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Dissertations / Theses on the topic 'Geometric algebra for conics'

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Published: 28 June 2021

Last updated: 3 February 2022

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1

Machálek, Lukáš. "Aplikace geometrických algeber." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2021. http://www.nusl.cz/ntk/nusl-445454.

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Tato diplomová práce se zabývá využitím geometrické algebry pro kuželosečky (GAC) v autonomní navigaci, prezentované na pohybu robota v trubici. Nejprve jsou zavedeny teoretické pojmy z geometrických algeber. Následně jsou prezentovány kuželosečky v GAC. Dále je provedena implementace enginu, který je schopný provádět základní operace v GAC, včetně zobrazování kuželoseček zadaných v kontextu GAC. Nakonec je ukázán algoritmus, který odhadne osu trubice pomocí bodů, které umístí do prostoru pomocí středů elips, umístěných v obrazu, získaných obrazovým filtrem a fitovacím algoritmem.

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2

Ellis, Amanda. "Classification of conics in the tropical projective plane /." Diss., CLICK HERE for online access, 2005. http://contentdm.lib.byu.edu/ETD/image/etd1104.pdf.

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3

Ellis, Amanda. "Classifcation of Conics in the Tropical Projective Plane." BYU ScholarsArchive, 2005. https://scholarsarchive.byu.edu/etd/697.

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This paper defines tropical projective space, TP^n, and the tropical general linear group TPGL(n). After discussing some simple examples of tropical polynomials and their hypersurfaces, a strategy is given for finding all conics in the tropical projective plane. The classification of conics and an analysis of the coefficient space corresponding to such conics is given.

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4

Lopes, Wilder Bezerra. "Geometric-algebra adaptive filters." Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/3/3142/tde-22092016-143525/.

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This document introduces a new class of adaptive filters, namely Geometric- Algebra Adaptive Filters (GAAFs). Those are generated by formulating the underlying minimization problem (a least-squares cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well-suited for the description of geometric transformations. Also, differently from the usual linear algebra approach, Geometric Calculus (the extension of Geometric Algebra to differential calculus) allows to apply the same derivation techniques regardless of the type (subalgebra) of the data, i.e., real, complex-numbers, quaternions etc. Exploiting those characteristics, among others, a general leastsquares cost function is posed, from which two types of GAAFs are designed. The first one, called standard, provides a generalization of regular adaptive filters for any subalgebra of GA. From the obtained update rule, it is shown how to recover the following least-mean squares (LMS) adaptive filter variants: real-entries LMS, complex LMS, and quaternions LMS. Mean-square analysis and simulations in a system identification scenario are provided, showing almost perfect agreement for different levels of measurement noise. The second type, called pose estimation, is designed to estimate rigid transformations { rotation and translation - in n-dimensional spaces. The GA-LMS performance is assessed in a 3-dimensional registration problem, in which it is able to estimate the rigid transformation that aligns two point clouds that share common parts.
Este documento introduz uma nova classe de filtros adaptativos, entitulados Geometric-Algebra Adaptive Filters (GAAFs). Eles s~ao projetados via formulação do problema de minimização (uma função custo de mínimos quadrados) do ponto de vista de álgebra geométrica (GA), uma abrangente linguagem matemática apropriada para a descrição de transformações geométricas. Adicionalmente, diferente do que ocorre na formulação com álgebra linear, cálculo geométrico (a extensão de álgebra geométrica que possibilita o uso de cálculo diferencial) permite aplicar as mesmas técnicas de derivação independentemente do tipo de dados (subálgebra), isto é, números reais, números complexos, quaternions etc. Usando essas e outras características, uma função custo geral de mínimos quadrados é proposta, da qual dois tipos de GAAFs são gerados. O primeiro, chamado standard, generaliza filtros adaptativos da literatura concebidos sob a perspectiva de subálgebras de GA. As seguintes variantes do filtro least-mean squares (LMS) s~ao obtidas como casos particulares: LMS real, LMS complexo e LMS quaternions. Uma análise mean-square é desenvolvida e corroborada por simulações para diferentes níveis de ruído de medição em um cenário de identificação de sistemas. O segundo tipo, chamado pose estimation, é projetado para estimar transformações rígidas - rotação e translação { em espaços n-dimensionais. A performance do filtro GA-LMS é avaliada em uma aplicação de alinhamento tridimensional na qual ele estima a tranformação rígida que alinha duas nuvens de pontos com partes em comum.

