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Geometric Computing in Computer Graphics and Robotics using Conformal Geometric Algebra

Hildenbrand, Dietmar :
Geometric Computing in Computer Graphics and Robotics using Conformal Geometric Algebra.
[Online-Edition]
TU Darmstadt
[Ph.D. Thesis], (2007)

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Item Type: Ph.D. Thesis
Title: Geometric Computing in Computer Graphics and Robotics using Conformal Geometric Algebra
Language: English
Abstract:

In computer graphics and robotics a lot of different mathematical systems like vector algebra, homogenous coordinates, quaternions or dual quaternions are used for different applications. Now it seems that a change of paradigm is lying ahead of us based on Conformal Geometric Algebra unifying all of these different approaches in one mathematical system. Conformal Geometric Algebra is a very powerful mathematical framework. Due to its geometric intuitiveness, compactness and simplicity it is easy to develop new algorithms. Algorithms based on Conformal Geometric Algebra lead to enhanced quality, a reduction of development time, better understandable and better maintainable solutions. Often a clear structure and greater elegance results in lower runtime performance. However, it will be shown that algorithms based on Conformal Geometric Algebra can even be faster than conventional algorithms. The main contribution of this thesis is the geometrically intuitive and - nevertheless - efficient algorithm for a computer animation application, namely an inverse kinematics algorithm for a virtual character. This algorithm is based on an embedding of quaternions in Conformal Geometric Algebra. For performance reasons two optimization approaches are used in a way to make the application now three times faster than the conventional solution. With these results we are convinced that Geometric Computing using Conformal Geometric Algebra will become more and more fruitful in a great variety of applications in computer graphics and robotics.

Alternative Abstract:
Alternative AbstractLanguage
In der Computergraphik und in der Robotik werden eine ganze Reihe von unterschiedlichen mathematischen Systemen eingesetzt wie Vektoralgebra, homogene Koordinaten, Quaternionen und duale Quaternionen. Jetzt scheint ein Paradigmenwechsel vor uns zu liegen, der auf konformer geometrischer Algebra basiert, die in der Lage ist, diese unterschiedlichen Ansätze in einem mathematischen System zu vereinheitlichen. Die konforme geometrische Algebra ist ein sehr mächtiges mathematisches Werkzeug. Dank ihrer geometrischen Intuitivität und Kompaktheit ist es einfach, mit ihr neue Algorithmen zu entwickeln. Algorithmen, die auf konformer geometrischer Algebra beruhen, führen zu erhöhter Qualität, einer Reduzierung von Entwicklungszeit, besser verständlichen und besser wartbaren Lösungen. Oft ist es jedoch so, dass eine klare Struktur und größere Eleganz zu einer geringeren Laufzeit führen. Demgegenüber kann aber in dieser Arbeit gezeigt werden, dass Algorithmen in konformer geometrischer Algebra sogar schneller sein können als herkömmliche Algorithmen.German
Uncontrolled Keywords: Inverse Kinematik, geometrische Algebra
Alternative keywords:
Alternative keywordsLanguage
Inverse Kinematik, geometrische AlgebraGerman
inverse kinematics, geometric algebraEnglish
Classification DDC: 000 Allgemeines, Informatik, Informationswissenschaft > 004 Informatik
Divisions: Fachbereich Informatik
Date Deposited: 17 Oct 2008 09:22
Last Modified: 07 Dec 2012 11:52
Official URL: http://elib.tu-darmstadt.de/diss/000764
URN: urn:nbn:de:tuda-tuprints-7647
License: Simple publication rights for ULB
Referees: Alexa, Prof. Dr. Marc and Straßer, Prof. Dr.- Wolfgang and Fellner, Prof. Dr. Dieter W.
Advisors: Alexa, Prof. Dr.- Marc
Refereed: 13 December 2006
URI: http://tuprints.ulb.tu-darmstadt.de/id/eprint/764
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