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s This paper proposes an analytical method based on conformal geometrical algebra to implement and unify type synthesis of one translational and two rotational (1T2R) parallel mechanisms and two rotational and one translational (2R1T) parallel mechanisms. Firstly, conformal geometrical algebra is introduced in this paper to describe finite motions and carry out type synthesis of parallel mechanisms in an algebraic way. Secondly, the finite motions of 1T2R and 2R1T parallel mechanisms are described and their relations are explored, and this results in unifying the type synthesis of these two types of parallel mechanisms in the same procedure. Then, a parametric and algebraic generation method is proposed to obtain available limbs and axes layout of joints with these limbs. Finally, the assembly conditions among available limbs are given to synthesize systematically 2R1T parallel mechanisms with non-overconstrained or overconstrained property, as well as 1T2R parallel mechanisms with the same properties. This method proposed in this paper is valid in carrying out finite motion representation and type synthesis of parallel mechanisms in an algebraic and analytical way, and then is beneficial to an automatic manner using computer programming languages. Furthermore, the method may solve type synthesis of special parallel mechanisms, such as parallel mechanisms with varied output rotational axes.

This paper proposes a new procedure for the construction of Voronoi diagrams using spheres. The auxiliary spheres that assist in the diagram construction are defined following the concept of a “hollow sphere”, and their geometrical properties, within the Conformal Geometric Algebra model, turn out to be crucial elements for increasing the global efficiency procedure with complexity $$O(n \log n)$$ . Several examples are presented, using the CLUCalc software, that show the effectiveness and ability of the proposed procedure. Also, we argue that the Conformal Geometric Algebra model provides straightforward and intuitive concepts to the field of Computational Geometry, that prove invaluable for allowing the incremental definition and construction of Voronoi diagrams.

Free-form motions in B-spline form can be created from a number of prescribed control poses using the de Casteljau algorithm. With poses defined using conformal geometric algebra, it is natural to combine poses multiplicatively. Additive combinations offer alternative freedoms in design and avoid dealing with noninteger exponents. This paper investigates additive combinations and shows how to modify the conventional conformal geometric algebra definitions to allow such combinations to be well-defined. The additive and multiplicative approaches are compared and in general they generate similar motions, with the additive approach offering computational simplicity.

International audience; This paper presents an efficient implementation of geometric algebra, based on a recursive representation of the algebra elements using binary trees. The proposed approach consists in restructuring a state of the art recursive algorithm to handle parallel optimizations. The resulting algorithm is described for the outer product and the geometric product. The proposed implementation is usable for any dimensions, including high dimension (e.g. algebra of dimension 15). The method is compared with the main state of the art geometric algebra implementations , with a time complexity study as well as a practical benchmark. The tests show that our implementation is at least as fast as the main geometric algebra implementations.

We apply the conformal geometric algebra (CGA) to the generalized binocular vision problem. More precisely, we reconstruct a 3D line from its images on the image planes of two cameras whose mutual position is specified by a given Euclidean transformation which depends on an arbitrary number of parameters. We represent all transformations by CGA elements which allows us to derive the general equations of 3D line reconstruction by formal CGA elements manipulation. The transformation equations can be solved w.r.t. either motor or projection unknown parameters. We present two specific examples, show the explicit form of two particular motors and solve the appropriate equations completely.

Representations of geometric (Clifford) algebras with real square matrices are reviewed by providing the general theorem as well as examples of lowest dimensions. New definitions for isometry and norm are proposed. Direct and indirect isometries are identified respectively with automorphisms and antiautomorphisms of the geometric algebra, while the norm of every element is defined as the $n^\textit{th}$-root of the absolute value of the determinant of its matrix representation of order $n$. It is deduced in which geometric algebras direct isometries are inner automorphisms (similarity transformations of matrices). Indirect isometries need reversion too. Finally, the most common isometries are reviewed in order to write them in this way.

In this paper we present a novel method for nonlinear rigid body motion estimation from noisy data using heterogeneous sets of objects of the conformal model in geometric algebra. The rigid body motions are represented by motors. We employ state-of-the-art nonlinear optimization tools and compute gradients and Jacobian matrices using forward-mode automatic differentiation based on dual numbers. The use of automatic differentiation enables us to employ a wide range of cost functions in the estimation process. This includes cost functions for motor estimation using points, lines and planes. Moreover, we explain how these cost functions make it possible to use other geometric objects in the conformal model in the motor estimation process, e.g., spheres, circles and tangent vectors. Experimental results show that we are able to successfully estimate rigid body motions from synthetic datasets of heterogeneous sets of conformal objects including a combination of points, lines and planes. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

We demonstrate the theory on the 1-link trident snake and the functionality in the CLUCalc software designed for the computations in Clifford algebra. Local control of a general trident snake robot is solved by means of conformal geometric algebra. It is shown that the model modifications are much easier to handle in this setting. Within this paper, we present an alternative model description only, while all its kinematic properties remain. The equations of the direct and differential kinematics, the Pfaff constraints, the inverse kinematics and the singular postures are discussed and translated into the language of CGA.