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5

Kessaris, Haris. "Geometric algebra and applications." Thesis, University of Cambridge, 2001. https://www.repository.cam.ac.uk/handle/1810/251756.

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6

MOREIRA, JOHANN SENRA. "CONSTRUCTION OF THE CONICS USING THE GEOMETRIC DRAWING AND CONCRETE INSTRUMENTS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2017. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=33061@1.

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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
PROGRAMA DE MESTRADO PROFISSIONAL EM MATEMÁTICA EM REDE NACIONAL
O presente trabalho tem como objetivo facilitar o estudo das cônicas e ainda despertar o interesse do aluno para o desenho geométrico. Será apresentado que as curvas cônicas estão em nosso dia a dia, não só como beleza estética, mas também provocando fenômenos físicos amplamente utilizado pela arquitetura e engenharia civil, como acústica e reflexão da luz. Utilizaremos instrumentos para desenhar curvas que despertem a curiosidade dos alunos e faremos uso das equações e lugares geométricos a fim de demostrar tais recursos. Pretende-se assim que ao adquirir tais conhecimentos o aluno aprimore seu entendimento matemático e amplie seu horizonte cultural.
The present research aims to facilitate the study of the conics and also to arouse the interest of the student for the geometric drawing. The conic curves will be presented not only as they are in our day to day as aesthetic beauty but also as responsible for the physical phenomena widely used by architecture and civil engineering as well as acoustics and reflection of light. We will use instruments to draw curves that arouse the curiosity of the students, making use of the equations and locus in order to demonstrate such resources. It is intended that the student acquire this knowledge, improving his mathematical understanding and broadening his cultural horizon.

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7

Minh, Tuan Pham, Tomohiro Yoshikawa, Takeshi Furuhashi, and Kaita Tachibana. "Robust feature extractions from geometric data using geometric algebra." IEEE, 2009. http://hdl.handle.net/2237/13896.

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8

Khalfallah, Hazem. "Mordell-Weil theorem and the rank of elliptical curves." CSUSB ScholarWorks, 2007. https://scholarworks.lib.csusb.edu/etd-project/3119.

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The purpose of this thesis is to give a detailed group theoretic proof of the rank formula in a more general setting. By using the proof of Mordell-Weil theorem, a formula for the rank of the elliptical curves in certain cases over algebraic number fields can be obtained and computable.

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Wu, Junhua. "Geometric structures and linear codes related to conics in classical projective planes of odd orders." Access to citation, abstract and download form provided by ProQuest Information and Learning Company; downloadable PDF file, 105 p, 2009. http://proquest.umi.com/pqdweb?did=1654490971&sid=2&Fmt=2&clientId=8331&RQT=309&VName=PQD.

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10

Wareham, Richard James. "Computer graphics using conformal geometric algebra." Thesis, University of Cambridge, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.612753.

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11

Hildenbrand, Dietmar. "Geometric computing in computer graphics and robotics using conformal geometric algebra." Phd thesis, [S.l.] : [s.n.], 2007. http://elib.tu-darmstadt.de/diss/000764.

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Ashdown, M. A. J. "Geometric algebra, group theory and theoretical physics." Thesis, University of Cambridge, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.596181.

