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A Hybrid Approach for Computing Products of High-dimensional Geometric Algebras

Breuils, Stéphane and Nozick, Vincent and Fuchs, Laurent and Hildenbrand, Dietmar and Benger, Werner and Steinmetz, Christian (2017):
A Hybrid Approach for Computing Products of High-dimensional Geometric Algebras.
In: CGI '17, New York, NY, USA, ACM, In: Proceedings of the Computer Graphics International Conference, Yokohama, Japan, June 27 - 30, 2017, pp. 43:1-43:6, ISBN 978-1-4503-5228-4,
DOI: 10.1145/3095140.3097284,
[Online-Edition: http://doi.acm.org/10.1145/3095140.3097284],
[Conference or Workshop Item]

Abstract

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 solutions. For example, 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 high dimension of their representations in Geometric Algebra, implementations of Geometric Algebra that are currently available do not allow efficient realizations of these objects. This paper presents a Geometric Algebra implementation dedicated for both low and high dimensions. The proposed method is a hybrid solution for precomputed code with fast execution and runtime computations with low memory requirement. More specifically, the proposed method combines a precomputed table approach with a recursive method using binary trees. 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. The resulting implementation is well suited for high dimensional spaces (e.g. algebra of dimension 15) as well as for lower dimensional space. This paper details the integration of this hybrid method as a plug-in into Gaalop, which is a very advanced optimizing code generator. This paper also presents some benchmarks to show the performances of our method, especially in high dimensional spaces.

Item Type: Conference or Workshop Item
Erschienen: 2017
Creators: Breuils, Stéphane and Nozick, Vincent and Fuchs, Laurent and Hildenbrand, Dietmar and Benger, Werner and Steinmetz, Christian
Title: A Hybrid Approach for Computing Products of High-dimensional Geometric Algebras
Language: English
Abstract:

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 solutions. For example, 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 high dimension of their representations in Geometric Algebra, implementations of Geometric Algebra that are currently available do not allow efficient realizations of these objects. This paper presents a Geometric Algebra implementation dedicated for both low and high dimensions. The proposed method is a hybrid solution for precomputed code with fast execution and runtime computations with low memory requirement. More specifically, the proposed method combines a precomputed table approach with a recursive method using binary trees. 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. The resulting implementation is well suited for high dimensional spaces (e.g. algebra of dimension 15) as well as for lower dimensional space. This paper details the integration of this hybrid method as a plug-in into Gaalop, which is a very advanced optimizing code generator. This paper also presents some benchmarks to show the performances of our method, especially in high dimensional spaces.

Series Name: CGI '17
Place of Publication: New York, NY, USA
Publisher: ACM
ISBN: 978-1-4503-5228-4
Uncontrolled Keywords: geometric algebra, high dimensional space, implementation
Divisions: 20 Department of Computer Science
Event Title: Proceedings of the Computer Graphics International Conference
Event Location: Yokohama, Japan
Event Dates: June 27 - 30, 2017
Date Deposited: 06 Jul 2017 11:15
DOI: 10.1145/3095140.3097284
Official URL: http://doi.acm.org/10.1145/3095140.3097284
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