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Geometric-algebra based adaptive filters have been successfully employed in many applications such as computer vision, data fusion and linear prediction where the unknown parameters of interest are high-dimensional multivectors. However, conventional geometric-algebra adaptive filters, such as the strictly linear geometric-algebra least mean square (SL-GA-LMS) algorithm, are only applicable to circular multivector-valued inputs with rotation-invariant probability distribution functions. To remove this limitation, we propose new semi-widely linear and widely linear GA-LMS algorithms. As geometric-algebra adaptive filters can have extremely high computational complexity, partial-update variants of these algorithms with reduced complexity are also developed employing stochastic, sequential and -max partial updating strategies. Steady-state and transient performances of the proposed partial-update algorithms are analysed. As an isomorphism to the partial-update GA-LMS algorithms, widely linear, semi-widely linear and strictly linear quaternion LMS algorithms with partial updates are proposed and analysed for noncircular quaternion inputs. Finally, numerical studies are carried out to confirm the advantages of the proposed methods and the convergence analysis results for multivector and quaternion-valued inputs. Refereed/Peer-reviewed

Funder: Engineering and Physical Sciences Research Council; Id: http://dx.doi.org/10.13039/501100000266; Grant(s): AgriForwards CDT (C. Matsantonis): EP/S023917/1, Funder: The Mathworks; Id: http://dx.doi.org/10.13039/100014600, If an initial frame of vectors { e i } $$ \left\{{e}_i\right\} $$ is related to a final frame of vectors { f i } $$ \left\{{f}_i\right\} $$ by, in geometric algebra (GA) terms, a rotor, or in linear algebra terms, an orthogonal transformation, we often want to find this rotor given the initial and final sets of vectors. One very common example is finding a rotor or orthogonal matrix representing rotation, given knowledge of initial and transformed points. In this paper, we discuss methods in the literature for recovering such rotors and then outline a GA method, which generalises to cases of any signature and any dimension, and which is not restricted to orthonormal sets of vectors. The proof of this technique is both concise and elegant and uses the concept of characteristic multivectors as discussed in the book by Hestenes and Sobczyk, which contains a treatment of linear algebra using geometric algebra. Expressing orthogonal transformations as rotors, enables us to create fractional transformations and we discuss this for some classic transforms. In real applications, our initial and/or final sets of vectors will be noisy. We show how to use the characteristic multivector method to find a ‘best fit’ rotor between these sets and compare our results with other methods.

14 pags., 13 figs., 5 tabs. -- MSC Classification: 68T40, 68T45, 68U05, Many problems in computer vision today are solved via deep learning. Tasks like pose estimation from images, pose estimation from point clouds or structure from motion can all be formulated as a regression on rotations. However, there is no unique way of parametrizing rotations mathematically: matrices, quaternions, axis-angle representation or Euler angles are all commonly used in the field. Some of them, however, present intrinsic limitations, including discontinuities, gimbal lock or antipodal symmetry. These limitations may make the learning of rotations via neural networks a challenging problem, potentially introducing large errors. Following recent literature, we propose three case studies: a sanity check, a pose estimation from 3D point clouds and an inverse kinematic problem. We do so by employing a full geometric algebra (GA) description of rotations. We compare the GA formulation with a 6D continuous representation previously presented in the literature in terms of regression error and reconstruction accuracy. We empirically demonstrate that parametrizing rotations as bivectors outperforms the 6D representation. The GA approach overcomes the continuity issue of representations as the 6D representation does, but it also needs fewer parameters to be learned and offers an enhanced robustness to noise. GA hence provides a broader framework for describing rotations in a simple and compact way that is suitable for regression tasks via deep learning, showing high regression accuracy and good generalizability in realistic high-noise scenarios., This work has been supported by the Signal Processing & Communications Group of Cambridge University EngineeringDepartment (CUED)

Theory of Everything

This paper explores formalizing Geometric (or Clifford) algebras into the Lean 3 theorem prover, building upon the substantial body of work that is the Lean mathematics library, mathlib. As we use Lean source code to demonstrate many of our ideas, we include a brief introduction to the Lean language targeted at a reader with no prior experience with Lean or theorem provers in general. We formalize the multivectors as the quotient of the tensor algebra by a suitable relation, which provides the ring structure automatically, then go on to establish the universal property of the Clifford algebra. We show that this is quite different to the approach taken by existing formalizations of Geometric algebra in other theorem provers; most notably, our approach does not require a choice of basis. We go on to show how operations and structure such as involutions, versors, and the $\mathbb{Z}_2$-grading can be defined using the universal property alone, and how to recover an induction principle from the universal property suitable for proving statements about these definitions. We outline the steps needed to formalize the wedge product and $\mathbb{N}$-grading, and some of the gaps in mathlib that currently make this challenging., Comment: 26 pages. This is the version accepted for publication on 2021-07-22, after peer review, in "Advances in Applied Clifford Algebras". This is not the Version of Record and does not reflect post-acceptance improvements, or corrections by the journal. Note that the web version of this article provided by the journal has typesetting issues not present here or in the journal's PDF