Local controllability of an n-link robotic snake with variable wheel positions is solved by means of the conformal geometric algebra, more precisely by the Clifford algebra of signature (3, 1). The non-holonomic kinematic equations are assembled, their role in the geometric control theory is discussed and the singular positions are elaborated. Within this paper, we present an alternative model description only, while all its kinematic properties remain.

This paper is an introduction to geometric algebra and geometric calculus, presented in the simplest way I could manage, without worrying too much about completeness or rigor. An understanding of linear algebra and vector calculus is presumed. This should be sufficient to read most of the paper.

Geometric algebra has been called a “unified language for mathematics and physics.” Sometimes known as Clifford algebra, it is based on the notion of an invertible product of vectors that captures the geometric relationship between two vectors, i.e., their relative magnitudes and the angle between them. This seemingly simple concept leads to a rich system of algebra and calculus that encompasses the diverse areas of complex numbers, quaternions, vectors, tensors, spinors, and differential forms. This tutorial provides a basic introduction to geometric algebra and presents formulations of known electrical engineering and signal processing concepts to illustrate some inherent advantages of geometric algebra for formulating and solving problems involving vectors. Being introductory, the goal of the tutorial is to introduce this emerging area that, although old as a mathematics discipline, has only recently started to garner significant attention in engineering communities. Geometric algebra should give another potentially powerful tool for pursuing research in any area that uses vectors.

This chapter gives a detailed outline of geometric algebra and explains the related traditional algebras in common use by mathematicians, physicists, computer scientists, and engineers.

To be a qualified engineer, or to pursue further studies in electronics and information engineering, undergraduate students should attempt to broaden their horizons in their professional fields. Particularly important are emerging mathematical theories or tools that produce relevant mathematical formalism for modelling the nature of physical space as simply and compactly as possible. In this article, we illustrate that geometric algebra provides just such a simplified and elegant representation or description of physical space. The basics of geometric algebra are explained and concrete examples in the plane and in 3D space are given. This article also provides two examples where geometric algebra has been found to provide an improved description of coordinate rotations and the Maxwell equations.

Conformal Geometric Algebra has been introduced into geographic information science as a mathematical theory because of its advantages in terms of uniform multidimensional representation and computation. The traditional intersection computation between two geometric objects of different types is not unified. In this study, we propose algorithms based on Conformal Geometric Algebra to determine the spatial relationships between geographic objects in a unified manner. The unified representation and intersection computation can be realized for geometric objects of different dimensions. Different basic judgment rules are provided for different simple geometries. The algorithms are designed and implemented using MapReduce to improve the efficiency of the algorithms. From the results of several experiments we provide, the correctness and effectiveness of the algorithms can be verified.

Clifford's Geometric Algebra provides an elegant formulation of Maxwell's equations in the space-time setting. Its clear geometric interpretation is used to derive a goal function, whose minimization results in Hodge-optimized material matrices being diagonal or diagonal-dominant. Effectively it is an optimization of the primal/dual mesh pair of a finite difference based discretization scheme taking into account the material properties. As a research example, a standing wave in 2D cavity filled with an anisotropic material is investigated. Convergence of the scheme for various choices of mesh pairs is discussed. The limitations of the method in the 3D case are presented.

To analyze the electromagnetic wave propagation in a medium the Maxwell equations should be supplemented by constitutive relations. At present the classification of linear constitutive relations is well established in tensorial-matrix and exterior p -form calculus. Here the constitutive relations are found in the context of Clifford geometric algebra. For this purpose Cl 1,3 algebra that conforms with relativistic 4D Minkowskian spacetime is used. It is shown that the classification of linear optical phenomena with the help of constitutive relations in this case comes from the structure of Cl 1,3 algebra itself. Concrete expressions for constitutive relations which follow from this algebra are presented. They can be applied in calculating the propagation properties of electromagnetic waves in any anisotropic, linear and nondissipative medium.

