Use Of Geometric Algebra Research Articles

Analysis of non-active power in non-sinusoidal circuits using geometric algebra

A new approach for the definition of non-active power in electrical systems is presented in this paper. Through the use of geometric algebra, it is possible to define a new term called geometric non-active power, which is applicable to both sinusoidal and non-sinusoidal systems, and to both linear and nonlinear loads. The classic definitions of distortion and reactive power are compared and discussed in our proposal. We verify how geometric non-active power can appear in both purely resistive and purely reactive systems. The superiority of geometric algebra is revealed through several examples of electrical circuits previously analysed in specialised literature. Furthermore, a new geometrical current decomposition is proposed, for the first time, to provide a greater physical sense to existing geometric power. The results obtained confirm that classic concepts based on apparent power S are based on a lack of physical meaning, which is why geometric algebra theory should be adopted instead.

Geometric Algebra Computing for Heterogeneous Systems

Starting from the situation 15 years ago with a great gap between the low symbolic complexity on the one hand and the high numeric complexity of coding in Geometric Algebra on the other hand, this paper reviews some applications showing, that, in the meantime, this gap could be closed, especially for CPUs. Today, the use of Geometric Algebra in engineering applications relies heavily on the availability of software solutions for the new heterogeneous computing architectures. While most of the Geometric Algebra tools are restricted to CPU focused programming languages, in this paper, we introduce the new Gaalop (Geometric Algebra algorithms optimizer) Precompiler for heterogeneous systems (CPUs, GPUs, FPGAs, DSPs ...) based on the programming language C++ AMP (Accelerated Massive Parallelism) of the HSA (Heterogeneous System Architecture) Foundation. As a proof-of-concept we present a raytracing application together with some computing details and first performance results.

Using geometric algebra to represent and interpolate tool poses

In conventional computer-aided manufacturing techniques, a tool path is sent to the machine tool controller as a sequence of precision points. This means that errors are introduced when the controller interpolates between the points to recover the intended path. For 4- and 5-axis machining, a sequence of precision poses (positions and orientations) is needed. The use of geometric algebra in representing poses is discussed. The form of algebra used can represent rigid-body transforms exactly and can interpolate, in a natural way, between two or more poses using spherical linear interpolation. Sequences of poses from free-form surfaces are investigated. Tool motions are generated and compared with tool paths derived from purely positional information. It is found that the motion paths more naturally follow the original surface so that the interpolation errors are smaller.

The Non-Degenerate Dupin Cyclides in the Space of Spheres Using Geometric Algebra

Dupin cyclides are algebraic surfaces of degree 4 discovered by the French mathematician Pierre-Charles Dupin early in the 19th century and were introduced in CAD by R. Martin in 1982. A Dupin cyclide can be defined, in two different ways, as the envelope of a oneparameter family of oriented spheres. So, it is very interesting to model the Dupin cyclides in the space of spheres, space wherein each family of spheres can be seen as a conic curve. In this paper, we model the nondegenerate Dupin cyclides and the space of spheres using Conformal Geometric Algebra. This new approach permits us to benefit from the advantages of the use of Geometric Algebra.

The Use of Geometric Algebra for 3D Modeling and Registration of Medical Data

We present a new approach to model 2D surfaces and 3D volumetric data, as well as an approach for non-rigid registration; both are developed in the geometric algebra framework. The approach for modeling is based on marching cubes idea using however spheres and their representation in the conformal geometric algebra; it will be called marching spheres. Note that before we can proceed with the modeling, it is needed to segment the object we are interested in; therefore, we include an approach for image segmentation, which is based on texture and border information, developed in a region-growing strategy. We compare the results obtained with our modeling approach against the results obtained with other approach using Delaunay tetrahedrization, and our proposed approach reduces considerably the number of spheres. Afterward, a method for non-rigid registration of models based on spheres is presented. Registration is done in an annealing scheme, as in Thin-Plate Spline Robust Point Matching (TPS-RPM) algorithm. As a final application of geometric algebra, we track in real time objects involved in surgical procedures.