In this paper, I examine aspects of the methodological debate that originated in 2010, when the distinguished historian of mathematics Sabetai Unguru reviewed Roshdi Rashed’s edition of the Arabic translation of Apollonius’ Conics. In his review, Unguru criticized what Rashed calls “l’usage instrumental d’une autre mathématique pour commenter une oeuvre ancienne”. I consider this debate very important and will try to place it within in the discussion of the so-called “geometric algebra” that goes back to the seventies, by tracing the contributions of the main figures who took part in it. Published Online (2021-04-30)Copyright © 2021 by Aldo Brigaglia Article PDF Link: https://jps.library.utoronto.ca/index.php/aestimatio/article/view/37627/28622 Corresponding Author: Aldo Brigaglia,University of PalermoE-Mail: aldo.brigaglia@gmail.com

Geometric Algebra (GA) has proven to be an advanced language for mathematics, physics, computer science, and engineering. This review presents a comprehensive study of works on Quaternion Algebra and GA applications in computer science and engineering from 1995 to 2020. After a brief introduction of GA, the applications of GA are reviewed across many fields. We discuss the characteristics of the applications of GA to various problems of computer science and engineering. In addition, the challenges and prospects of various applications proposed by many researchers are analyzed. We analyze the developments using GA in image processing, computer vision, neurocomputing, quantum computing, robot modeling, control, and tracking, as well as improvement of computer hardware performance. We believe that up to now GA has proven to be a powerful geometric language for a variety of applications. Furthermore, there is evidence that this is the appropriate geometric language to tackle a variety of existing problems and that consequently, step-by-step GA-based algorithms should continue to be further developed. We also believe that this extensive review will guide and encourage researchers to continue the advancement of geometric computing for intelligent machines.

The current work considers the sprefield wave functions received as special g-qubit solutions of Maxwell equations in the terms of geometric algebra. I will call such g-qubits spreons or sprefields. The purpose of this article is to analyze behavior of such wave functions in scattering and measurements. It is shown that sprefields are defined through the whole three-dimensional space at all values of the time parameter. They instantly change all their values when get scattered, that is subjected to Clifford translation. In “measurements”, when a sprefield acts on a static geometric algebra element through the Hopf fibration, sprefield collapses and new geometric algebra non static, rotating element is thereby created.

In this paper, a new approach for instantaneous reactive power analysis in the geometric algebra (GA) environment is presented. The different formulations of the instantaneous reactive power theory (IRPT) proposed, to date, have been developed in three-phase systems. There, an instantaneous power variable, and two/three reactive power variables, all handled independently, were introduced. Thanks to GA, it is possible to carry out a global treatment where an instantaneous power multivector is defined. Thus, in the same multidimensional entity all the power variables are included. From the instantaneous power multivector, the instantaneous power current and the instantaneous reactive current are determined. It should be noted that in this mathematical framework there is no limitation on the number of phases, and the extension of the IRPT to the analysis of multi-phase systems appears in a natural manner. In this study, a systematic approach with the most relevant definitions and theorems corresponding to the proposed methodology has been established. Two practical cases of five-phase and three-phase systems have been included to apply the new established formulation.

The following paper talks about the dynamic modeling and control of robot manipulators using Hamilton’s equations in the screw theory framework. The difference between the proposed work with diverse methods in the literature is the ease of obtaining the laws of control directly with screws and co-screws, which is considered modern robotics by diverse authors. In addition, geometric algebra (GA) is introduced as a simple and iterative tool to obtain screws and co-screws. On the other hand, such as the controllers, the Hamiltonian equations of motion (in the phase space) are developed using co-screws and screws, which is a novel approach to compute the dynamic equations for robots. Regarding the controllers, two laws of control are designed to ensure the error’s convergence to zero. The controllers are computed using the traditional feedback linearization and the sliding mode control theory. The first one is easy to program and the second theory provides robustness for matched disturbances. On the other hand, to prove the stability of the closed loop system, different Lyapunov functions are computed with co-screws and screws to guarantee its convergence to zero. Finally, diverse simulations are illustrated to show a comparison of the designed controllers with the most famous approaches.

