| Both sides previous revisionPrevious revisionNext revision | Previous revisionNext revisionBoth sides next revision |
| geometric_algebra [2020/11/02 09:50] – [Modern] pbk | geometric_algebra [2020/11/19 01:14] – [Historical] pbk |
|---|
| * [[https://www.researchgate.net/publication/228955605_A_brief_introduction_to_Clifford_algebra|A brief introduction to Clifford Algebra]] (2010) - //Silvia Franchini, Giorgio Vassallo, Filippo Sorbello// | * [[https://www.researchgate.net/publication/228955605_A_brief_introduction_to_Clifford_algebra|A brief introduction to Clifford Algebra]] (2010) - //Silvia Franchini, Giorgio Vassallo, Filippo Sorbello// |
| Geometric algebra (also known as Clifford algebra) is a powerful mathematical tool that offers a natural and direct way to model geometric objects and their transformations. It is gaining growing attention in different research fields as physics, robotics, CAD/CAM and computer graphics. Clifford algebra makes geometric objects (points, lines and planes) into basic elements of computation and defines few universal operators that are applicable to all types of geometric elements. This paper provides an introduction to Clifford algebra elements and operators. | Geometric algebra (also known as Clifford algebra) is a powerful mathematical tool that offers a natural and direct way to model geometric objects and their transformations. It is gaining growing attention in different research fields as physics, robotics, CAD/CAM and computer graphics. Clifford algebra makes geometric objects (points, lines and planes) into basic elements of computation and defines few universal operators that are applicable to all types of geometric elements. This paper provides an introduction to Clifford algebra elements and operators. |
| | |
| | * [[https://vixra.org/pdf/1203.0011v1.pdf|A Very Brief Introduction to Clifford Algebra]] (2012) - //Stephen Crowley// |
| | This article distills many of the essential definitions from the very thorough book, Clifford Algebras: An Introduction, by Dr D.J.H. Garling, with some minor additions. |
| |
| * [[http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/bookGA.pdf|An Introduction to Geometric Algebra and Calculus]] (2014) - //Alan Bromborsky// | * [[http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/bookGA.pdf|An Introduction to Geometric Algebra and Calculus]] (2014) - //Alan Bromborsky// |
| This thesis investigates the emerging field of Conformal Geometric Algebra (CGA) as a new basis for a CG framework. Computer Graphics is, fundamentally, a particular application of geometry. From a practical standpoint many of the low-level problems to do with rasterising triangles and projecting a three-dimensional world onto a computer screen have been solved and hardware especially designed for this task is available. | This thesis investigates the emerging field of Conformal Geometric Algebra (CGA) as a new basis for a CG framework. Computer Graphics is, fundamentally, a particular application of geometry. From a practical standpoint many of the low-level problems to do with rasterising triangles and projecting a three-dimensional world onto a computer screen have been solved and hardware especially designed for this task is available. |
| |
| * [[http://home.deib.polimi.it/tubaro/Journals/Journal_2008_DA.pdf|3D Motion from structures of points, lines and planes]] (2007) - //Andrea Dell'Acqua, Augusto Sarti, Stefano Tubaro// | * [[https://tubaro.faculty.polimi.it/Journals/Journal_2008_DA.pdf|3D Motion from structures of points, lines and planes]] (2007) - //Andrea Dell'Acqua, Augusto Sarti, Stefano Tubaro// |
| In this article we propose a method for estimating the camera motion from a video-sequence acquired in the presence of general 3D structures. Solutions to this problem are commonly based on the tracking of point-like features, as they usually back-project onto viewpoint-invariant 3D features. In order to improve the robustness, the accuracy and the generality of the approach, we are interested in tracking and using a wider class of structures. In addition to points, in fact, we also simultaneously consider lines and planes. In order to be able to work on all such structures with a compact and unified formalism, we use here the Conformal Model of Geometric Algebra, which proved very powerful and flexible. | In this article we propose a method for estimating the camera motion from a video-sequence acquired in the presence of general 3D structures. Solutions to this problem are commonly based on the tracking of point-like features, as they usually back-project onto viewpoint-invariant 3D features. In order to improve the robustness, the accuracy and the generality of the approach, we are interested in tracking and using a wider class of structures. In addition to points, in fact, we also simultaneously consider lines and planes. In order to be able to work on all such structures with a compact and unified formalism, we use here the Conformal Model of Geometric Algebra, which proved very powerful and flexible. |
