Please read the abstract in the section front of this document. / Thesis (PhD)--University of Pretoria, 2010. / Mathematics and Applied Mathematics / unrestricted

Session ID: FR-G2-2 / Joint 4th International Conference on Soft Computing and Intelligent Systems and 9th International Symposium on advanced Intelligent Systems, September 17-21, 2008, Nagoya University, Nagoya, Japan

No description available.

2011 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2011), June 27-30, 2011, Grand Hyatt Taipei, Taipei, Taiwan

Statistical analysis of 3-dimensional motions of humans, animals or objects is instrumental to establish how these motions differ, depending on various influences or parameters. When such motions involve no stretching or tearing, they may be described by the elements of a Lie group called the Special Euclidean Group, denoted SE(3). Statistical analysis of trajectories lying in SE(3) is complicated by the basic properties of the group, such as non-commutativity, non-compactness and lack of a bi-invariant metric. This necessitates the generalization of the ideas of “mean” and “variance” to apply in this setting. We describe how to exploit the unique properties of a formalism called Conformal Geometric Algebra to express these generalizations and carry out such statistical analyses efficiently; we introduce a practical method of visualizing trajectories lying in the 6-dimensional group manifold of SE(3); and we show how this methodology can be applied, for example, in testing theoretical claims about the influence of an attended object on a competing action applied to a different object. The two prevailing views of such movements differ as to whether mental action-representations evoked by an object held in working memory should perturb only the early stages of subsequently reaching to grasp another object, or whether the perturbation should persist over the entire movement. Our method yields “difference trajectories” in SE(3), representing the continuous effect of a variable of interest on an action, revealing statistical effects on the forward progress of the hand as well as a corresponding effect on the hand’s rotation. / Graduate / 0405 / 0541 / 0623

In this paper the Clifford Algebra is introduced and proposed as analternative to Gibbs' vector algebra as a unifying language for geometricoperations on vectors. Firstly, the algebra is constructed using a quotientof the tensor algebra and then its most important properties are proved,including how it enables division between vectors and how it is connected tothe exterior algebra. Further, the Clifford algebra is shown to naturallyembody the complex numbers and quaternions, whereupon its strength indescribing rotations is highlighted. Moreover, the wedge product, is shown asa way to generalize the cross product and reveal the true nature ofpseudovectors as bivectors. Lastly, we show how replacing the cross productwith the wedge product, within the Clifford algebra, naturally leads tosimplifying Maxwell's equations to a single equation.

L'algèbre géométrique est un outil permettant de représenter et manipuler les objets géométriques de manière générique, efficace et intuitive. A titre d'exemple, l'Algèbre Géométrique Conforme (CGA), permet de représenter des cercles, des sphères, des plans et des droites comme des objets algébriques. Les intersections entre ces objets sont incluses dans la même algèbre. Il est possible d'exprimer et de traiter des objets géométriques plus complexes comme des coniques, des surfaces quadriques en utilisant une extension de CGA. Cependant due à leur représentation requérant un espace vectoriel de haute dimension, les implantations de l'algèbre géométrique, actuellement disponible, n'autorisent pas une utilisation efficace de ces objets. Dans ce manuscrit, nous présentons tout d'abord une implantation de l'algèbre géométrique dédiée aux espaces vectoriels aussi bien basses que hautes dimensions. L'approche suivie est basée sur une solution hybride de code pré-calculé en vue d'une exécution rapide pour des espaces vectoriels de basses dimensions, ce qui est similaire aux approches de l'état de l'art. Pour des espaces vectoriels de haute dimension, nous proposons des méthodes de calculs ne nécessitant que peu de mémoire. Pour ces espaces, nous introduisons un formalisme récursif et prouvons que les algorithmes associés sont efficaces en termes de complexité de calcul et complexité de mémoire. Par ailleurs, des règles sont définies pour sélectionner la méthode la plus appropriée. Ces règles sont basées sur la dimension de l'espace vectoriel considéré. Nous montrons que l'implantation obtenue est bien adaptée pour les espaces vectoriels de hautes dimensions (espace vectoriel de dimension 15) et ceux de basses dimensions. La dernière partie est dédiée à une représentation efficace des surfaces quadriques en utilisant l'algèbre géométrique. Nous étudions un nouveau modèle en algèbre géométrique de l'espace vectoriel $mathbb{R}^{9,6}$ pour manipuler les surfaces quadriques. Dans ce modèle, une surface quadrique est construite par l'intermédiaire de neuf points. Nous montrerons que ce modèle permet non seulement de représenter de manière intuitive des surfaces quadriques mais aussi de construire des objets en utilisant les définitions de CGA. Nous présentons le calcul de l'intersection de surfaces quadriques, du vecteur normal, du plan tangent à une surface en un point de cette surface. Enfin, un modèle complet de traitement des surfaces quadriques est détaillé / Geometric Algebra is considered as a very intuitive tool to deal with geometric problems and it appears to be increasingly efficient and useful to deal with computer graphics problems. The Conformal Geometric Algebra includes circles, spheres, planes and lines as algebraic objects, and intersections between these objects are also algebraic objects. More complex objects such as conics, quadric surfaces can also be expressed and be manipulated using an extension of the conformal Geometric Algebra. However due to the high dimension of their representations in Geometric Algebra, implementations of Geometric Algebra that are currently available do not allow efficient realizations of these objects. In this thesis, we first present a Geometric Algebra implementation dedicated for both low and high dimensions. The proposed method is a hybrid solution that includes precomputed code with fast execution for low dimensional vector space, which is somehow equivalent to the state of the art method. For high dimensional vector spaces, we propose runtime computations with low memory requirement. For these high dimensional vector spaces, we introduce new recursive scheme and we prove that associated algorithms are efficient both in terms of computationnal and memory complexity. Furthermore, some rules are defined to select the most appropriate choice, according to the dimension of the algebra and the type of multivectors involved in the product. We will show that the resulting implementation is well suited for high dimensional spaces (e.g. algebra of dimension 15) as well as for lower dimensional spaces. The next part presents an efficient representation of quadric surfaces using Geometric Algebra. We define a novel Geometric Algebra framework, the Geometric Algebra of $mathbb{R}^{9,6}$ to deal with quadric surfaces where an arbitrary quadric surface is constructed by merely the outer product of nine points. We show that the proposed framework enables us not only to intuitively represent quadric surfaces but also to construct objects using Conformal Geometric Algebra. In the proposed framework, the computation of the intersection of quadric surfaces, the normal vector, and the tangent plane of a quadric surface are provided. Finally, a computational framework of the quadric surfaces will be presented with the main operations required in computer graphics

