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We show that the gradient of a strongly differentiable function at a point is the limit of a single coordinate‐free Clifford quotient between a multidifference pseudo‐vector and a pseudo‐scalar, or of a sum of Clifford quotients between scalars (as numerators) and vectors (as denominators), both evaluated at the vertices of a same non‐degenerate simplex contracting to that point. Such result allows to fix an issue with a defective definition of pseudo‐scalar field in Sobczyck's simplicial calculus. Then, we provide some consequences and conjectures implied by the foregoing results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a generalization of Vieta theorem (Vieta's formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta formulas with the ordinary Vieta formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand–Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. We introduce the notion of a simple basis‐free formula for a determinant in geometric algebra and prove that a formula of this type exists in the case of arbitrary dimension. Using this notion, we present and prove generalized Vieta theorem in geometric algebra of arbitrary dimension. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering. [ABSTRACT FROM AUTHOR]

Oriented elements are part of geometry, and they come in two complementary types: intrinsic and extrinsic. Those different orientation types manifest themselves by behaving differently under reflection. Projective dualization in geometric algebras can encode them, or conversely, orientation types inform the interpretation of dualization. Oriented elements may be combined using the meet operation, and the dual‐join (which is here introduced for that purpose). We discuss algebra‐based implementation techniques on how to double the representational power of a geometric algebra, without explicitly doubling the algebra. [ABSTRACT FROM AUTHOR]

Protein structure prediction (PSP) is the prediction of the three‐dimensional (3D) folding of a protein (its tertiary structure) starting from its amino acid sequence (its primary structure). The state of the art in PSP is achieved by deep learning pipelines that require several input features extracted from amino acid sequences. It has been demonstrated that features that grasp the relative orientation of amino acids positively impact the prediction accuracy of the 3D coordinates of atoms in the protein backbone. In this paper, we demonstrate the relevance of geometric algebra (GA) in instantiating orientational features for PSP problems. We do so by proposing two novel GA‐based metrics which contain information on relative orientations of amino acid residues. We then employ these metrics as additional input features to a graph transformer (GT) architecture to aid the prediction of the 3D coordinates of a protein, and compare them to classical angle‐based metrics. We show how our GA features yield comparable results to angle maps in terms of accuracy of the predicted coordinates. This is despite being constructed from less initial information about the protein backbone. The features are also fewer and more informative and can be (i) closely associated to protein secondary structures and (ii) more easily predicted compared to angle maps. We hence deduce that GA can be employed as a tool to simplify the modeling of protein structures and pack orientational information in a more natural and meaningful way. [ABSTRACT FROM AUTHOR]

Using the classification of Clifford algebras and Bott periodicity, we show how higher geometric algebras can be realized as matrices over classical low dimensional geometric algebras. This matrix representation allows us to use standard geometric algebra software packages more easily. As an example, we express the geometric algebra for conics (GAC) as a matrix over the Compass ruler algebra (CRA). [ABSTRACT FROM AUTHOR]

To achieve 3D skeleton-based human motion prediction, attention-based methods have encouraged performance due to the observation that attention parts in the past state influence future actions. The model can identify the most relevant information for motion prediction by introducing an attention mechanism. However, existing methods tend to address the general allocation of attention without further exploiting the precision to pinpoint the exact location of the most relevant information. This oversight subsequently curtails the potential of prediction performance. To solve this problem, we propose a novel Gradient Multi-Foci Network (GMFnet), which leverages two-stage foci: spectral focus and spatial focus, to find the most pertinent information in a manner that emulates natural cognitive processes. The core idea of the proposed GMFnet is based on two aspects: spectral focus to model the repeatability of the observation action sequence by deploying an attention-based Related Sequences Directing Block (RSDB), spatial focus to capture the most valuable parts between motion joints by using Attention Feature Computational Unit (AFCU). Extensive experiments are conducted to reveal that GMFnet can capture the precision to pinpoint the exact attention location, thus enhancing the prediction performance. The proposed GMFnet outperforms state-of-the-art methods by 10.7 and 7.4% of MPJPEs for short-term and long-term prediction in Human 3.6M and by 14.3 and 4.8% of MPJPEs for short-term and long-term forecast in CMU-Mocap. Moreover, GMFnet outperforms even more in short term by 24.9% in AMASS. The code is available at https://github.com/JunyuShi02/GMFNet. [ABSTRACT FROM AUTHOR]