ВВ криптографических системах шифрования информации используются гиперкомплексные числа: кватернионы и октонионы. В качестве ключа применяется кватернион, который производит вращения группы выборок информации. Кватернионы и бикватернионы являются частными случаями геометрической алгебры Клиффорда. Использование векторов и мультивекторов геометрической алгебры для шифрования информации позволяет расширить разнообразие этих векторов. Для шифрования информации, представленной совокупностью векторов геометрической алгебры, эти векторы умножаются на мультивектора, которые осуществляют операцию ротор (rotor). В качестве ключа используется мультивектор (ротор). Для дешифрования информации применяется операция, которая соответствует обратному ротору. Алгоритмы геометрической алгебры повышают безопасность шифрования информации за счет повышения размерности алгебры. Для повышения производительности шифрования предлагается коэффициенты информационного вектора и мультивектора вращения выбирать из поля Z256. Предлагается вектор информации с коэффициентами из Z256 складывать со случайным вектором с коэффициентами из Z256 и считать эти коэффициенты ключами шифрования. Приведены базисные векторы применяемых геометрических алгебр и таблицы геометрических произведений базисных векторов.

Omar Khayyam (“Khayyam” meaning the tent maker) was born on May 18, 1048, in Nishapur, a city in the state of Khorasan in northeastern Iran, and died on December 4, 1131. Khayyam, while widely know...

Ray-tracing is an efficient asymptotic computational electromagnetic method for studying wave propagation. The major bottleneck of ray-tracing algorithms is the computational complexity of the geom...

The authors solve the problem of estimating the best rotation aligning two sets of corresponding vectors (also known as Wahba's problem or point cloud registration). The proposed method is among the fastest methods reported in recent literatures, moreover it is robust to noise, accurate and simpler than most other methods. It is based on solving the linear equations derived from the formulation of the problem in Euclidean Geometric Algebra. The authors show its efficiency in two applications: the as-rigid-as-possible (ARAP) surface modelling and the more smooth rotation enhanced ARAP mesh animation which is the only method capable of deforming surface modes with quality of tetrahedral models. Mesh deformation is a key technique in games, automated construction and robotics. The ARAP technique along with its improved variants, although have been extensively studied, can still not be achieved efficiently. Linear geometric algebra based rotor solution proposed in this study gives another perspective of the kernel problem. This, however, not only improves the real performance of the three-dimensional mesh deformation, but also provides a brand new computationally efficient solution to the Wahba's problem and point cloud registration, which has been closely related to the automation science and engineering.

Recently, skeleton-based action recognition approaches achieve great improvement by providing various features from skeleton sequences as the inputs of deep neural networks for classification. Compared with other works which directly utilize the locations of joints as the representation of each action for recognition, our paper describes the relative relations embedded in skeletons to alleviate the impact of viewpoint diversity. Specifically, we propose a novel feature descriptor based on the rotation relations in skeletons to represent a certain action. Geometric algebra (GA) is introduced to calculate and derive the rotation relations according to the particular operator, namely the rotor in GA. In order to exploit the spatial and temporal characteristics in one skeleton sequence, we design two different rotor-based feature descriptors, respectively, from one frame of skeleton and two skeletons of consecutive frames. Then an efficient feature encoding strategy is proposed to transform each kind of feature descriptor into a RGB image. Afterward, we propose a two-stream convolutional neural network(CNN) based framework to learn the RGB images generated by each skeleton sequence and then fuse the scores of two networks to provide final recognition accuracy. Extensive experimental results on NTU RGB+D, Northwestern-UCLA, Gaming 3D, SYSU and UTD-MHAD datasets have demonstrated the superiority of our method.

With the increasing demand for multidimensional data processing, Geometric algebra (GA) has attracted more and more attention in the field of geographical information systems. GA unifies and generalizes real numbers and complex, quaternion, and vector algebra, and converts complicated relations and operations into intuitive algebra independent of coordinate systems. It also provides a solution for solving multidimensional information processing with a high correlation among the dimensions and avoids the loss of information. Traditional methods of computer vision and artificial intelligence (AI) provide robust results in multidimensional processing after being combined with GA and give additional feature analysis facility to remote sensing images. In this paper, we provide a detailed review of GA in different fields of AI and computer vision regarding its applications and the current developments in geospatial research. We also discuss the Clifford-Fourier transform (CFT) and quaternions (sub-algebra of GA) because of their necessity in remote sensing image processing. We focus on how GA helps AI and solves classification problems, as well as improving these methods using geometric algebra processing. Finally, we discuss the issues, challenges, and future perspectives of GA with regards to possible research directions.

