Abstract
Manipulating objects using geometric algebra may involve several associative products in a single expression. For example, an object can be constructed by the outer product of multiple points. This number of products can be small for some conformal algebra and high for higher dimensional algebras such as quadric conformal geometric algebras. In these situations, the order of products (i.e. the choice of the parenthesis in the expression) should not change the final result but may change the overall computational cost, according to the grade of the intermediate multivectors. Indeed, the usual left to right way to evaluate the expression may not be most computationally efficient. Studies on the number of arithmetic operations of geometric algebra expressions have been limited to products of only two homogeneous multivectors. This paper shows that there exists an optimal order in the evaluation of an expression involving geometric and outer products, and presents a dynamic programming framework to find it.
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Breuils, S., Nozick, V., Sugimoto, A. (2020). Optimal Parenthesizing of Geometric Algebra Products. In: Magnenat-Thalmann, N., et al. Advances in Computer Graphics. CGI 2020. Lecture Notes in Computer Science(), vol 12221. Springer, Cham. https://doi.org/10.1007/978-3-030-61864-3_42
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DOI: https://doi.org/10.1007/978-3-030-61864-3_42
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