The study of signal processing has recently devoted significantly more attention to adaptive filtering techniques. By addressing the shortcoming of the conventional geometric algebra-based fixed step-size least mean square algorithm that is unable to satisfy in terms of both reducing steady-state error and a faster convergence speed, simultaneously, this study presents an improved logarithmic function-based variable step-size least mean square geometric algebra adaptive filtering algorithm by establishing the step-size factor μ and error signal e(n) nonlinear function relationship. The instantaneous values of a current error estimate e(n) and the previous error estimate e (n - 1) are used to determine the step size of the defined algorithm. Besides, an extensive discussion is given on the performance of algorithm under influence of parameters γ and T as well as comparative analysis with other existing geometric algebra-based adaptive filters. Computer simulation reveals that the proposed approach not only has a low steady-state error, robustness against impulsive noise, and fast convergence speed, but it also overcomes some existing algorithm's instability under steady-state phase. [ABSTRACT FROM AUTHOR]

With regards to the problem of multidimensional signal processing in the field of adaptive filtering, geometric algebra based higher-order statistics algorithms were proposed. For instance, to express a multidimensional signal as a multi-vector, the adaptive filtering algorithms described in this work leverage all of the benefits of GA theory in multidimensional signal processing. GA space is employed to extend the traditional least-mean absolute third (LMAT) and newly deduced least-mean mixed third-fourth (LMMTF) adaptive filtering methods for multi-dimensional signal processing. The objective of the presented GA-based least-mean absolute third (GA-LMAT) and GA-based least-mean mixed third-fourth (GA-LMMTF) algorithms is to minimize the cost functions by using higher-order statistics of the error signal e(n) in GA space. The simulation's results revealed that at significantly smaller step size, the given GA-LMAT algorithm is better than the others in terms of steady-state error and convergence rate. Besides, the defined GA-LMMTF algorithm mitigates for the instability of GA-LMAT as the step size increases and illustrates an improved performance relative to mean absolute error and convergence rate. [ABSTRACT FROM AUTHOR]

The Geometric Algebra Fulcrum Library (GA-FuL) version 1.0 is introduced in this paper as a comprehensive computational library for geometric algebra (GA) and Clifford algebra (CA), in addition to other classical algebras. As a sophisticated software system, GA-FuL is useful for practical applications requiring numerical or symbolic prototyping, optimized code generation, and geometric visualization. A comprehensive overview of the GA-FuL design is provided, including its core design intentions, data-driven programming characteristics, and extensible layered design. The library is capable of representing and manipulating sparse multivectors of any dimension, scalar kind, or metric signature, including conformal and projective geometric algebras. Several practical and illustrative use cases of the library are provided to highlight its potential for mathematical, scientific, and engineering applications. The metaprogramming code optimization capabilities of GA-FuL are found to be unique among other software systems. This allows for the automated production of highly efficient code, based on powerful geometric modeling formulations provided by geometric algebra. [ABSTRACT FROM AUTHOR]