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This dissertation applies the language of geometric algebra to group theory and theoretical physics. Geometric algebra, which is introduced in Chapter 2, provides a natural extension of the concept of multiplication from real numbers to geometric objects such as line segments and planes. It is based on Clifford algebra and augmented by auxiliary definitions which give it a geometric interpretation. Since geometric algebra provides a natural encoding of the concepts of directed quantities, it has the potential to unify many of the disparate systems of notation that are used in mathematics. In Chapter 3, the properties of multilinear functions are investigated and the theory is developed to make them useful for formulating the representation of groups. It will be found that multilinear functions are more flexible than their tensor or matrix counterparts in traditional linear algebra. Multilinear functions can be classified according to the symmetry class of their arguments and their behaviour under the monogenic or harmonic decomposition. It is found that the previous definitions of monogenic and harmonic functions need some modification if they are to be defined consistently. Polynomial projection is also discussed, a technique that is useful in constructing non-linear functions from linear functions, an operation outside the scope of conventional linear algebra. In Chapter 4, multilinear functions are used to construct the irreducible representations of the three regular classes of classical groups; rotation groups, the special unitary and special linear group, and the symplectic group. In each case it is found that a decomposition must be applied to the multilinear functions in order to find the irreducible representations of the groups. For the representations of some of the groups this entails finding the harmonic or monogenic parts of the functions. The groups can be realised as subgroups of the spin group of some dimension and signature. However, geometric algebra provides such a rich algebraic structure that the representations of the groups can be realised in more than one way. In Chapter 7 a brief review is given of computer software for performing symbolic calculations with geometric algebra. A new software package which performs semi-symbolic manipulation of multivectors in spaces of any dimension and signature is presented.

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13

Setiawan, Sandi. "Applications of geometric algebra to black holes." Thesis, University of Cambridge, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.621990.

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Perwass, Christian Bernd Ulrich. "Applications of geometric algebra in computer vision." Thesis, University of Cambridge, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.621941.

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15

Biggs, Adam Marc. "Geometric structures on the algebra of densities." Thesis, University of Manchester, 2014. https://www.research.manchester.ac.uk/portal/en/theses/geometric-structures-on-the-algebra-of-densities(c2356ed0-2fe5-45e9-9807-c920535a8f04).html.

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The algebra of densities can be seen to have origins dating back to the 19th century where densities were used to find invariants of the modular group. Since then they have continued to be a source of projective invariants and cocycles related with the projective group, most notably the Schwarzian derivative. One of the first times that the algebra of densities appears in the literature in a similar guise to the way we shall introduce it, is in the work of T.Y. Thomas. He showed that a projective connection on a manifold allows one to determine a canonical affine connection on the total space of a certain bundle which is now known as Thomas' bundle. More recently they have appeared, with the definition we shall use, by H. Khudaverdian and Th. Voronov when studying second order operators generating certain brackets. Of prime importance in this situation is the case of Gerstenhaber algebras and in particular the Batalin-Vilkovisky operator on the odd cotangent bundle. They have also been used by V.Y. Ovsienko and his group in the area of equivariant quantization which is a topic we shall come across in the text. Densities also regularly appear in physics. For example the correct interpretation of a wavefunction is a half-density on a manifold, and this explains their transformation properties under the Galilean group. These results motivate a study into the geometric structure of the algebra of densities as an object in their own right. We shall see that by considering them as a whole algebra many classical results have a clear geometrical picture. Moreover one finds that there are a wealth of areas within this algebra still to explore. We focused on two fundamental classes of objects, differential operators and Poisson structures. The results we find lead to interesting formula for certain equivariantly defined differential operators which can be applied to gain a wide class of cocycles similar to the Schwarzian derivative. We also find very intimate links with Batalin-Vilkovisky geometry and the methods we use show that it may be useful to consider the full algebra of densities when entering into this arena.

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Lundkvist, Christian. "Moduli spaces of zero-dimensional geometric objects." Doctoral thesis, Stockholm : Matematik, Kungliga Tekniska högskolan, 2009. http://www.diva-portal.org/smash/record.jsf?searchId=1&pid=diva2:223079.

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Furuhashi, Takeshi, Tomohiro Yoshikawa, Kanta Tachibana, and Minh Tuan Pham. "A Clustering Method for Geometric Data based on Approximation using Conformal Geometric Algebra." IEEE, 2011. http://hdl.handle.net/2237/20706.

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18

Challinor, A. "Applications of geometric algebra in physics and cosmology." Thesis, University of Cambridge, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.597396.

