Geometric Algebra and Quaternion Techniques in Computer Algebra Systems for Describing Rotations in Eucledean Space

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Abstract

Tensor formalism (and its special case—vector formalism) is a mathematical technique that is widely used in physical and engineering problems. Even though this formalism is fairy universal and suitable for describing many spaces, the application of other special mathematical techniques is sometimes required. For example, the problem of rotation in a 3D space is not very well described in tensor representation, and it is reasonable to use the formalism of Clifford algebra, in particular, quaternions and geometric algebra representations for its solution. In this paper, computer algebra is used to demonstrate the solution of the problem of rotation in a 3D space using both the quaternion and geometric algebra formalisms. It is shown that although these formalisms are fundamentally similar, the latter one seems to be clearer both for computations and interpretation of results.

About the authors

T. R. Velieva

Peoples’ Friendship University of Russia (RUDN University)

Email: velieva-tr@rudn.ru
117198, Moscow, Russia

M. N. Gevorkyan

Peoples’ Friendship University of Russia (RUDN University)

Email: gevorkyan-mn@rudn.ru
117198, Moscow, Russia

A. V. Demidova

Peoples’ Friendship University of Russia (RUDN University)

Email: demidova-av@rudn.ru
117198, Moscow, Russia

A. V. Korol’kova

Peoples’ Friendship University of Russia (RUDN University)

Email: korolkova-av@rudn.ru
117198, Moscow, Russia

D. S. Kulyabov

Peoples’ Friendship University of Russia (RUDN University); Joint Institute for Nuclear Research

Author for correspondence.
Email: kulyabov-ds@rudn.ru
117198, Moscow, Russia; 141980, Dubna, Moscow oblast, Russia

References

  1. Kanatani K. 3D Rotations: Parameter Computation and Lie Algebra based Optimization. Chapman and Hall/CRC, 2020.
  2. Vince J. Rotation transforms for computer graphics. London : Springer-Verlag, 2011.
  3. Kuipers J.B. Quaternions and rotation sequences. Princeton University Press, 1999.
  4. Grassmann H.G. Die Mechanik nach den Principien der Ausdehnungslehre // Mathematische Annalen. 1877. Bd. 12. S. 222–240.
  5. Clifford W.K. Applications of Grassmann’s Extensive Algebra // American Journal of Mathematics. 1878. V. 1. № 4. P. 350–358.
  6. Казанова Г. Векторная алгебра / Под ред. М.К. Поливанов. Современная математика. М.: Мир, 1979. 120 с.
  7. Hestenes D., Sobczyk G. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Fundamental Theories of Physics. Netherlands Springer, 1987.
  8. Delanghe R., Sommen F., Soucek V. Clifford algebra and spinor-valued functions. Mathematics and Its Applications. Netherlands: Kluwer Academic Publishers, 1992.
  9. Doran C., Lasenby A. Geometric Algebra for Physicists. Morgan Kaufmann Publishers, 2003.
  10. Dorst L., Fontijne D., Mann S. Geometric algebra for computer science. The Morgan Kaufmann Series in Computer Graphics. Morgan Kaufmann, 2007.
  11. Vince J. Geometric algebra for computer graphics. Springer-Verlag, 2008.
  12. Lengyel E. Mathematics. Lincoln, California : Terathon Software LLC, 2016. Vol. 1. 195 p.
  13. Kanatani K. Understanding Geometric Algebra. Taylor and Francis Group/CRC, 2015.
  14. ten Bosch M. Let’s remove Quaternions from every 3D Engine. Access mode: https://marctenbosch.com/quaternions/.
  15. SymPy. 2022. Access mode: http://www.sympy.org/ru/index.html.
  16. Van Rossum G., Drake F.L. Python 3 Reference Manual. Scotts Valley, CA: CreateSpace, 2009.
  17. GAlgebra – Symbolic Geometric Algebra/Calculus package for SymPy. 2022. Access mode: https://galgebra.readthedocs.io/en/latest/index.html.
  18. Dickson L.E. On Quaternions and Their Generalization and the History of the Eight Square Theorem // The Annals of Mathematics. 1919. 3. V. 20. № 3. P. 155–171.
  19. Cayley A. IV. A sixth memoir upon quantics // Philosophical Transactions of the Royal Society of London. 1859. 1. V. 149. P. 61–90.
  20. Klein F.C. Ueber die sogenannte Nicht-Euklidische Geometrie // Gauβ und die Anfänge der nicht-euklidischen Geometrie. Wien: Springer-Verlag Wien, 1985. Bd. 4 von Teubner-Archiv zur Mathematik. S. 224–238.
  21. Klein F.C. A comparative review of recent researches in geometry // Bulletin of the American Mathematical Society. 1893. V. 2. № 10. P. 215–249. 0807.3161.
  22. Яглом И.М. Комплексные числа и их применение в геометрии // Математика, ее преподавание, приложения и история. 1961. Т. 6 из Математическое просвещение, сер. 2. С. 61–106.
  23. Gevorkyan M.N., Korolkova A.V., Kulyabov D.S. Approaches to the implementation of generalized complex numbers in the Julia language // Workshop on information technology and scientific computing in the framework of the X International Conference Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems (ITTMM-2020) / Ed. by D.S. Kulyabov, K.E. Samouylov, L.A. Sevastianov. Vol. 2639 of CEUR Workshop Proceedings. Aachen, 2020. 4. P. 141–157. arXiv: 2007.09737.
  24. Митюшов Е.А., Мисюра Н.Е., Берестова С.А. К 175-летию открытия кватернионов // Вестник Удмуртского университета. Матем. Мех. Компьют. науки. Т. 28. С. 611–617.
  25. Hamilton W.R. On a New Species of Imaginary Quantities, Connected with the Theory of Quaternions // Proceedings of the Royal Irish Academy (1836-1869). 1840. V. 2. P. 424–434.
  26. Конвей Д.Х., Смит Д.А. О кватернионах и октавах / Под ред. В.В. Доценко. М.: МЦНМО, 2009. 184 с.
  27. Кострикин А.И. Линейная алгебра. М. : МЦНМО, 2009. Т. 2. 368 с.
  28. Quaternions in numpy. 2022. Access mode: https://quaternion.readthedocs.io.
  29. Pyquaternion. 2022. Access mode: https://kieranwynn.github.io/pyquaternion/.
  30. Геворкян М.Н., Демидова А.В., Велиева Т.Р., Королькова А.В., Кулябов Д.С. Аналитико-численная реализация алгебры поливекторов на языке Julia // Программирование. 2022. № 1. С. 54–64.
  31. Пенроуз Р., Риндлер В. Спиноры и пространство-время. М. : Мир, 1988. Т. 2. 573 с.
  32. Пенроуз Р., Риндлер В. Спиноры и пространство-время. М. : Мир, 1987. Т. 1. 527 с.

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Copyright (c) 2023 Т.Р. Велиева, М.Н. Геворкян, А.В. Демидова, А.В. Королькова, Д.С. Кулябов