Historically and to date, the continuity equation (C.E.) has served as a consistency criterion for the development of physical theories. In this paper, we study the C.E. employing the mathematical framework of space–time algebra (STA), showing how common equations in mathematical physics can be identified and derived from the C.E.'s structure. We show that, in STA, the nabla equation given by the geometric product between the vector derivative operator and a generalized multivector can be identified as a system of scalar and vectorial C.E.—and, thus, another form of the C.E. itself. Associated with this continuity system, decoupling conditions are determined, and a system of wave equations and the generalized analogous quantities to the energy–momentum vectors and the Lorentz force density (and their corresponding C.E.) are constructed. From the symmetry transformations that make the C.E. system's structure invariant, a system with the structure of Maxwell's field equations is derived. This indicates that a Maxwellian system can be derived not only from the nabla equation and the generalized continuity system as special cases, but also from the symmetries of the C.E. structure. Upon reduction to well-known simpler quantities, the results found are consistent with the usual STA treatment of electrodynamics and hydrodynamics. The diffusion equation is explored from the continuity system, where it is found that, for decoupled systems with constant or explicitly dependent diffusion coefficients, the absence of external vector sources implies a loss in the diffusion equation structure, transforming it into Helmholtz-like and wave equations. [ABSTRACT FROM AUTHOR]

Multi-loop coupling mechanisms (MCMs) have been widely used in architectural design antennas and space-deployable. This paper proposes a general method for actuation selection of MCMs using geometric algebra. The constraint space is obtained by collineation and dual operators of twist space. This method determines additional constraints by analyzing the linear correlation between the constrained spaces of the two limbs within the closed-loop and making corrections to the constraint space of the basic limb. Through outer and dual operators, the input screw corresponding to the transmission wrench screw and output twist screw are obtained. The input transmission index (ITI) and output transmission index (OTI) corresponding to this input are calculated, which leads to generation of a local transmission index (LTI). The actuation selection with the best performance can be obtained by comparing and analyzing the motion/force transfer performance of all input combinations. The proposed method has the advantages of simple computation and clear physical significance, and three typical MCMs are applied to validate this method. Finally, an experimental analysis compared the power consumption of a test motor at different actuation positions, further proving the accuracy of the method. [ABSTRACT FROM AUTHOR]

In this article I argue that i is a quantity associated with the two-dimensional real number plane, whether as a vector, a bi-vector, a point or a transformation (rotation). This position provides a foundation for the complex numbers and accounts for complex numbers in some equations of applied mathematics and physics. I also argue that complex numbers are fundamentally geometrical and can be described by geometric algebra, and that moreover the meaning of complex numbers in physics varies with dimension and geometry of the manifold. [ABSTRACT FROM AUTHOR]

In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related structures such as Hermitian conjugation, Euclidean space, and Lie groups in geometric algebras. The results can be used in various applications of geometric algebras in computer science, engineering, and physics. [ABSTRACT FROM AUTHOR]

From viewpoints of crystallography and of elementary particles, we explore symmetries of multivectors in the geometric algebra Cl(3, 1) that can be used to describe space-time. [ABSTRACT FROM AUTHOR]

Detecting the concavity and convexity of three-dimensional (3D) geometric objects is a well-established challenge in the realm of computer graphics. Serving as the cornerstone for various related graphics algorithms and operations, researchers have put forth numerous algorithms for discerning the concavity and convexity of such objects. The majority of existing methods primarily rely on Euclidean geometry, determining concavity and convexity by calculating the vertices of these objects. However, within the realm of Euclidean geometric space, there exists a lack of uniformity in the expression and calculation rules for geometric objects of differing dimensions. Consequently, distinct concavity and convexity detection algorithms must be tailored for geometric objects with varying dimensions. This approach inevitably results in heightened complexity and instability within the algorithmic structure. To address these aforementioned issues, this paper introduces geometric algebra theory into the domain of concavity and convexity detection within 3D spatial objects. With the algorithms devised in this study, it becomes feasible to detect concavity and convexity for geometric objects of varying dimensions, all based on a uniform set of criteria. In comparison to concavity-convexity detection algorithms grounded in Euclidean geometry, this research effectively streamlines the algorithmic structure. [ABSTRACT FROM AUTHOR]

