ABSTRACT. Quaternionic polynomials occur naturally in applications of quaternions in science and engineering, and normalization of quaternionic polynomials is a basic manipulation. This paper presents a method of computing the normal form of quaternionic polynomials under the conjugate-alternating order, and establishes order-preserving Cayley expressions for low-degree elements of the reduced Gr\"obner basis of the defining ideal of quaternionic polynomial algebra under the conjugate-separating order.

ABSTRACT. This paper describes the Geometric Algebra for Quadrics (GAQ) with a focus on elements representing Euclidean transformations, particularly translations and rotations. We provide general methods of deriving the generators of such transformations and verify their correctness on examples. We also show the way of the actual versors calculations numerically, yet their exact form is rather technical and extensive because of high dimension of GAQ.

ABSTRACT. In this paper, we present an explicit formula for the inverse and determinant in geometric (Clifford) algebras over vector spaces of dimension $n=7$. We generalize the concept of conjugation to basis conjugation operations, allowing us to express the determinant formula independently of any specific algebra isomorphism. This construction provides a practical computational tool for determining invertibility and calculating inverses of multivectors in geometric algebras associated with seven-dimensional vector spaces. The resulting formulas extend previous results for lower dimensions and offer new insights for applications in mathematical physics and computational geometry.

ABSTRACT. While the Characteristic Multivector (CM) method within Geometric Algebra provides accurate 3D rotation estimation, its computational expense motivated the use of GAALOP (Geometric Algebra Algorithms Optimizer) symbolic optimization to create efficient standard and Common Subexpression Elimination (CSE) variants. We evaluated these optimized versions against the original MATLAB CM implementation provided by the Clifford Multivector Toolbox for the Absolute Orientation (AO) problem and benchmarked them against both the original CM and a standard SVD based method within the Iterative Closest Point (ICP) framework. The evaluation revealed significant runtime improvements with GAALOP and demonstrated speeds competitive with SVD, while preserving the inherent high accuracy of the Geometric Algebra approach and establishing it as a viable 3D registration alternative.

ABSTRACT. This paper introduces GeloVec, a new CNN-based attention smoothing framework that combines Chebyshev distance metrics and multispatial transformations to enhance semantic segmentation accuracy through stabilized feature extraction. Conventional attention-backed approaches to segmentation suffer from boundary instability and contextual discontinuities during feature mapping processes. We propose a higher-dimensional geometric smoothing method that operates in n-dimensional feature space, establishing robust manifold relationships between visually coherent regions. By applying a modified Chebyshev distance calculation with adaptive sampling weights, our approach achieves superior edge preservation while maintaining intra-class homogeneity. Experimental validation across multiple benchmark datasets (Caltech Birds-200, LSDSC, FSSD) demonstrates mean Intersection over Union (mIoU) 2.1, 2.7, 2.4 percent improvements compared to state-of-the-art methods. The multispatial transformation matrix incorporates tensorial projections with orthogonal basis vectors, creating more discriminative feature representations. Additionally, computational efficiency is maintained through parallelized implementation of the proposed geodesic transformations. GeloVec's mathematical foundation in Riemannian geometry provides theoretical guarantees on segmentation stability. Our framework is generalizable across disciplines due to absence of information loss during transformations.

ABSTRACT. Subtitle: Perspectives on Dynamical System between Geometric Vector Time-Series

Quantifying how motion flows from one body segment to another is a long–standing challenge in sports biomechanics, rehabilitation, and human–robot interaction, especially for non-periodic movements such as over-arm pitching. We introduce a convolution–based dissimilarity metric that treats each pair of time-series as the input and output of a Linear Time-Invariant (LTI) system. Replacing the scalar inner product with the geometric product yields a framework that is agnostic to dimensionality: for planar motions the convolution coefficients form a vector in $\mathbb{C}^{M+N-1}$, whereas for spatial motions they inhabit $\mathbb{H}^{\,M+N-1}$. Similarity is then quantified with the complex or quaternion $L_{2}$ norm, preserving scale invariance while embedding rich directional information. Experiments on video / motion-capture data of baseball pitching motions show that the proposed metric (i) discriminates subtle ankle-to-wrist coordination patterns, and (ii) remains stable under camera shooting angle in horizontal plane of 40 deg. These findings establish a direct bridge between classical LTI analysis and Geometric Vector Time-Series (GVTS), suggesting a unified language for multi-modal sensor fusion, explainable graph neural networks, and real-time feedback applications.