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Geometric algebra - a unified language for mathematics and physics which takes Clifford algebra as its grammar - is slowly gaining the recognition it deserves in the physics community. The advantages of geometric algebra over existing techniques are being demonstrated continually through its application to a wide variety of topics in modern physics. This thesis adds to the growing body of applications of geometric algebra by considering problems in cosmology and relativistic quantum theory. The aim is to bring fresh insight and novel resolutions to the problems considered by exploiting the conceptual and computational advantages afforded by formulating the underlying theory in the language of geometric algebra. The applications to relativistic quantum theory include a reconsideration of the tunnelling time problem, where a resolution is offered which has significant implications for fields such as quantum measurement theory and solid state device physics. Also included is the developed of models of the early universe based on spin - ½ fields coupled to gravity through their inertia and quantum spin, where new, exact solutions to the Einstein-Cartan-Dirac equations are given. The treatment here employs a gauge-theoretic approach to gravity, developed recently in Cambridge by Lasenby, Doran, & Gull, which has a very elegant formulation in geometric algebra, and simplifies the treatment of many problems in general relativity and its spin-torsion extension. As a prelude to the discussion of spin-1/2 fields in the early universe, the torsion sector of the gauge theory of Lasenby et al. is explored thoroughly, and a number of new results are obtained. The applications to cosmology focus on the development of covariant methods for the description of inhomogeneity and anisotropy in the universe. A reformulation of the covariant approach to cosmology of Ehlers and Ellis, and the perturbation theory of Ellis & Bruni derived from it, is given using the gauge theory of gravity, resulting in a powerful set of tools for later application.

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Voiculescu, Irina Dana. "Implicit function algebra in set-theoretic geometric modelling." Thesis, University of Bath, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.392056.

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Pesonen, Janne. "Application of geometric algebra to theoretical molecular spectroscopy." Helsinki : University of Helsinki, 2001. http://ethesis.helsinki.fi/julkaisut/mat/kemia/vk/pesonen/.

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Satchell, Marcel John Francis. "Geometric algebra & the quantum theory of fields." Thesis, University of Cambridge, 2014. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.708105.

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Arcaute, Elsa. "Spinors, wave-functions and twistors within geometric algebra." Thesis, University of Cambridge, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.613700.

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Mishra, Biswajit. "Investigation into a Floating Point Geometric Algebra Processor." Thesis, University of Southampton, 2007. https://eprints.soton.ac.uk/266009/.

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The widespread use of Computer Graphics and Computer Vision applications has led to a plethora of hardware implementations that are usually expressed using linear algebraic methods. There are two drawbacks with this approach that are posing fundamental challenges to engineers developing hardware and software applications in this area. The first is the complexity and size of the hardware blocks required to practically realize such applications – particularly multiplication, addition and accumulation operations. Whether the platform is Field Programmable Gate Arrays (FPGA) or Application Specific Integrated Circuits (ASICs), in both cases there are significant issues in efficiently implementing complex geometric functions using standard mathematical techniques, particularly in floating point arithmetic. The second major issue is the complexity required for the effective solution of complex multi-dimensional problems either for scientific computation or for advanced graphical applications. Conventional algebraic techniques do not scale well in hardware terms to more than 3 dimensional problems, so a new approach is desirable to handle these situations. Geometric Algebra (GA) promises to unify the different approaches used in vector algebra, trigonometry, homogeneous coordinates and quaternion algebra into a single framework. Geometric Algebra provides a rich set of geometric primitives to describe points, lines, planes, circles and spheres along with simple algebraic operations instead of points and lines alone as in a conventional algebra. This ability to carry out direct operations on this rich set of primitives enables GA to be a powerful tool for solving a wide variety of problems in computer vision, graphics and robotics. In all these areas, performance is a key issue, therefore hardware architecture of GA is considered essential to meet the stringent performance requirements for these applications. In this thesis, a detailed review of the influential research in the development of GA along with the necessary fundamentals of GA is given. Subsequently a review of background relating different implementation strategies provides an important element in understanding the specific requirements and thereby developing the hardware architecture. Based on this study, an architecture was developed that is modular and scalable to higher dimensions for geometric algebra processing. In this architecture, the designer can easily specify the floating point resolution, the order of the computation and also configure the trade-offs between the hardware area and speed. The modularity and the flexibility of the interface of the architecture also provides a platform where the designer can quantify the clock cycles to the number of resources that they may have in hand for any GA based application. This architecture has been designed not only to be a stand alone core, but can also be configured and used as a coprocessor to a larger system. To demonstrate the performance and flexibility of the GA architecture presented in this thesis, the hardware has been tested extensively using a standard image processing application. The performance results obtained from these experiments are comparable to the results obtained using existing methods. It is also shown through derivations and also from the experiments that the convolution operation in the image processing application with the GA based rotor masks, belong to a class of linear vector filters. This linear vector filter can be applied to image or speech signals where vector filtering is of fundamental interest. This opens up a range of research opportunities to the growing field of color image processing. This work has explored the totally new area of GA hardware with novel aspects including the grade tracking, configurability and linearity of the hardware. From a software point of view and application development, this work has explored the development of a platform with compiler support and easier programming methods specific to the GA hardware in an FPGA based platform. This has further enabled and increased the practical significance of the work by verifying the GA techniques in a variety of real world designs. Therefore, from both points of view it has advanced the state-of-art and has opened up opportunities for further research in GA hardware.