Hidden geometric character of relativistic quantum mechanics

Geometry can be an unsuspected source of equations with physical relevance, as everybody is aware since Einstein formulated the general theory of relativity. However, efforts to extend a similar type of reasoning to other areas of physics, namely, electrodynamics, quantum mechanics, and particle physics, usually had very limited success; particularly in quantum mechanics the standard formalism is such that any possible relation to geometry is impossible to detect; other authors have previously trod the geometric path to quantum mechanics, some of that work being referred to in the text. In this presentation we will follow an alternate route to show that quantum mechanics has indeed a strong geometric character. The paper makes use of geometric algebra, also known as Clifford algebra, in five-dimensional space-time. The choice of this space is given the character of first principle, justified solely by the consequences that can be derived from such choice and their consistency with experimental results. Given a metric space of any dimension, one can define monogenic functions, the natural extension of analytic functions to higher dimensions; such functions have null vector derivative and have previously been shown by other authors to play a decisive role in lower dimensional spaces. All monogenic functions have null Laplacian by consequence; in a hyperbolic space this fact leads inevitably to a wave equation with planelike solutions. This is also true for five-dimensional space-time and we will explore those solutions, establishing a parallel with the solutions of the free particle Dirac equation. For this purpose we will invoke the isomorphism between the complex algebra of 4×4 matrices, also known as Dirac’s matrices. There is one problem with this isomorphism, because the solutions to Dirac’s equation are usually known as spinors (column matrices) that do not belong to the 4×4 matrix algebra and as such are excluded from the isomorphism. We will show that a solution in terms of Dirac spinors is equivalent to a plane wave solution. Just as one finds in the standard formulation, monogenic functions can be naturally split into positive∕negative energy together with left∕right ones. This split is provided by geometric projectors and we will show that there is a second set of projectors providing an alternate fourfold split. The possible implications of this alternate split are not yet fully understood and are presently the subject of profound research.

Geometric algebra and transition‐selective implementations of the controlled‐NOT gate

AbstractGeometric algebra provides a complete set of simple rules for the manipulation of product operator expressions at a symbolic level, without any explicit use of matrices. This approach can be used not only to describe the state and evolution of a spin system, but also to derive the effective Hamiltonian and associated propagator in full generality. In this article, we illustrate the use of geometric algebra via a detailed analysis of transition‐selective implementations of the controlled‐NOT gate, which plays a key role in NMR‐based quantum information processing. In the appendices, we show how one can also use geometric algebra to derive tight bounds on the magnitudes of the errors associated with these implementations of the controlled‐NOT. © 2004 Wiley Periodicals, Inc. Concepts Magn Reson Part A 23A: 49–62, 2004

Use of geometric algebra: compound matrices and the determinant of the sum of two matrices

In this paper we demonstrate the capabilities of geometric algebra by the derivation of a formula for the determinant of the sum of two matrices in which both matrices are separated in the sense that the resulting expression consists of a sum of traces of products of their compound matrices. For the derivation we introduce a vector of Grassmann elements associated with an arbitrary square matrix, we recall the concept of compound matrices and summarize some of their properties. This paper introduces a new derivation and interpretation of the relationship between p-forms and the pth compound matrix, and it demonstrates the use of geometric algebra, which has the potential to be applied to a wide range of problems.

Geometric Algebra in Linear Algebra and Geometry

This article explores the use of geometric algebra in linear and multilinear algebra, and in affine, projective and conformal geometries. Our principal objective is to show how the rich algebraic tools of geometric algebra are fully compatible with and augment the more traditional tools of matrix algebra. The novel concept of an h-twistor makes possible a simple new proof of the striking relationship between conformal transformations in a pseudo-Euclidean space to isometries in a pseudo-Euclidean space of two higher dimensions. The utility of the h-twistor concept, which is a generalization of the idea of a Penrose twistor to a pseudo-Euclidean space of arbitrary signature, is amply demonstrated in a new treatment of the Schwarzian derivative.

A HAMILTONIAN FORMULATION OF GRAVITATIONAL THEORY THAT ALLOWS ONE TO CONSIDER CURVATURE AND TORSION AS CONJUGATE VARIABLES

We consider a quadratic Lagrangian, in both curvature and torsion, with the aim of exploring the possibility that torsion and curvature behave as conjugate variables satisfying the commutation relations. For that proposal we first show that torsion represents a quantum correction to the classical equations of motion. We then observe that we have to introduce the spin in the Einstein theory with two spaces: a real space-time where we describe the curvature with tensors; and a complex space-time, where we describe torsion with spinors. This is not satisfactory, and to overcome this contradiction we describe both mass and spin in a unique manifold, the real space-time, with the use of geometric algebra and geometric product. In that manner we are able to introduce a torsion trivector and curvature trivector that satisfy the commutation relations, opening the road for a quantum description of gravity theory.

An introduction to geometric algebra with an application in rigid body mechanics

This paper presents a tutorial of geometric algebra, a very useful but generally unappreciated extension of vector algebra. The emphasis is on physical interpretation of the algebra and motives for developing this extension, and not on mathematical rigor. The description of rotations is developed and compared with descriptions using vector and matrix algebra. The use of geometric algebra in physics is illustrated by solving an elementary problem in classical mechanics, the motion of a freely spinning axially symmetric rigid body.