Multi-loop coupling mechanisms (MCMs) have been widely used in spacedeployable antennas. However, the mobility of MCMs is difficult to analyze due to their complicated structure and coupled limbs. This paper proposes a general method for calculating the mobility of MCMs using geometric algebra (GA). For the independent limbs in the MCM, the twist spaces are constructed by the join operator. For coupled limbs coupled with closed loops in the MCM, the equivalent limbs can be found by solving the analytical expressions of the twist space on each closed loop’s output link. Then, the twist spaces of the coupled limbs can be easily obtained. The twist space of the MCM’s output link is the intersection of all the limb twist spaces, which can be calculated by the meet operator. The proposed method provides a simplified way of analyzing the mobility of MCMs, and three typical MCMs are chosen to validate this method. The analytical mobility of the MCM’s output link can be obtained, and it naturally indicates both the number and the property of the degrees of freedom (DOFs). [ABSTRACT FROM AUTHOR]

Hand dysfunction is a common symptom in stroke patients. This paper presents a robotic device which assists the rehabilitation process in order to reduce the need of physical therapy, i.e., a 3UPS/S parallel robotic device is employed for repetitive robot-assisted rehabilitation. Euler angle representation was used to solve the robot's inverse kinematics. The robot's joint space and rotational workspace were determined for two scenarios. In the first scenario, the workspace was obtained considering the actuator's stroke limitations, while in the second scenario, the workspace was determined by adding a second condition, i.e., the range of motion of the spherical joints. Singularity analysis was performed using the geometric algebra approach. The robot was manufactured using additive manufacturing technology. The solution of the inverse kinematic problem was employed to control the robot. The robot can perform a full range of motion during wrist ulnar deviation and radial deviation motions, with the exception of limited wrist flexion and extension motions. The robot has singular configurations within its workspace. Although the spherical joints have roles in reducing the workspace, the primary causes are actuator selection, radii of the base and moving platforms, and the length of the central leg. These factors can be considered to improve the workspace. Singularity can be avoided by carefully selecting the rotation of the moving platform about the Z-axis and avoiding same leg lengths. [ABSTRACT FROM AUTHOR]

Using the powerful language of geometric algebra, we present an observationally symmetric derivation of the strong correlations predicted by the entangled singlet state in a deterministic and locally causal model, usually also referred to as a local-realistic model, in which the physical space is assumed to be a quaternionic 3-sphere, or S 3 , available as the spatial part of a solution of Einstein’s field equations of general relativity, and compare it in quantitative detail with Bell’s local-realistic model for the singlet correlations set within a flat Euclidean space I R 3 . Since the quantitatively detailed expressions of relative-angle-dependent probabilities of observing measurement outcomes for Bell’s local model do not seem to have been fully articulated before, our novel analysis exploiting the non-commutative properties of quaternions, in addition to allowing the comparison with the quaternionic 3-sphere model, may also provide useful comparisons for other less compelling local-realistic models, such as those relying on retrocausality or superdeterminism. Apart from the conservation of zero spin angular momentum, the key attribute underlying the strong singlet correlations within S 3 in comparison with Bell’s local model turns out to be the spinorial sign changes intrinsic to quaternions that constitute the 3-sphere. In addition, we also discuss anew a macroscopic experiment that can, in principle, test our 3-sphere hypothesis. [ABSTRACT FROM AUTHOR]

Quantum simulation qubit models of electronic Hamiltonians rely on specific transformations in order to take into account the fermionic permutation properties of electrons. These transformations (principally the Jordan–Wigner transformation (JWT) and the Bravyi–Kitaev transformation) correspond in a quantum circuit to the introduction of a supplementary circuit level. In order to include the fermionic properties in a more straightforward way in quantum computations, we propose to use methods issued from Geometric Algebra (GA), which, due to its commutation properties, are well adapted for fermionic systems. First, we apply the Witt basis method in GA to reformulate the JWT in this framework and use this formulation to express various quantum gates. We then rewrite the general one and two-electron Hamiltonian and use it for building a quantum simulation circuit for the Hydrogen molecule. Finally, the quantum Ising Hamiltonian, widely used in quantum simulation, is reformulated in this framework. [ABSTRACT FROM AUTHOR]