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Doran, Christopher John Leslie. "Geometric algebra and its application to mathematical physics." Thesis, University of Cambridge, 1994. https://www.repository.cam.ac.uk/handle/1810/251691.

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Clifford algebras have been studied for many years and their algebraic properties are well known. In particular, all Clifford algebras have been classified as matrix algebras over one of the three division algebras. But Clifford Algebras are far more interesting than this classification suggests; they provide the algebraic basis for a unified language for physics and mathematics which offers many advantages over current techniques. This language is called geometric algebra - the name originally chosen by Clifford for his algebra - and this thesis is an investigation into the properties and applications of Clifford's geometric algebra. The work falls into three broad categories: - The formal development of geometric algebra has been patchy and a number of important subjects have not yet been treated within its framework. A principle feature of this thesis is the development of a number of new algebraic techniques which serve to broaden the field of applicability of geometric algebra. Of particular interest are an extension of the geometric algebra of spacetime (the spacetime algebra) to incorporate multiparticle quantum states, and the development of a multivector calculus for handling differentiation with respect to a linear function. - A central contention of this thesis is that geometric algebra provides the natural language in which to formulate a wide range of subjects from modern mathematical physics. To support this contention, reformulations of Grassmann calculus, Lie algebra theory, spinor algebra and Lagrangian field theory are developed. In each case it is argued that the geometric algebra formulation is computationally more efficient than standard approaches, and that it provides many novel insights. - The ultimate goal of a reformulation is to point the way to new mathematics and physics, and three promising directions are developed. The first is a new approach to relativistic multiparticle quantum mechanics. The second deals with classical models for quantum spin-I/2. The third details an approach to gravity based on gauge fields acting in a fiat spacetime. The Dirac equation forms the basis of this gauge theory, and the resultant theory is shown to differ from general relativity in a number of its features and predictions.

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Silva, Luiz Eduardo Landim. "Inequalities between arithmetic and geometric averages and Cauchy-Schwarz." Universidade Federal do CearÃ, 2013. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=9538.

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CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior
Este trabalho trata de duas das mais importantes desigualdades da MatemÃtica: a desigualdade entre as mÃdias geomÃtrica e aritmÃtica e a desigualdade de Cauchy-Schwarz. Apresentamos inicialmente diversas demonstraÃÃes para o caso n = 2, apÃs as quais seguem muitas demonstraÃÃes para o caso geral. Nessas demonstraÃÃes utilizamos Ãlgebra elementar, geometria euclidiana, construÃÃes geomÃtricas, geometria analÃtica, induÃÃo matemÃtica, convexidade de funÃÃes, multiplicadores de Lagrange entre outros assuntos. AlÃm disso foram selecionados vinte problemas que visam dar ao leitor uma melhor compreensÃo de como estas desigualdades podem ser aplicadas em diversos assuntos e de diversas formas, estimulando a criatividade dos alunos na resoluÃÃo de problemas.
This paper deals with two of the most important inequalities of Mathematics: the inequality between the geometric and arithmetic and Cauchy-Schwarz. Here several first statements for the case n = 2, after which many statements following for the general case. In these statements we use algebra elementary Euclidean geometry, geometric constructions, analytical geometry, mathematical induction, convexity of functions, Lagrange multipliers among other issues. Also selected were twenty problems that aim to give the reader a better understanding of how these inequalities can be applied in various subjects and in many ways, stimulating students' creativity in problem solving.