ABSTRACT. Current machine learning methods for predicting sea level change often neglect the intrinsic geometric constraints and physical relationships between multidimensional vector field components, resulting in the loss of spatio-temporal coupling information. To address this problem, this study proposes a geometric algebra (GA)-based method for predicting the rate of sea level variability.Salinity, potential temperature, eastward and northward seawater velocities are uniformly represented as multidimensional geometric objects. Compared with traditional methods, the framework utilizes the unified mathematical structure of geometric algebra to represent scalars and vectors in a holistic manner, avoiding the segmented treatment of temperature, salinity and velocity fields and fully preserving their intrinsic geometric and physical relationships. This effectively captures the complex spatial and temporal dynamics of sea level variability. First, TSGAConvGRU organizes ocean stereodynamic data (such as potential temperature, salinity, and zonal and meridional velocities) into multivector inputs. It then employs GAConvGRU to capture spatial and temporal features while accounting for the geometric relationships among components like potential temperature, salinity, and seawater velocities. Second, Series Embedding is introduced to incorporate time markers into temporal features, enhancing the model’s awareness of temporal context. A Dual Path approach with different receptive fields adapts to local and global dynamic features of sea level variability, using three GRU layers to capture long- and short-term dependencies in time series, further modeling the periodicity and trends of sea level variability. A final linear layer maps to the target prediction. Experimental results demonstrate that the proposed method significantly outperforms existing models in predicting sea level variability in the Northeast Pacific, exhibiting higher accuracy and robustness.Furthermore, by establishing a unified GA-based framework, this approach provides a novel scientific perspective for future studies to explore the physical mechanisms underlying variable interactions, thereby enhancing the understanding of the driving factors of sea level changes.

ABSTRACT. hree-dimensional cadastral update refers to the process of performing data update operations on cadastral parcels in the cadastral database in response to changes in information such as spatial scope, parcel ownership, land use patterns, and land uses. It is an important means to maintain the authority, accuracy, and currency of cadastral data. In the daily cadastral management process, the change of parcels mainly involves two parts: attribute data update and spatial data update. The update of spatial data of cadastral parcels is mainly manifested as the adjustment of the spatial ownership scope of cadastral parcels, specifically as the update and maintenance of the geometric shapes of cadastral parcels in the cadastral database, which mainly includes the division and merger of parcels. Based on the previous multi-dimensional unified expression of three-dimensional cadastre using geometric algebra, this study proposes a self-adaptive update method for cadastral parcels based on the multi-vector structure. This method can achieve the self-adaptive update of cadastral topological relations in different dimensions during the update process, and reduce the complexity of maintaining topological relations during the update of cadastral parcels.

ABSTRACT. The complex Clifford algebra CG(n,n) is constructed using a split signature metric. This results in a different representation of gates compared to the classical case with Euclidean signatures CG(2n). The use of split signature, along with Boot periodicity, enables an efficient matrix representation and facilitates implementation.

ABSTRACT. In this work we look at a hypercomplex algebra prominent in quantum chromo (color) dynamics, introduced by S. Okubo, generated by the eight $3\times 3$ traceless hermition matrices of M. Gell-Mann. We go beyond the familiar algebras of W.K. Clifford (geometric algebra), W.R. Hamilton (quaternions and biquaternions) and J.T. Graves and A. Caley (octonions) and introduce several properties of the eight dimensional Okubo algebra, known to be a division algebra, not unital, not associative, not alternative but flexible with a positive definite norm that is associative and compositional. We give an easy to interpret full multiplication table, show how two units can generate the whole algebra, study several subalgebras and look at powers and exponentials of Okubo algebra units.