Traditional methods for geometric entities resort to the Hough transform and tensor voting schemes for detect lines and circles. In this work, the authors extend these approaches using representations in terms of k-vectors of the Conformal Geometric Algebra. Of interest is the detection of lines and circles in images, and planes, circles, and spheres in the 3-D visual space; for that, we use the randomized Hough transform, and by means of k-blades we code such geometric entities. Motivated by tensor voting, we have generalized this approach for any kind of geometric entities or geometric flags formulating the perceptual saliency function involving k-vectors. The experiments using real images show the performance of the algorithms.

A complex vector is a sum of a vector and a bivector and forms a natural extension of a vector. The complex vectors have certain special geometric properties and considered as algebraic entities. These represent rotations along with specified orientation and direction in space. It has been shown that the association of complex vector with its conjugate generates complex vector space and the corresponding basis elements defined from the complex vector and its conjugate form a closed complex four dimensional linear space. The complexification process in complex vector space allows the generation of higher n-dimensional geometric algebra from (n — 1)-dimensional algebra by considering the unit pseudoscalar identification with square root of minus one. The spacetime algebra can be generated from the geometric algebra by considering a vector equal to square root of plus one. The applications of complex vector algebra are discussed mainly in the electromagnetic theory and in the dynamics of an elementary particle with extended structure. Complex vector formalism simplifies the expressions and elucidates geometrical understanding of the basic concepts. The analysis shows that the existence of spin transforms a classical oscillator into a quantum oscillator. In conclusion the classical mechanics combined with zeropoint field leads to quantum mechanics.

Clustering is one of the most useful methods for understanding similarity among data. However, most conventional clustering methods do not pay sufficient attention to the geometric distributions of data. Geometric algebra (GA) is a generalization of complex numbers and quaternions able to describe spatial objects and the geometric relations between them. This paper uses conformal GA (CGA), which is a part of GA. This paper transforms data from a real Euclidean vector space into a CGA space and presents a new clustering method using conformal vectors. In particular, this paper shows that the proposed method was able to extract the geometric clusters which could not be detected by conventional methods.

A parallel mechanism with redundancy can be regarded as a means for not only maximizing the benefits of parallel mechanisms but also overcoming their drawbacks. We proposed a novel parallel mechanism by eliminating an unnecessary degree of freedom of the configuration space. Because of redundancy, however, the solution for the inverse kinematics of the developed parallel mechanism is infinite. Therefore, we defined a cost function that can minimize the movement time to the target orientation and found the solution for the inverse kinematics by using a numerical method. In addition, we proposed a method for determining the boundary of the geometric singularity in order to avoid singularities.

A generalized model of geometry based upon the idea of using blades of a geometric algebra as representatives of geometry is developed. Similar to the idea of an algebraic set, the main object of study becomes the notion of a geometric set. Results about geometric sets are obtained, and their implications discussed. Being a generalization, results obtained are immediately applicable to well-known models of geometry, including the homogeneous and conformal models, as well as any specific future model that falls under the generalization.

In this paper, we propose a novel solution process for the forward kinematics of general 3-RPS parallel mechanisms based on conformal geometric algebra (CGA). First of all, the position of one of the three spherical joints is expressed in the rotational angle about the axis of one kinematic chain RPS. Secondly, the other two spherical joints can be determined by this angle via CGA operation. Thirdly, an explicit 16th-degree univariate polynomial equation is reduced from two geometric constraint equations. At last, one numerical example is employed to verify the solution procedure. The novelties of this paper lie in that (1) the modeling formulation and the elimination procedure have intrinsic geometric intuition due to the use of CGA and (2) the 16th-degree polynomial equation without extraneous roots is explicitly formulated and suitable for implementing the mathematics mechanization of this problem.

In this paper, we explicate the suggested benefits of Clifford's geometric algebra (GA) when applied to the field of electrical engineering. Engineers are always interested in keeping formulas as simple or compact as possible, and we illustrate that geometric algebra does provide such a simplified representation in many cases. We also demonstrate an additional structural check provided by GA for formulas in addition to the usual checking of physical dimensions. Naturally, there is an initial learning curve when applying a new method, but it appears to be worth the effort, as we show significantly simplified formulas, greater intuition, and improved problem solving in many cases.

This paper introduces a new non-Euclidean geometry, that consist in a generalization of conformal geometry G 4,1. In this geometry, it is possible to handle not only spheres, but also quadratic surfaces and their intersections easily as well. The Clifford algebra G 6,3 is being used as the framework, which allows the creation of a nine dimensional geometry with some additional transformations, i.e. anisotropic dilatation, allowing rotations for all G 4,1 entities. It also eases the use of quadratic surfaces including conics in the 3D space.