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Hedén, Isac. "Russell’s hypersurface from a geometric point of view." Licentiate thesis, Uppsala universitet, Algebra, geometri och logik, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-144688.

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Hadi, Shazia. "Applications of the Pauli algebra and other geometric algebras." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp01/MQ52560.pdf.

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Kallfelz, William Michael. "Clifford algebra a case for geometric and ontological unification /." College Park, Md. : University of Maryland, 2008. http://hdl.handle.net/1903/8079.

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Thesis (Ph. D.) -- University of Maryland, College Park, 2008.
Thesis research directed by: Dept. of Philosophy. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.

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Somaroo, Shyamal Sewlal. "Applications of the geometric algebra to relativistic quantum theory." Thesis, University of Cambridge, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.627593.

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Riedel, Gårding Elias. "Geometric algebra, conformal geometry and the common curves problem." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-210866.

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This bachelor’s thesis gives a thorough introduction to geometric algebra (GA), an overview of conformal geometric algebra (CGA) and an application to the processing of single particle data from cryo-electron microscopy (cryo-EM). The geometric algebra over the vector space Rp;q, i.e. the Clifford algebra over an orthogonal basis of the space, is a strikingly simple algebraic construction built from the geometric product, which generalizes the scalar and cross products between vectors. In terms of this product, a host of algebraically and geometrically meaningful operations can be defined. These encode linear subspaces, incidence relations, direct sums, intersections and orthogonal complements, as well as reflections and rotations. It is with good reason that geometric algebra is often referred to as a universal language of geometry. Conformal geometric algebra is the application of geometric algebra in the context of the conformal embedding of R3 into the Minkowski space R4;1. By way of this embedding, linear subspaces of R4;1 represent arbitrary points, lines, planes, point pairs, circles and spheres in R3. Reflections and rotations in R4;1 become conformal transformations in R3: reflections, rotations, translations, dilations and inversions. The analysis of single-particle cryo-electron microscopy data leads to the common curves problem. By a variant of the Fourier slice theorem, this problem involves hemispheres and their intersections. This thesis presents a rewriting, inspired by CGA, into a problem of planes and lines.

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Lewis, Antony Martin. "Geometric algebra and covariant methods in physics and cosmology." Thesis, University of Cambridge, 2001. https://www.repository.cam.ac.uk/handle/1810/251774.

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Bojorquez, Betzabe. "Geometric Constructions from an Algebraic Perspective." CSUSB ScholarWorks, 2015. https://scholarworks.lib.csusb.edu/etd/237.

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Many topics that mathematicians study at times seem so unrelated such as Geometry and Abstract Algebra. These two branches of math would seem unrelated at first glance. I will try to bridge Geometry and Abstract Algebra just a bit with the following topics. We can be sure that after we construct our basic parallel and perpendicular lines, bisected angles, regular polygons, and other basic geometric figures, we are actually constructing what in geometry is simply stated and accepted, because it will be proven using abstract algebra. Also we will look at many classic problems in Geometry that are not possible with only straightedge and compass but need a marked ruler.

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Cameron, J. I. "Aspects of conformal geometric algebra with applications in motion capture." Thesis, University of Cambridge, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.597235.

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The body of this thesis has three main sections. The first introduces Oriented Conformal Geometric Algebra (OCGA), an extension to conventional Conformal Geometric Algebra (CGA), related closely to Stolfi’s Oriented Projective Geometry. In OCGA the sign of previously homogeneous elements is used to represent a concept of directionality, allowing the definition of half spaces of oriented n-planes or spheres with the boundary element as an oriented n – 1 plane or sphere. Oriented intersection tests allow the order in which a ray intersects a sphere to be identified without a specific test. The utility of this framework is shown in the analysis of a sphere interpolation method of A. Lasenby. The second major section is concerned with the development of a numerically stable general exponential library for CGA bivectors. This work builds on that of Wareham et al. in which a suitable pose bivector exponential expansion was identified. The fully general bivector exponential is derived and the piecewise interpolation paths using pose and dilation bivector exponentials is shown to correspond to a conical helix. The third section considers the problem of taking optical motion capture data, which provides approximate marker locations attached to clothing, to identify and parameterize the underlying human skeleton structure and motion over time. The first of two related chapters is concerned with real-time algorithms suitable for use within a real-time visual feedback system, an example of which is presented. The algorithms presented require 3 markers on each limb segment. Following this is a chapter considering algorithms for offline analysis, either to improve on the accuracy of what is possible in real-time, or to provide estimates with less available data. A new class of symmetric sphere fitting algorithm is introduced.