For color image fusion, existing traditional algorithms regard each color image pixel as a scalar which may lose the relationships among spectral channels. In this paper, a novel multi-focus image fusion algorithm using geometric algebra based Discrete Fourier Transform (GA-DFT) is proposed. First, a novel Discrete Fourier Transform based on geometric algebra (GA) is provided. Second, the proposed algorithm represents the color image as a GA multivector by mapping the RGB channels into the GA space. Third, the variance is selected as a contrast measure to fuse the source images in GA space. The experimental results demonstrate that the proposed algorithm can achieve the comparable performance with its counterparts in terms of subjective quality evaluation, and outperform most of the traditional algorithms in terms of objective quality metrics, indicating good performance in image fusion applications.

Three-dimensional image modeling is essential in many scientific disciplines, including computer vision and precision agriculture. So far, various methods of creating three-dimensional (3D) models have been considered. However, the processing of transformation matrices of each input image data is not controlled. Site-specific crop mapping is essential because it helps farmers determine yield, biodiversity, energy, crop coverage, etc. Clifford Geometric Algebraic understanding of signaling and image processing has become increasingly important in recent years. Geometric Algebraic treats multi-dimensional signals in a holistic way to maintain relationship between side sizes and prevent loss of information. This article has used agricultural images acquired by unmanned aerial vehicles (UAVs) to construct three-dimensional models using Clifford geometric algebra. The qualitative and quantitative performance evaluation results show that Clifford geometric algebra can generate a three-dimensional geometric statistical model directly from drones' RGB images. Through peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and visual comparison, the proposed algorithm's performance is compared with latest algorithms. Experimental results show that proposed algorithm is better than other leading 3D modeling algorithms.

The research considers wavelike objects that are elements of even subalgebra of geometric algebra in three dimensions. The used formalism particularly eliminates long existing confusion about the reasons behind the appearance of the imaginary unit in quantum mechanics and introduces clear definition of wave functions. When a wave function acts through the Hopf fibration on a localized geometric algebra element, that is executing a measurement, the result can be named as “collapse” of the wave function.

espanolAmpliamos la introduccion al algebra geometrica bidimensional AG(2) presentada en el Volumen 05(02) de SELECCIONES MATEMATICAS. Se trata de una introduccion amigable a un tema de interes comun para docentes y estudiantes universitarios de Fisica y Matematica. EnglishWe enlarge a previus introduction to the two-dimensional geometric algebra published in Volume 05(02) of SELECCIONES MATEMATICAS. It is a friendly introduction of a theme of common interest for university proffessors and students of physics and mathematics.

Geometric algebras known as a generalization of Grassmann algebras complex numbers and quaternions are presented by Clifford (1878). Geometric algebra describing the geometric symmetries of both physical space and spacetime is a strong language for physics. Groups generated from `Clifford numbers` are firstly defined by Lipschitz (1886). They are used for defining rotations in a Euclidean space. In this work, Clifford algebra is identified. The energy of classic particles with Clifford algebra are defined. This calculations are applied to some Archimedean solids. Also, the vertices of Archimedean solids presented in the Cartesian coordinates are calculated.

Speeded-Up Robust Features (SURF) can be used for tasks such as object recognition, image registration, classification or 3D reconstruction. However, existing SURF algorithm cannot be directly applied to deal with multispectral images. In this paper, based on SURF and the theory of Geometric Algebra (GA), a novel feature extraction algorithm named GA-SURF is proposed for multispectral images. First, a Hessian matrix based on GA is calculated for locating interest points in spatial and spectral space. The box filters based on GA are used to simplify the calculation of Hessian matrix and generate image pyramids. Then, following the procedures of SURF, interest points are located by the image pyramids and described in GA space. Experimental results demonstrate that, compared with other feature extraction algorithms for multispectral images, GA-SURF can be computed much faster and are more robust and distinctive.

We describe a possibility for geometric calculation of specific conics’ intersections in Geometric Algebra for Conics (GAC) using its operations that may be expressed as sums of products. The advantage is that no solver for a system of quadratic equations is needed and thus no numerical error is involved. We also describe specific conics connected to intersections of conics in a general mutual position. Then we show how symbolic operations may be calculated directly in GAALOPWeb software, that the basis coefficients may be read off in the appropriate basis and, moreover, the result may be immediately and truly visualized. We compare the functionality with Maple package Clifford.