Is there more to Dirac's gamma matrices than meets the eye? It turns out that gamma zero can be factorized into a product of three operators. This revelation facilitates the expansion of Dirac's space-time algebra to Clifford algebra Cl (0 , 6). The resultant rich geometric structure can be leveraged to establish a combined framework of the standard model and gravity, wherein a gravi-weak interaction between the vierbein field and the extended weak gauge field is allowed. Inspired by the composite Higgs model, we examine the vierbein field as an effective description of the fermion–antifermion condensation. The compositeness of space-time manifests itself at an energy scale which is different from the Planck scale. We propose that all the regular classical Lagrangian terms are of quantum condensation origin, thus possibly addressing the cosmological constant problem provided that we exercise extreme caution in the renormalization procedure that entails multiplications of divergent integrals. The Clifford algebra approach also permits a weaker form of charge conjugation without particle–antiparticle interchange, which leads to a Majorana-type mass that conserves lepton number. Additionally, in the context of spontaneous breaking of two global U (1) symmetries, we explore a three-Higgs-doublet model which could explain the fermion mass hierarchies. [ABSTRACT FROM AUTHOR]

The instantaneous reactive power theory (IRPT) has long been utilized for load compensation through active power line conditioners (APLCs). IRPT effectively divides the current vector into two components: the instantaneous power current and the instantaneous reactive current. Among these components, only the former is necessary to transfer instantaneous real power to the load. However, the commonly adopted approach faces challenges when extended to multi-phase systems and their compensation strategy design. This challenge is particularly pertinent today due to the consensus that treating a three-phase system with a neutral wire as a four-conductor system is more appropriate. This paper introduces a formulation of IRPT tailored for multi-phase systems within the framework of geometric algebra (GA). GA is a mathematical structure that defines a single power variable, the instantaneous power multivector, encompassing both instantaneous real power and instantaneous reactive power within a unified mathematical entity. The current components can be directly derived from this power multivector. Additionally, this paper establishes a connection with the original p-q formulations and lays the foundations for time-instantaneous compensation (TIC) and time-average compensation (TAC). Finally, to validate the proposed model, simulation and experimental results from a three-phase four-wire industrial system using a novel approach are presented.

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In this work, we propose a geometric algebra‐based variation of a well‐known threshold secret‐sharing scheme introduced by Adi Shamir in 1979. Secret sharing is a cryptographic primitive which allows a secret input to be divided into multiple shares which are then sent to a collection of parties. The shares are generated so that only "authorized" sets of shares can reconstruct the secret. In Shamir's scheme, any sufficiently large set of shares can reconstruct the secret. The minimum number of shares which can obtain the secret is called the threshold, and any number of shares smaller than the threshold reveals nothing about the secret. The shares are generated such that each party can perform computations, generating a new set of shares that, when reconstructed, are equivalent to performing those exact computations directly on the secret input data. Our variant changes the domain from which secrets are taken: A finite field with prime order is replaced by a geometric algebra over a finite field of prime order. This change preserves the important security properties of Shamir's scheme, namely, idealness (secrets and shares are chosen from the same space) and perfectness ("unauthorized" sets of shares learn nothing about the secret). Our scheme allows secret sharing to be seamlessly added to the arsenal of GA‐based applications. Our extension of Shamir's secret scheme was first worked out for geometric algebras. It appears, however, that in fact it works for other algebras, a situation worthy to be explored in future work. For definiteness, in this paper, we restrict the analysis to the case of geometric algebras. [ABSTRACT FROM AUTHOR]

If an initial frame of vectors {ei}$$ \left\{{e}_i\right\} $$ is related to a final frame of vectors {fi}$$ \left\{{f}_i\right\} $$ 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. [ABSTRACT FROM AUTHOR]

