic control approach that can be used with point clouds, based on heat equation-driven area coverage (HEDAC). Our method enables closed-loop exploration by measuring the actual coverage using vision. Unlike existing approaches, we approximate the potential field from non-stationary diffusion using spectral acceleration, which does not require complex preprocessing steps and achieves real-time closed-loop control frequencies. We exploit geometric algebra to stay in contact with the target surface by tracking a line while simultaneously exerting a desired force along that line. Our approach is suitable for fully autonomous and human-robot interaction settings where the robot can either directly measure the coverage of the target with its sensors or by being guided online by markings or annotations of a human expert. We tested the performance of the approach in kinematic simulation using point clouds, ranging from the Stanford bunny to a variety of kitchen utensils. Our real-world experiments demonstrate that the proposed approach can successfully be used to wash kitchenware with curved surfaces, by cleaning the dirt detected by vision in an online manner. Website: https://geometric-algebra.tobiloew.ch/tactile_ergodic_control, Comment: Submitted to the special issue for IEEE Transactions on Robotics (T-RO) on Tactile Robotics