A rigorous proof of no finite time blowup of the 3D Incompressible Navier Stokes equations in R3/Z3 is hereby shown using Geometric Algebra and subsequently the Gagliardo-Nirenberg and Pr kopa-Leindler inequalities are used to prove that the integrand of the integral form of the solution obtained can be set to zero everywhere in space and time, as well as results on the velocity-pressure distribution using Debreu's, Brouwer's, Lusin's and a final theorem proving no blowup on [0,\(\infty\)]. A complex equation's values for positive and t tending to large values indicates with the help of five theorems that as approaches positive infinity, approaches infinity.

In this paper, two new geometric algebra (GA) based adaptive filtering algorithms in non-Gaussian environment are proposed, which are deduced from the robust algorithms based on the minimum error entropy (MEE) criterion and the joint criterion of the MEE and the mean square error (MSE) with the help of GA theory. Some experiments validate the effectiveness and superiority of the GA-MEE and GA-MSEMEE algorithms in α-stable noise environment. At the same time, the GA-MSEMEE algorithm has faster convergence speed compared with the GA-MEE.

The currents' physical component power theory (CPC-PT) is used here to show the improperness of extrapolating definitions of the power phenomena in sinusoidal conditions to non-sinusoidal settings. We prove, through seven examples, that the fundamentals of circuit theory are sine qua non for assessing: 1) the acceptability of a power theory (PT), 2) the appropriates of power quantities such as the reactive power and 3) the reason PTs fail. We show how the Geometric Algebra-based PT overcomes the contradictions found with the CPC-PT and unveils the reason PTs fail. Finally, we show that once the appropriate definition of the reactive power is obtained, the definition of $S$ based on the geometric addition of the active and non-active powers is a more reliable that the definition of $S$ based on the product of the root mean square values of the voltage and the current.

This paper derives a linear, first-order, partial differential field equation (a Dirac-like equation) in the geometric calculus of the geometric algebra $${\mathcal {G}}_{4,1}$$ that has free plane-wave solutions distinct from one another that correspond to the left and right chiral states of the electron and the neutrino. Besides the usual spacetime dependence of plane waves, the solutions have a multivector structure yielding a ladder of states with raising and lowering operators appropriate to electroweak theory and having an $$SU(2)_L$$ relationship among the chiral electron and neutrino states. The required Dirac-like equation in $${\mathcal {G}}_{4,1}$$ results from a systematic review of Dirac-like equations (i.e., first-order field equations whose solutions also satisfy the Klein–Gordon equation) in geometric algebras of lower dimension.

A 2D metric space has a limited number of properties through which it can be described. This metric space may comprise objects such as a scalar, a vector, and a rank-2 tensor. The paper provides a comprehensive description of relations between objects in 2D space using the matrix product of vectors, geometric product, and dot product of complex numbers. These relations are also an integral part of the Lagrange’s identity. The entire structure of derived theoretical relationships describing properties of 2D space draws on the Lagrange’s identity. The description of how geometric algebra and tensor calculus are interconnected is given here in a comprehensive and essentially clear manner, which is the main contribution of this paper. A new term in this regard is the total geometric and matrix product, which—in a simple manner—predetermines and defines the existence of differential relations such as the gradient, the divergence, and the curl of a vector field. In addition, geometric interpretation of tensors is pointed out, expressed through angular parameters known from the literature as a tensor glyph. This angular interpretation of the tensor has an unequivocal analytical form, and the paper shows how it is linked to the classical tensor denoted by indices.

In this paper we tackle the problem of constrained rigid body dynamics in the Conformal and Projective Geometric Algebras (CGA, PGA). First we construct a screw-theory based formulation of dynamics in CGA and note the equivalence between this and the PGA dynamics presented by Gunn in [1]. After verifying the formulation via simulation, we move on to the challenge of adding constraints. First we apply the standard mechanical engineering technique of virtual power to the constraint problem in our Geometric Algebra (GA) framework. We then discuss a novel technique for ‘pinning’ dynamic rigid bodies to geometric primitives, a technique that relies on the invariance of certain multivectors and functions of multivectors to specific rotor transformations.