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34

Tachibana, Kanta, Takeshi Furuhashi, Tomohiro Yoshikawa, Eckhard Hitzer, and MINH TUAN PHAM. "Clustering of Questionnaire Based on Feature Extracted by Geometric Algebra." 日本知能情報ファジィ学会, 2008. http://hdl.handle.net/2237/20676.

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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

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35

Milne, Philip S. "On the algorithms and implementation of a geometric algebra system." Thesis, University of Bath, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.236564.

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36

Simpson, Leon. "Geometric algebra as applied to freeform motion design and improvement." Thesis, University of Bath, 2012. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.558894.

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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.

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37

Chislenko, Julia. "On geometric constructions of the universal enveloping algebra U(slnÌ³)." Thesis, Massachusetts Institute of Technology, 1994. http://hdl.handle.net/1721.1/28097.

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Daza, John Elber Gómez. "Superfícies mínimas e curvatura de gauss de conóides em espaços de finsler com (α,β) - métricas." Universidade Federal de Goiás, 2014. http://repositorio.bc.ufg.br/tede/handle/tede/3634.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
We consider(α,β)−metric F=αφ(β α), whereα is the euclidean metric,φ is a smooth positive function on a symmetric interval I=(−b0,b0) and β is a 1-form with the norm b,0 ≤bNeste trabalho consideramos (α,β)−métricas do tipo F=αφ(β α), ondeα é a métrica euclidiana,φ é uma função positiva suave sobre um intervalo simétrico I=(−b0,b0) e β é uma 1-forma de norma b,0 ≤ b < b0, sobre uma variedade de Finsler M. Estudamos superfícies mínimas nestes espaços (M,F) com respeito à forma volume de Holmes-Thompson e apresentamos uma equação que caracteriza as hipersuperfícies mínimasemumespaçogeral(α,β)−Minkowski.Mostramosqueosconóidesnoespaço tridimensional comβ na direção do eixo ˜y3 são mínimas se, e somente se, é um helicóide ou um plano, provamos também que a curvatura de Gauss do conóide em um espaço tridimensional de Randers-Minkowski pode ser positiva em superfícies mínimas. Finalmente apresentamos uma equação diferencial ordinária que caracteriza superfícies mínimas de rotação eum exemplo de superfíciemínimade rotação.

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Stanley, Adrian. "The geometric phase from imaginary time and Clifford algebra space translations." Thesis, University of Kent, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.240169.

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40

Miller, Richard A. "Geometric algebra| An introduction with applications in Euclidean and conformal geometry." Thesis, San Jose State University, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=1552269.

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This thesis presents an introduction to geometric algebra for the uninitiated. It contains examples of how some of the more traditional topics of mathematics can be reexpressed in terms of geometric algebra along with proofs of several important theorems from geometry. We introduce the conformal model. This is a current topic among researchers in geometric algebra as it is finding wide applications in computer graphics and robotics. The appendices provide a list of some of the notational conventions used in the literature, a reference list of formulas and identities used in geometric algebra along with some of their derivations, and a glossary of terms.

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41

Wells, Stephen Anthony. "Real-space rigid unit analysis of framework structures using geometric algebra." Thesis, University of Cambridge, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.619904.

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42

Candy, Liam Patrick. "Kinematics in conformal geometric algebra with applications in strapdown inertial navigation." Thesis, University of Cambridge, 2012. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.610522.

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Wimelaratna, Ramasinghege. "Multi dimensional geometric moduli and exterior algebra of a Banach space /." The Ohio State University, 1988. http://rave.ohiolink.edu/etdc/view?acc_num=osu148759830383865.