Freeform curve design has existed in various forms for at least two millennia, and is important throughout computer-aided design and manufacture. With the increasing importance of animation and robotics, coupled with the increasing power of computers, there is now interest in freeform motion design, which, in part, extends techniques from curve design, as well as introducing some entirely distinct challenges. There are several approaches to freeform motion construction, and the first step in designing freeform motions is to choose a representation. Unlike for curves, there is no "standard" way of representing freeform motions, and the different tools available each have different properties. A motion can be viewed as a continuously-varying pose, where a pose is a position and an orientation. This immediately presents a problem; the dimensions of rotations and translations are different, and it is not clear how the two can be compared, such as to define distance along a motion. One solution is to treat the rotational and translational components of a motion separately, but this is inelegant and clumsy. The philosophy of this thesis is that a motion is not defined purely by rotations and translations, but that the body following a motion is a part of that motion. Specifically, the part of the body that is accounted for is its inertia tensor. The significance of the inertia tensor is that it allows the rotational and translational parts of a motion to be, in some sense, compared in a dimensionally- consistent way. Using the inertia tensor, this thesis finds the form of kinetic energy in <;1'4, and also discusses extensions of the concepts of arc length and curvature to the space of motions, allowing techniques from curve fairing to be applied to motion fairing. Two measures of motion fairness are constructed, and motion fairing is the process of minimizing the measure of a motion by adjusting degrees of freedom present in the motion's construction. This thesis uses the geometric algebra <;1'4 in the generation offreeform motions, and the fairing of such motions. <;1'4 is chosen for its particular elegance in representing rigid-body transforms, coupled with an equivalence relation between elements representing transforms more general than for ordinary homogeneous coordinates. The properties of the algebra germane to freeform motion design and improvement are given, and two distinct frameworks for freeform motion construction and modification are studied in detail.

Made available in DSpace on 2015-04-22T22:16:03Z (GMT). No. of bitstreams: 1 ieda.pdf: 330424 bytes, checksum: a628e679a50a8f50be962ac95af15d1f (MD5) Previous issue date: 2009-02-27 / This work will present the Plane Geometric Algebra, according Grassmann postulate, starting the axioms of plane euclidean geometry. / Este trabalho apresenta a Álgebra Geométrica do Plano, de acordo com a proposta de Grassmann, a partir dos axiomas da geometria euclidiana plana.

A quarter of the world's population experience wheezing. These sounds have been used for diagnosis since the time of the Ebers Papyrus (ca. 1500 BC), but the underlying physical mechanism responsible for the sounds is still poorly understood. The main purpose of this thesis is to change this, developing a theory for the onset of wheezing using both experimental and analytical approaches, with implications for both scientific understanding and clinical diagnosis. Wheezing is caused by a fluid structure interaction between the airways and the air flowing through them. We have developed the first systematic set of experiments of direct relevance to this physical phenomena. We have also developed new tools in shell theory using geometric algebra to improve our physical understanding of the self-excited oscillations observed when air flows through flexible tubes. In shell theory, the use of rotors from geometric algebra has enabled us to develop improved physical understanding of how changes of curvature, which are of direct importance to constitutive laws, come about. This has enabled a scaling analysis to be applied to the self-excited oscillations of flexible tubes, showing for the first time that bending energy is dominated by strain energy. We made novel use of multiple camera reconstruction to validate this scaling analysis by directly measuring the bending and strain energies during oscillations. The dominance of strain energy allows a simplification of the governing shell equations. We have developed the first theory for the onset of self-excited oscillations of flexible tubes based on a flutter instability. This has been validated with our experimental work, and provides a predictive tool that can be used to understand wheezing in the airways of the lung. Our theory for the onset of wheezing relates the frequency of oscillation to the airway geometry and material properties. This will allow diagnoses based on wheezing sounds to become more specific, which will allow the stethoscope, which has changed little in the last 200 years, to be brought into the 21st century.