ABSTRACT. This review provides a comprehensive overview of the applications of Geometric Algebra (GA) in electrical engineering, with a particular focus on power systems. The paper covers the fundamental concepts of GA and its advantages over traditional mathematical approaches in power system analysis. Key topics explored include: an introduction to GA and its relevance to electrical engineering; AC Circuit Theory using GA, covering GA formulations for power flow analysis in non-sinusoidal conditions including harmonic and interharmonic analysis for single-phase and multi-phase systems; power-free current compensation techniques that move beyond traditional reactive power concepts; parameter identification in electrical circuits using GA; GA-based perspectives on electrical transformations (Clarke or Fortescue); and the analysis of electrical curves and their geometric invariants. The review synthesizes recent advancements, including contributions from the author and colleagues, discusses future directions, and highlights the potential of GA to address current and emerging challenges in power system analysis, control, smart grids, and renewable energy systems. We aim to provide both a theoretical foundation and practical insights for researchers and practitioners in the field.

ABSTRACT. This paper presents Neural-GASh, a novel real-time shading pipeline for 3D meshes, that leverages a neural network architecture to perform image-based rendering (IBR) using Conformal Geometric Algebra (CGA)-encoded vertex information as input. Unlike traditional Precomputed Radiance Transfer (PRT) methods, that require expensive offline precomputations, our learned model directly consumes CGA-based representations of vertex positions and normals, enabling dynamic scene shading without precomputation. Integrated seamlessly into the Unity engine, Neural-GASh facilitates accurate shading of animated and deformed 3D meshes—capabilities essential for dynamic, interactive environments. The shading of the scene is implemented within Unity, where rotation of scene lights in terms of Spherical Harmonics is also performed optimally using CGA. This neural field approach is designed to deliver fast and efficient light transport simulation across diverse platforms, including mobile and VR, while preserving high rendering quality. Additionally, we evaluate our method on scenes generated via 3D Gaussian splats, further demonstrating the flexibility and robustness of Neural-GASh in diverse scenarios. Performance is evaluated in comparison to conventional PRT, demonstrating competitive rendering speeds even with complex geometries.

ABSTRACT. The applications of hypercomplex algebras, such as complex numbers and quaternions, to machine learning as an alternative to real-valued architectures have been an established area of research. Clifford algebra can be viewed as a structure generalising hypercomplex number systems. More recently, Clifford algebra has attracted attention in the context of neural networks, with applications for example in the areas of partial differential equation modelling or image analysis. In neural networks, the choice of a particular Clifford algebra used in the architecture along with the embedding of input data into the algebra can be thought of as a hyperparemeter of the model. In this article, we investigate the applications of Clifford algebra to multidimensional medical imaging data through a modification of the U-Net architecture at the embedding level. We selected the breast cancer segmentation as a case study, considering dynamic contrast-enhanced magnetic resonance images (DCE-MRI). This acquisition consists of a sequence of MRI, in which MRI is acquired after the administration of a contrast agent. Such multidimensional input was analysed by modifying the traditional U-Net architecture integrating a Clifford algebra block to exploit correlations between MRI timestamps.

ABSTRACT. Geometric Algebra (GA) presents challenges to learners due to its highly abstract mathematical structure and complex operational rules, translating algebraic manipulations into concrete geometric interpretations a non-intuitive process when developing related code. Currently, some existing GA tools rely on manually written scripts for code generation and visualization, but their high learning curve hinders widespread adoption. Meanwhile, methods based on Large Language Models (LLMs) often produce logical errors when generating specific GA scripts, such as GaalopScript, resulting in generally low accuracy. To address these issues, this paper proposes GA-VisAgent—a multi-agent interactive learning application for GA code generation and visualization—building upon GAGPT. Integrating task planning mechanisms with ReAct reasoning strategies, GA-VisAgent can decompose complex operations into five standardized subtasks, including core operations like geometric products, rotations, and reflections. It supports natural language and mathematical formulas as input to automatically generate executable code, accompanied by interactive visualizations to aid user comprehension. Experimental results show that GA-VisAgent achieved a 90\% code generation success rate across 40 typical Conformal Geometric Algebra tasks, representing a 70\% improvement over GPT-4o. This application introduces an extensible new paradigm for teaching GA and developing visualization tools for related mathematical concepts. The online service for this project will be available at http://gagis.cn/gacrac.