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44

Fujita, Ryo. "A geometric study of Dynkin quiver type quantum affine Schur-Weyl duality." Kyoto University, 2019. http://hdl.handle.net/2433/242573.

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Kaiser, Tim Benjamin. "From data tables to general geometric structures." Aachen Shaker, 2008. http://d-nb.info/992124212/04.

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46

Tian, Xijin. "Modeling of planar elastically coupled rigid bodies: Geometric algebra methods and applications." Diss., The University of Arizona, 2002. http://hdl.handle.net/10150/280214.

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This study presents two new, generic methods to modeling planar elastically coupled rigid body systems using Geometric Algebra. The two methods are twist-based potential energy function method and twistor-based potential energy function method. In this research, the rigid body motion in the plane is modeled as a twist or twistor motion in which the rotational motion and translational motion happen simultaneously. The twist is denoted as a bivector using Geometric Algebra which facilitates the notation and computation. A twistor is defined in an intermediate frame half way between two displacement frames. The twistor parameters intuitively represent the relative displacement between two frames. Both twist-based and twistor-based potential energy functions are shown to be frame-independent and body-independent. The kinematics is studied using twist and twistor parameters. The constitutive equations are derived in which the wrench exerted by a pair of elastic bodies is computable given twist or twistor displacements. To analyze large displacements, this study also provides two higher order polynomial potential energy functions of twist parameters and twistor parameters. The polynomial potential energy functions are also shown to be frame-independent and body-independent. They are generally applicable to analyze large displacements of elastically coupled rigid body systems. Several case studies are provided in this research to demonstrate the utility of the presented modeling methods. A micropositioning stage device is modeled as a flexural mechanism with 6 rigid bodies and 7 flexural joints. Simulation is performed using Scilab software. The simulation results show good agreement with actual experimental data. The methods are also applied to simulate the displacement of flexural four-bar linkages with various geometry and various flexural hinges. This case study shows that the presented methods in this research are generic and case-independent. In another case study, the higher order polynomial function method is applied to fit some randomly generated data which demonstrates the generality of the method and the applicability of the method in cases when only experimental data is available without knowing the geometry parameters of a mechanism. The case study of modeling electrostatic potential energy between liquid water molecules using polynomial function of twistor shows the potential utility of the method in the analysis of large displacement. The methods presented in this research have been shown to be generic, easily applicable, and easily computable.

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47

Hiniduma, Udugama Gamage Sahan Sajeewa. "Theoretical and experimental analysis of articulated rigid body motion using geometric algebra." Thesis, University of Cambridge, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.620516.

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Gebken, Christian [Verfasser]. "Conformal geometric algebra in stochastic optimization problems of 3D-vision applications / Christian Gebken." Kiel : Universitätsbibliothek Kiel, 2009. http://d-nb.info/1019870109/34.

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Lemay, Joel. "Geometric Realizations of the Basic Representation of the Affine General Linear Lie Algebra." Thesis, Université d'Ottawa / University of Ottawa, 2015. http://hdl.handle.net/10393/32866.

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The realizations of the basic representation of the affine general linear Lie algebra on (r x r) matrices are well-known to be parametrized by partitions of r and have an explicit description in terms of vertex operators on the bosonic/fermionic Fock space. In this thesis, we give a geometric interpretation of these realizations in terms of geometric operators acting on the equivariant cohomology of certain Nakajima quiver varieties.

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Koyuncu, Fatih. "A Geometric Approach To Absolute Irreducibility Of Polynomials." Phd thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/3/12604873/index.pdf.

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This thesis is a contribution to determine the absolute irreducibility of polynomials via their Newton polytopes. For any field F
a polynomial f in F[x1, x2,..., xk] can be associated with a polytope, called its Newton polytope. If the polynomial f has integrally indecomposable Newton polytope, in the sense of Minkowski sum, then it is absolutely irreducible over F
i.e. irreducible over every algebraic extension of F. We present some new results giving integrally indecomposable classes of polytopes. Consequently, we have some new criteria giving infinitely many types of absolutely irreducible polynomials over arbitrary fields.

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