Orientador: Carlile Campos Lavor Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica Resumo: Neste trabalho apresentaremos um método alternativo à resolução de Problemas de Geometria de Distâncias, em que suas instâncias podem ser discretizadas. Essencialmente, estes problemas consistem em determinar um conjunto de pontos num dado espaço geométrico, no qual algumas distâncias entre dois desses pontos são conhecidas. A abordagem considerada difere da tradicional por tratar os objetos geométricos em um espaço projetivo de uma imersão conformal, do espaço euclidiano em um espaço de Minkowski (Álgebra Geométrica Conforme). Seu modelo de geometria proporciona uma interpretação natural dos objetos relevantes ao problema. Além do estudo teórico apresentado, propomos uma aplicação do método que resolve a principal classe de problemas com esta característica, conhecida na literatura por Discretizable Molecular Distance Geometry Problem - DMDGP, referente ao cálculo de estrutura tridimensional de proteínas (conformação de proteínas Abstract: In this work we present an alternative method to solve distances geometry problems in that their instances can be discretized. Essentially, these problems consists in determining a set of points in a given geometric space in which certain distances between two of these points are known. The considered approach differs from the traditional because it deals with geometric objects in a projective space of a conformal immersion of Euclidean space into a Minkowski space (Conformal Geometric Algebra). This geometric model provides a natural interpretation of the relevant objects to the problem. In addition, for the theoretical study, we propose a method for solve the most important class of this kind of problem known in the literature as Discretizable Molecular Distance Geometry Problem - DMDGP. Essentially, this class concerns the calculation of three-dimensional structure protein (conformation of proteins) Doutorado Matemática Aplicada Doutor em Matemática Aplicada

Orientador: Jayme Vaz Jr. Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica Resumo: Nessa dissertação apresentamos a Álgebra Geométrica do Espaço Euclidiano e estudamos algumas de suas propriedades. Para exemplificar suas aplicações, estudamos a Transformação Kustaanheimo-Stiefel em termos de Álgebra Geométrica. Para isso apresentamos inicialmente a Transformação KS, que regulariza o movimento de Kepler em três dimensões removendo uma singularidade na origem, da forma como foi originalmente formulada, baseando-se em álgebra de matrizes. Feito isso, a Transformação KS é apresentada com Álgebra Geométrica, o que torna o seu entendimento geométrico mais claro e seu desenvolvimento mais simplificado. Para tal o uso do conceito de spinors é de grande importância Abstract: In this dissertation we presented the Geometric Algebra of Euclidean Space and studied some of its properties. To exemplify its applications, we studied the Kustaanheimo-Stiefel Transformation in terms of Geometric Algebra. This purpose we presented initially the KS Transformation which regularizes the Kepler motion in three dimensions by removing a singularity at the origin, as it was originally formulated, based on matrix algebra. Done, the KS transformation is presented with Geometric Algebra, making clearer its geometric understanding and its development more simplified. With this goal the spinors concept use is of great importance Mestrado Física-Matemática Mestre em Matemática Aplicada

A robotic manipulator is a networked assemblage of kinematic pairs, mobile platform(s) and/or tool(s) that generate motion and forces at its end-effector. A manipulator’s theoretical dynamic, kinematic and workspace capabilities are governed by the design parameters (mass, geometry, dimensions, etc.) of its kinematic pairs and its architecture (number of limbs, degrees of freedom, actuation ability, etc.). By adjusting or changing the manipulator’s parameters (both architectural and kinematic pair-related), either from a design standpoint or physically, the robot undergoes what the authors broadly refer to as a ‘reassembling transformation’. By effecting reassembling transformations on a robot, we may generate desired changes in the workspace, kinematic and dynamic capabilities for a specific task manoeuvre. This work presents a novel methodology that marries quaternion rotors, the Lagrangian formulation, geometric algebra and associated mathematical apparatus, to tackle the problem of analysing and modelling robotic manipulators or mechanisms that undergo such ‘reassembling transformations’. Upon presenting a quaternion-based methodology for position, kinematics and Lagrangian dynamics, reassembling transformation types are introduced and detailed. We apply reassembling transformations to a Delta manipulator, in two separate case studies, each with unique target changes to the dynamic capabilities of the post transformed robot in its chosen task manoeuvres. The framework presented in this paper is generalised and its methods can be used to analyse and transform any parallel manipulator and also extended to the study of non-linear systems of equations for modelling multi-body dynamics.

Orientador: Carlile Campos Lavor Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica Resumo: O Algoritmo de Grover é um algoritmo quântico de busca em um conjunto desordenado. Com o uso de propriedades da mecânica quântica, ele apresenta um ganho quadrático em relação a um algoritmo clássico. Neste trabalho, apresentamos uma outra visão deste algoritmo, através da Álgebra Geométrica, motivados pela interpretação geométrica dos operadores, e verificamos que é possível escrevê-lo com uma nova linguagem, e ainda apresentar uma expressão mais simples para o operador de Grover (G) além de expressões gerais para estados resultantes de aplicações sucessivas deste operador Abstract: Grover¿s algorithm is a quantum algorithm for searching in unstructured databases. Due to the properties of quantum mechanics, it provides a quadratic speedup over their classical counterparts. Using the Geometric Algebra, we present a new way to understand and simplify the operators of Grover¿s algorithm Mestrado Computação Quântica Mestre em Matemática Aplicada

A formulation of the laws of reflection and refraction of optical rays by the interface of two isotropic media using the mathematical apparatus of geometric algebra is given. The cases of specular reflection, positive and negative refraction, and refraction in the presence of a metasurface are considered.

