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Analytical-Numerical Implementation of Polyvector Algebra in Julia

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Abstract

Geometric algebra is based on the works by Grassmann and Clifford. Its main objects of research are polyvectors (p-vectors) and multivectors. Polyvectors, together with the exterior product, implement the Grassmann algebra, while multivectors with the geometric product implement the Clifford algebra. Multivector algebra generalizes many operations and objects of analytic geometry and differential geometry (e.g., vector and mixed products, normal vectors and binormals, etc.) to the multidimensional case, as well as provides their geometric interpretation. Complex numbers and quaternions are isomorphic to multivectors of a special kind. This paper applies certain ideas of geometric algebra to solve problems that occur in computer geometry. For this purpose, the Grassmann.jl package for Julia is used.

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Notes

  1. It should be noted, however, that the principle of one language in Julia initially implied the use of one language for both prototyping and developing high-performance programs [7].

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Funding

This work was supported by the RUDN University Strategic Academic Leadership Program (M.N. Gevorkyan and A.V. Demidova), the Russian Science Foundation, grant no. 19-71-30008 (T.R. Velieva), and the Russian Foundation for Basic Research, project no. 19-01-00645 (A.V. Korol’kova and D.S. Kulyabov).

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Authors and Affiliations

  1. Peoples’ Friendship University of Russia, ul. Miklukho-Maklaya 6, 117198, Moscow, Russia

    M. N. Gevorkyan, A. V. Demidova, T. R. Velieva, A. V. Korol’kova & D. S. Kulyabov

  2. Plekhanov Russian University of Economics, Stremyannyi per. 36, 117997, Moscow, Russia

    T. R. Velieva

  3. Laboratory of Information Technologies, Joint Institute for Nuclear Research, ul. Zholio-Kyuri 6, 141980, Dubna, Moscow oblast, Russia

    D. S. Kulyabov

Authors
  1. M. N. Gevorkyan
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  2. A. V. Demidova
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  3. T. R. Velieva
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  4. A. V. Korol’kova
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  5. D. S. Kulyabov
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Corresponding authors

Correspondence to M. N. Gevorkyan, A. V. Demidova, T. R. Velieva, A. V. Korol’kova or D. S. Kulyabov.

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Translated by Yu. Kornienko

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Gevorkyan, M.N., Demidova, A.V., Velieva, T.R. et al. Analytical-Numerical Implementation of Polyvector Algebra in Julia. Program Comput Soft 48, 49–58 (2022). https://doi.org/10.1134/S0361768822010054

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