It is found that all polynomials of up to degree m have an encoding as m-vectors in a geometric algebra referred to as the Mother Minkowski algebra of order m. It is then shown that all conformal transformations may be applied to these m-vectors, the results of which, when converted back into polynomial form, give us the transformed surfaces in terms of the zero sets of the original and final polynomials.

Clifford algebra is introduced as a theoretical foundation for network topology expression and algorithm construction. Network nodes are coded with basis vectors in a vector space , and the edges and k-walk routes can be expressed by 2-blades and k-blades, respectively, in the Clifford algebra Cl(n,0). The topologies among nodes, edges, and routes of networks can be directly calculated, and the network routes can be extended and traversed with oriented join products. The network algorithm construction processes based on Clifford algebra are instantiated by the single source shortest path algorithm. The experimental results on different scale random networks suggest that Clifford algebra is suited for network expression and relation computation. The Clifford algebra-based shortest path algorithm is vivid and clear in geometric meaning and has great advantage on temporal and spatial complexity. Copyright © 2013 John Wiley & Sons, Ltd.

We present various methods that have reduced the finite set of missing geometric configurations (n"4). We show that there is no geometric configuration (17"4), and we provide examples for the former unknown cases n=18, n=29, n=31 - there do exist geometric configurations (n"4). To construct these we use methods from computer algebra and optimization.

Linear algebra is the language describing systems in finite (or countably infinite) dimensions, where dimension represents the number of variables at hand.

In the subsequent sections we survey results from combinatorics, discrete geometry and commutative algebra concerning invariants and properties of subdivisions of simplicial complexes. For most of the time we are interested in deriving results that hold for specific subdivision operations that are motivated from combinatorics, geometry and algebra. In particular, we study barycentric, edgewise and interval subdivisions (see Sect. 3 for the respective definitions). Even though we mention some suspicion that part of the results we present may only be a glimpse of what is true for general subdivision operations we do not focus on this aspect. In particular, we are quite sure that some asymptotic results and some convergence results from Sect. 9 are just instances of more general phenomena. Overall, retriangulations are subtle geometric operations and we refer the reader to the book De Loera et al. (Algorithms and Computation in Mathematics. Springer, Heidelberg, 2010) for a comprehensive introduction. Since our focus lies on specific constructions we make only little use of the theory from De Loera et al. (Algorithms and Computation in Mathematics. Springer, Heidelberg, 2010). Nevertheless, we are convinced that if one wants to go beyond specific subdivision operations it will become inevitable to dig deeper into the theory of triangulations.

The purpose of this paper is to provide a general algebraic and geometric structure for the set of matrices. First, the identity equivalence is introduced to the set. Then two operations STA and STP are introduced, which are the generalizations of conventional matrix addition and matrix product respectively. Using them, the set of matrices is partitioned into several vector spaces. A topological structure is proposed to make each vector space a second countable, Hausdorff space. Then a differential structure is posed to make each vector space a differential manifold with multi-dimensions. Finally, the multi-dimensional Lie group and Lie algebra structures are obtained for the set of all square matrices. Some applications are also explored.

We introduce Clifford geometric algebra based multivector modeling of quartic and general quadric surfaces, Darboux cyclides, Dupin cyclies, tori and pairs of standard CGA objects. A computer visualization is briefly described.

Connection between optical Mueller matrices and geometrical (Clifford) algebra multivectors is established. It is shown that starting from 3-dimensional (3D) Cl 3,0 algebra and using isomorphism between Cl 3,0 and even Cl 3,1 + subalgebra one can generate canonical Mueller matrices and their combinations that describe an optical system. It appears that representation of polarization devices in terms of geometric algebra is very compact and, in contrast to Mueller matrix approach, there is no need for speculative physical restrictions. If needed, properties of media can be logically introduced into Maxwell equation in a form of Clifford algebra via constitutive relations. Since representation of polarization by Cl 3,1 algebra is Lorentz invariant it allows to include relativistic effects of moving bodies on light polarization as well. In this paper only simple examples of connection between Mueller matrices and geometric algebra multivectors is presented.