This paper presents an efficient graphical representation geometric algebra approach for the forward displacement analysis (FDA) of 3-PPS parallel mechanisms (PMs). Our approach based on conformal geometric algebra (CGA) reveals interesting intuitions and insight of the modelling and solution for FDA of this type PMs and reduces considerably computational burden and time. In this work, we first deduce the position of an arbitrary joint among three spherical joints on the moving platform through motors and represent it in terms of only one variable. Then, we formulate the position of either one joint of two remained spherical joints using the intersection of a sphere and two planes under the CGA framework. To obtain the geometric formulation of the third spherical joint position, we construct the outer product of two spatial balls with centers at two formulated spherical joints centers and two known planes. The positions of three spherical joints are all explicit geometric formulations of geometric entities of the parallel mechanisms. We achieve a coordinate-invariant equation expressed in terms of geometric entities via inner product of the last joint by itself. The symbolic 8th-degree univariate polynomial input-output equation is first proposed and all 8 sets of closed-form solutions can be obtained. The algorithm is the geometric algebra computation in a clear and coordinates-free way that avoids the use of rotation matrices, and complex algebraic modelling and nonlinear and multivariable elimination computations as most current approaches do. Finally, two numerical examples have been applied to demonstrate the efficiency of the algorithm comparing with the Dixon resultant method. The results show that this algorithm can simplify complexity of the problem and reduce computation time dramatically with strong geometrical intuition. The achieved symbolic univariate polynomial input-output equation can be applied directly to solve the DPA for all types of 3-PPS PMs instead of using any other methods. This work provides a novel geometric modeling and solution approach for the forward displacement analysis of spatial parallel mechanism and be of greatly useful for roboticists or engineers to design and develop 3-PPS PMs for kinds of applications.

Due to the increasing complexity of GIS data and service modes, there is an urgent need for the next generation of GIS with new representation and computation methods. A number of spatiotem...

This work introduces a modern and intuitive geometric language to support and to enhance the handling of different tasks in medical robot vision. By reformulating screw theory (generalization of quaternions) in the conformal geometric algebra framework, we address the hand eye calibration, 3D model registration using Kinect, interpolation, haptics, virtual reality, graphics engineering, navigation and guided surgery. The contribution of this work is the use of conformal geometric algebra to solve some key computational issues in medical robot vision without the need to leave the mathematical framework. The experimental analysis shows promising possibilities for the use of this powerful geometric language to handle multiple tasks in minimal invasive medical robotics. For this goal, we use the geometric algebra language as a vehicle between the surgeon, haptics and the organ in the virtual and real world, this language relate the surgeon approach stimulating the surgeon’s intuition based on the utilization of geometric entities and geometric properties of the organ and the surgery itself. Readers can use this geometric language for different applications in graphic engineering and robotics as well.

Presently, it is not possible to corroborate Tellegen's theorem or to articulate the compensation theorem in the frequency domain when considering all the harmonics simultaneously. The circuit analysis approach based on geometric algebra is used here to solve these two challenges. We show here the significance of representing harmonics by k -vectors and how k -vectors process the magnitude, the phase and the frequency of a sine wave. We take a tutorial approach and provide examples to demonstrate both, the simplicity of this approach and how a distinct representation of time-domain signals of different frequencies facilitates both, energy analysis and confirming the principle of superposition and Kirchhoff's circuits’ laws in non-sinusoidal conditions when considering all the harmonics simultaneously.

In modern days multi-dimensional signal processing has appeared as a trendy research area, including hyper-spectral image/video processing. It involves computationally complex algebraic equations, thereby considering conventional algebra incompetent. To alleviate this problem, Geometric Algebra (GA) establishes its competency uniquely due to its compact and less complex equations for operating in higher dimensional frameworks in contrast to the conventional one. Hardware realization is indispensible for any real time signal processing algorithm. As any GA based algorithm usually results in a significantly complex, outsized and sluggish digital circuit, it lags behind the conventional algebra in case of hardware implementation. Hence, in this article, a parallel-pipelined VLSI architecture of GA co-processor has been proposed to mitigate this intricacy, thereby becoming proficiently competent for computing a variety of GA operations. 4 channel parallelisms and 8-stage pipelining is the main attraction of our proposed design which does not necessitate any wait-state even in the situation of concurrent accessing of Memory. Supremacy of our design has been established over other state-of-the-art GA architectures in terms of number of processing cycle, latency and throughput. Superiority of our proposed design has also been proved in case of exemplary real time applications.