Different ways of representing the group SU(3)$$ SU(3) $$ within a Geometric Algebra approach are explored. As part of this, we consider characteristic multivectors for SU(3)$$ SU(3) $$ and how these are linked with decomposition of generators into commuting bivectors. The setting for this work is within a 6d Euclidean Clifford Algebra. We then go on to consider whether the fundamental forces of particle physics might arise from symmetry considerations in just the 4d geometric algebra of spacetime—the STA. As part of this, a representation of SU(3)$$ SU(3) $$ is found wholly within the STA, involving preservation of a bivector norm. We also show how Octonions can be fully represented within the Spacetime Algebra, which we believe will be useful in making them understandable and accessible to a new community in Physics and Engineering. The two strands of the paper are drawn together in showing how preserving the octonion norm is the same as preserving the timelike part of the Dirac current of a particle. This suggests a new model for the symmetries preserved in particle physics. Following on from work by Günaydin and Gürsey on the link between quarks, and octonions, and by Furey on chains of octonionic multiplications, we show how both of these fit well within our scheme and give some wholly STA versions of the operations involved, which in the cases considered have easily understandable equivalents in terms of 4d geometry. Links with larger groups containing SU(3)$$ SU(3) $$, such as G2$$ {G}_2 $$ and SU(8)$$ SU(8) $$, are also considered. [ABSTRACT FROM AUTHOR]

An algorithm for finding a control of a linear switched system by means of Geometric Algebra is designed. More precisely, we develop a switching path searching algorithm for a two‐dimensional linear dynamical switched system with a non‐singular matrix whose integral curves are formed by two sets of centralized ellipses. It is natural to represent them as elements of Geometric Algebra for Conics and construct the switching path by calculating switching points, i.e., intersections and contact points. For this, we use symbolic algebra operations or, more precisely, the wedge and inner products that are realizable by sums of products in the coordinate form. Therefore, no numerical solver to the system of equations is needed. Indeed, the only operation that may bring in an inaccuracy is vector normalization, i.e., square root calculation. The resulting switching path is formed by pieces of ellipses that are chosen, respectively, from the two sets of integral curves. The switching points are either intersections in the first or final step of our algorithm, or contact points. This choice guarantees the optimality of the switching path with respect to the number of switches. Two examples are provided to demonstrate the search for the intersections of the conics and, consequently, a description is presented of the construction of a switching path in both cases. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Geometric algebras of dimension n<6$$ n<6 $$ are becoming increasingly popular for the modeling of 3D and 3 + 1D geometry. With this increased popularity comes the need for efficient algorithms for common operations such as normalization, square roots, and exponential and logarithmic maps. The current work presents a signature‐agnostic analysis of these common operations in all geometric algebras of dimension n<6$$ n<6 $$ and gives efficient numerical implementations in the most popular algebras ℝ4,ℝ3,1,ℝ3,0,1$$ {\mathbb{R}}_4,{\mathbb{R}}_{3,1},{\mathbb{R}}_{3,0,1} $$, and ℝ4,1$$ {\mathbb{R}}_{4,1} $$, in the hopes of lowering the threshold for adoption of geometric algebra solutions by code maintainers. [ABSTRACT FROM AUTHOR]

This paper presents some new Lie groups preserving fixed subspaces of geometric algebras (or Clifford algebras) under the twisted adjoint representation. We consider the cases of subspaces of fixed grades and subspaces determined by the grade involution and the reversion. Some of the considered Lie groups can be interpreted as generalizations of Lipschitz groups and spin groups. The Lipschitz groups and the spin groups are subgroups of these Lie groups and coincide with them in the cases of small dimensions. We study the corresponding Lie algebras. [ABSTRACT FROM AUTHOR]

As an addition to proper points of the real plane, we introduce a representation of improper points, i.e. points at infinity, in terms of Geometric Algebra for Conics (GAC) and offer possible use of both types of points. More precisely, we present two algorithms fitting a conic to a dataset with a certain number of points lying on the conic precisely, referred to as the waypoints. Furthermore, we consider inclusion of one or two improper waypoints, which leads to the asymptotic directions of the fitted conic. The number of used waypoints may be up to four and we classify all the cases. [ABSTRACT FROM AUTHOR]