International audience; Multivectorial algebra is of both academic and technological interest. Its application, however, is not always easy. A distinction must be made between polar and axial vectors and between scalars and pseudo-scalars. Eight element types are often considered even if they are not always identified as multivectors. In some cases, for simplicity’s sake, only vectorial algebra or quaternion algebra is explicitly used for physical and mechanical applications. It would, however, be more convenient to use more complex algebra directly in order to have a wider range of mechanical applications. The aim of this paper is to examine one particular type of Clifford algebra that could solve this problem. The present study focusses on showing how these quantities can be used to model mechanical and engineering problems. First, continuum mechanics in a Cauchy medium is investigated for elastic transformations. Second, a specific type of shot-peening application is studied. Applications are then used to illustrate the scope and efficiency of this type of modeling based on geometric algebra.

The geometric significance of the imaginary unit in a complex geometric algebra has troubled the author for 40 years. In the unitary geometric algebra presented here, the imaginary i is a unit (pseudo) vector with square minus one which anti commutes with all of the real vectors. The resulting natural hermitian inner product and hermitian outer product induce a grading of the algebra into complex k-vectors. Basic orthogonality relationships are studied.

We introduce a geometric process algebra based on affine geometry, with the aim of describing the concurrent evolution of geometric structures in 3D space. We prove a relativity theorem stating that algebraic equations are invariant under rigid body transformations.

Clustering is one of the most useful methods for understanding similarity among data. However, most conventional clustering methods do not pay sufficient attention to the geometric properties of data. Geometric algebra (GA) is a generalization of complex numbers and quaternions able to describe spatial objects and the relations between them. This paper uses conformal GA (CGA), which is a part of GA, to transform a vector in a real vector space into a vector in a CGA space and presents a proposed new clustering method using conformal vectors. In particular, this paper shows that the proposed method was able to extract the geometric clusters which could not be detected by conventional methods., 2011 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2011), June 27-30, 2011, Grand Hyatt Taipei, Taipei, Taiwan

A new mathematical tool, the geometric algebra of Euclidean 3-space (G3), is introduced for electromagnetic vector-sensor array processing herein. This paper focuses on modeling the six-component outputs of a vector-sensor holistically by an entry called as a multivector in G3. A compact polarized model for the array, termed as a geometric algebra model (G-MODEL), is then presented. Using the G-MODEL, a novel data covariance matrix model is defined by the geometric products in G3 and then analyzed. The analytical results show that the six-component measurement noise of a vector-sensor can naturally be whitened if the noise cross-correlations between the different axial electric and magnetic components are equal to one another. Compared with the known best quad-quaternion model, the new covariance matrix model results in a reduction of half memory requirements while the amount of divisions is reduced to 1/2, multiplications and additions reduced to almost 1/7.

In this paper we investigate how, using the language of Geometric Algebra [7, 4], the common escape-time Julia and Mandelbrot set fractals can be extended to arbitrary dimension and, uniquely, non-Eulidean geometries. We develop a geometric analog of complex numbers and show how existing ray-tracing techniques [2] can be extended. In addition, via the use of the Conformal Model for Geometric Algebra, we develop an analog of complex arithmetic for the Poincare disc and show that, in non-Euclidean geometries, there are two related but distinct variants of the Julia and Mandelbrot sets.

In this paper, we discuss boundary behavior for the Cauchy type integrals with values in a universal Clifford algebra for certain distinguished boundary and obtain some Sochocki–Plemelj formulae and Privalov–Muskhelishvili theorems.

It is known that Clifford (geometric) algebra offers a geometric interpretation for square roots of –1 in the form of blades that square to –1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Research has been done [1] on the biquaternion roots of –1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra Cl3 of \({{\mathbb R^3}}\) . All these roots of –1 find immediate applications in the construction of new types of geometric Clifford Fourier transformations.

Introducing Clifford algebra as the mathematical foundation, a unified spatio-temporal data model and hierarchical spatio-temporal index are constructed by linking basic data objects, like pointclouds and Spatio-Temporal Hyper Cubes of different dimensions, within the multivector structure of Clifford algebra. The transformation from geographic space into homogeneous and conformal space means that geometric, metric and many other kinds of operators of Clifford algebra can be implemented and we then design the shortest path, high-dimensional Voronoi and unified spatial-temporal process analyses with spacetime algebra. Tests with real world data suggest these traditional GIS analysis algorithms can be extended and constructed under Clifford Algebra framework, which can accommodate multiple dimensions. The prototype software system CAUSTA (Clifford Algebra based Unified Spatial-Temporal Analysis) provides a useful tool for investigating and modeling the distribution characteristics and dynamic process of complex geographical phenomena under the unified spatio-temporal structure.