The magnetotelluric impedance tensor is analysed in the context of Clifford algebras. In this framework, the tensor is broken down into different parts, each one with a particular geometric algebra meaning, the simplicity of which allows us to deduce a number of known properties and opens up many other possibilities. As examples to show its capabilities, some of the algebraic relationships involving the impedance tensor, such as rotations, Mohr diagrams and phase tensor, are shown under this theory. Rotations are analysed in Clifford algebra Cl3 and Mohr diagrams, and phase tensor are expressed in Clifford algebra Cl2. The galvanic distortion matrix and its transformations are also seen in Clifford algebra Cl2, where a number of relationships allow us to recognize the galvanic distortion in a measured two‐dimensional/three‐dimensional impedance tensor. These relationships are useful as constraints to determine the galvanic distortion parameters.

This paper presents the practice of automated theorem proving in Euclidean geometry with null geometric algebra, a combination of Conformal Geometric Algebra and Grassmann-Cayley algebra. This algebra helps generating extremely short and readable proofs: The proofs generated are mostly one-termed or two-termed. Besides, the theorems are naturally extended from qualitative description to quantitative characterization by removing one or more geometric constraints from the hypotheses.

The union and the intersection of subspaces are fundamental operations in geometric algebra, while it is well known that both sum and intersection of linear subspaces in a vector space over a field are linear subspaces as well. A fundamental problem on a given family of linear subspaces is to determine the dimensions of their sum and intersection, as well as the orthogonal projectors onto the sum and the intersection of the family of linear subspaces. In this article, we first revisit an analytical formula for calculating the dimension of the intersection of a family of linear subspaces $${{\mathcal {M}}}_1, \, {{\mathcal {M}}}_2, \ldots , {\mathcal M}_k$$ in a finite-dimensional vector space, and establish some analytical expressions of the orthogonal projector onto the intersection using generalized inverses of matrices. We then establish a general formula for calculating the dimension of the sum of the family of linear subspaces. We also give some applications of these formulas in determining the maximum and minimum dimensions of the intersections of column spaces of matrix products involving generalized inverses.

Recently, Geometric Algebra (GA) has attracted more and more attention in the field of signal and image processing. GA can treat multi-dimensional signals in a holistic way to keep the correlations among multiple dimensions and avoid information loss. So when traditional signal and image processing algorithms are redefined in GA space, they will be more powerful and achieve better performance in multi-dimensional signal processing. In this paper, we provide a comprehensive survey covering various GA-based algorithms. In particular, we first review the mathematic theories of GA and Reduced Geometric Algebra (RGA). Then, advanced GA-based algorithms are elaborately analyzed and compared, including GA-based Sparse representation model, GA-based Dictionary Learning method, Clifford Support Vector Machine, GA-based Feature extraction algorithms, GA-based adaptive filtering algorithms, GA-based Fourier-type transform, and GA-based edge detection algorithm. Finally, we discuss several open issues and challenges of GA, and point out possible research directions in the future.

The Geometric Algebra formalism opens the door to developing a theory deeper than conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions, unambiguous definition of states, observables, measurements, Maxwell equations solution in that terms, bring into reality a kind of physical fields, states in the suggested theory, spreading through the whole three-dimensional space and values of the time parameter. The fields can be modified instantly in all points of space and time values, thus eliminating the concept of cause and effect and perceiving of one-directional time.

The Cell Method (CM) is an algebraic numerical method based on the use of global variables: the configuration, source and energetic global variables. The configuration variables with their topological equations, on the one hand, and the source variables with their topological equations, on the other hand, define two vector spaces that are a bialgebra and its dual algebra. The operators of these topological equations are generated by the outer product of the geometric algebra, for the primal vector space, and by the dual product of the dual algebra, for the dual vector space. The topological equations in the primal cell complex are coboundary processes on even exterior discrete p−forms, whereas the topological equations in the dual cell complex are coboundary processes on odd exterior discrete p−forms. Being expressed by coboundary processes in two different vector spaces, compatibility and equilibrium can be enforced at the same time, with compatibility enforced on the primal cell complex and equilibrium enforced on the dual cell complex. By way of example, in the present paper compatibility and equilibrium are enforced on a cantilever elastic beam with elastic inclusion. In effect, the CM shows its maximum potentialities right in domains made of several materials, as, being an algebraic approach, can treat any kind of discontinuities of the domain easily.