| |
| |
| ^ ^ Description ^ | ^ ^ Description ^ |
| | [[https://archive.org/details/dieausdehnungsl04grasgoog|{{:ga:die_ausdehnungslehre_von_1844-grassmann.jpg?400}}]] | **Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik [Die Ausdehnungslehre von 1844] (1878)**\\ //Hermann Grassmann//\\ The Prussian schoolmaster Hermann Grassmann taught a range of subjects including mathematics, science and Latin and wrote several secondary-school textbooks. Although he was never appointed to a university post, he devoted much energy to mathematical research and developed revolutionary new insights. Die lineale Ausdehnungslehre, published in 1844, is an astonishing work which was not understood by the mathematicians of its time but which anticipated developments that took a century to come to fruition - vector spaces, dimension, exterior products and many other ideas. Admired rather than read by the next generation, it was only fully appreciated by mathematicians such as Peano and Whitehead. | | | [[https://archive.org/details/dieausdehnungsl04grasgoog|{{:ga:die_ausdehnungslehre_von_1844-grassmann.jpg?600}}]] | **Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik [Die Ausdehnungslehre von 1844] (1878)**\\ //Hermann Grassmann//\\ The Prussian schoolmaster Hermann Grassmann taught a range of subjects including mathematics, science and Latin and wrote several secondary-school textbooks. Although he was never appointed to a university post, he devoted much energy to mathematical research and developed revolutionary new insights. Die lineale Ausdehnungslehre, published in 1844, is an astonishing work which was not understood by the mathematicians of its time but which anticipated developments that took a century to come to fruition - vector spaces, dimension, exterior products and many other ideas. Admired rather than read by the next generation, it was only fully appreciated by mathematicians such as Peano and Whitehead. | |
| | [[https://archive.org/details/dieausdehnugsle00grasgoog|{{:ga:die_ausdehnungslehre-grassmann.jpg?100}}]] | **Die Ausdehnungslehre (1864)**\\ //Hermann Grassmann//\\ In 1844, the Prussian schoolmaster Hermann Grassmann published Die Lineale Ausdehnungslehre. This revolutionary work anticipated the modern theory of vector spaces and exterior algebras. It was little understood at the time and the few sympathetic mathematicians, rather than trying harder to comprehend it, urged Grassmann to write an extended version of his theories. The present work is that version, first published in 1862. However, this also proved too far ahead of its time and Grassmann turned to historical linguistics, in which field his contributions are still remembered. His mathematical work eventually found champions such as Hankel, Peano, Whitehead and Élie Cartan, and it is now recognised for the brilliant achievement that it was in the history of mathematics. | | | [[https://archive.org/details/dieausdehnugsle00grasgoog|{{:ga:die_ausdehnungslehre-grassmann.jpg?100}}]] | **Die Ausdehnungslehre (1864)**\\ //Hermann Grassmann//\\ In 1844, the Prussian schoolmaster Hermann Grassmann published Die Lineale Ausdehnungslehre. This revolutionary work anticipated the modern theory of vector spaces and exterior algebras. It was little understood at the time and the few sympathetic mathematicians, rather than trying harder to comprehend it, urged Grassmann to write an extended version of his theories. The present work is that version, first published in 1862. However, this also proved too far ahead of its time and Grassmann turned to historical linguistics, in which field his contributions are still remembered. His mathematical work eventually found champions such as Hankel, Peano, Whitehead and Élie Cartan, and it is now recognised for the brilliant achievement that it was in the history of mathematics. | |
| | [[https://archive.org/details/bub_gb_bU9rkSdWlFAC|{{:ga:theorie_der_complexen_zahlensysteme-hankel.jpg?100}}]] | **Theorie Der Complexen Zahlensysteme (1867)**\\ //Hermann Hankel//\\ Insbesondere der Gemeinen Imaginären Zahlen und der Hamilton'schen Quaternionen Nebst Ihrer Geometrischen Darstellung. | | | [[https://archive.org/details/bub_gb_bU9rkSdWlFAC|{{:ga:theorie_der_complexen_zahlensysteme-hankel.jpg?100}}]] | **Theorie Der Complexen Zahlensysteme (1867)**\\ //Hermann Hankel//\\ Insbesondere der Gemeinen Imaginären Zahlen und der Hamilton'schen Quaternionen Nebst Ihrer Geometrischen Darstellung. | |
