Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)4750610.22363/2658-4670-2025-33-4-411-439HPZRYAModeling and SimulationМатематическое моделированиеResearch ArticleDual quaternion representation of points, lines and planesБикватернионное представление точек, прямых и плоскостейhttps://orcid.org/0000-0002-4834-489557190004380E-9214-2016GevorkyanMigran N.ГеворкянМ. Н.
Russian Federation

Docent, Ph.D. in Physics and Mathematics, Associate Professor of Department of Probability Theory and
Cyber Security

gevorkyan-mn@rudn.ruhttps://orcid.org/0009-0004-4410-8635VishnevskiyNikita A.ВишневскийН. А.
Russian Federation

PhD student of Department of Probability Theory and Cyber Security

1142240277@rudn.ruhttps://orcid.org/0000-0002-5622-8480DidusKirill V.ДидусьК. В.
Russian Federation

PhD student of Department of Probability Theory and Cyber Security

1142240434@rudn.ruhttps://orcid.org/0000-0001-7141-761036968057600I-3191-2013KorolkovaAnna V.КорольковаА. В.
Russian Federation

Docent, Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Probability Theory and Cyber Security

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korolkova-av@rudn.ruhttps://orcid.org/0000-0002-0877-706335194130800I-3183-2013KulyabovDmitry S.КулябовД. С.
Russian Federation

Professor, Doctor of Sciences in Physics and Mathematics, Professor of Department of Probability Theory and Cyber Security of RUDN University; Senior Researcher of Laboratory of Information Technologies, Joint Institute for Nuclear Research

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kulyabov_ds@pfur.ruRUDN UniversityРоссийский университет дружбы народовJoint Institute for Nuclear ResearchОбъединённый институт ядерных исследований07122025334VOL 33, No4 (2025)ТОМ 33, №4 (2025)41143906122025Copyright ©; 2025, Gevorkyan M.N., Vishnevskiy N.A., Didus K.V., Korolkova A.V., Kulyabov D.S.Copyright ©; 2025, Геворкян М.Н., Вишневский Н.А., Дидусь К.В., Королькова А.В., Кулябов Д.С.2025Gevorkyan M.N., Vishnevskiy N.A., Didus K.V., Korolkova A.V., Kulyabov D.S.Геворкян М.Н., Вишневский Н.А., Дидусь К.В., Королькова А.В., Кулябов Д.С.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/47506

Background. The bulk of the work on dual quaternions is devoted to their application to describe helical motion. Little attention is paid to the representation of points, lines, and planes (primitives) using them. Purpose. It is necessary to consistently present the dual quaternion theory of the representation of primitives and refine the mathematical formalism. Method. It uses the algebra of dual numbers, quaternions and dual quaternions, as well as elements of the theory of screws and sliding vectors. Results. Formulas have been obtained and systematized that use exclusively dual quaternionic operations and notation to solve standard problems of three-dimensional geometry. Conclusions. Dual quaternions can serve as a full-fledged formalism for the algebraic representation of a three-dimensional projective space.

Предпосылки. Основная масса работ по бикватернионам, посвящена их применению для описания винтового движения. Представлению с их помощью точек, прямых и плоскостей (примитивов) уделяется мало внимания. Цель. Необходимо последовательно изложить бикватернионную теорию представления примитивов и доработать математический формализм. Методы. Используется алгебра дуальных чисел, кватернионов и бикватернионов, а также элементы теории винтов и скользящих векторов. Результаты. Получены и систематизированы формулы которые использую исключительно бикватернионные операции и обозначения для решения стандартных задач трёхмерной геометрии. Выводы. Бикватернионы могут служить полноценным формализмом алгебраического представления трёхмерного проективного пространства.

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