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)4999110.22363/2658-4670-2026-34-1-70-97UOBPEGModeling and SimulationМатематическое моделированиеResearch ArticleDual quaternion representation of geometrical motion in 3D spaceБикватернионное представление движения в трёхмерном пространствеhttps://orcid.org/0009-0002-5236-0027AbakumovaOlesya M.АбакумоваО. М.

Student of Department of Probability Theory and Cyber Security

1132220832@rudn.ruhttps://orcid.org/0000-0002-4834-489557190004380E-9214-2016GevorkyanMigran N.ГеворкянМ. Н.

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

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

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

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

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

kulyabov-ds@rudn.ruRUDN UniversityРоссийский университет дружбы народов им. Патриса ЛумумбыJoint Institute for Nuclear ResearchОбъединённый институт ядерных исследований30042026341Vol 34, No 1 (2026)ТОМ 34, № 1 (2026)709729042026Copyright ©; 2026, Abakumova O.M., Gevorkyan M.N., Korolkova A.V., Kulyabov D.S.Copyright ©; 2026, Абакумова О.М., Геворкян М.Н., Королькова А.В., Кулябов Д.С.2026Abakumova O.M., Gevorkyan M.N., Korolkova A.V., Kulyabov D.S.Абакумова О.М., Геворкян М.Н., Королькова А.В., Кулябов Д.С.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/49991

\emph {Background} In a previous article we discussed the use of dual quaternions for modeling points, lines and planes and solving standard geometric problems. This article is a logical continuation and reveals the use of dual quaternions to describe isometries of three-dimensional space. \emph {Purpose} The derivation of all necessary formulas for the screw motion of points, straight lines and planes, as well as reflection relative to the plane. Refinement of notation and formalism. \emph {Method} The algebra of dual numbers, quaternions and dual quaternions is used, as well as elements of the theory of screws and sliding vectors. \emph {Results} Formulas for rotation, translation, reflection, helical motion, and mirror rotation are obtained and systematized. \emph {Conclusions} Dual quaternions can serve as a full-fledged tool for describing helical motion in space. Due to the possibility of expressing dual quaternion operations in terms of standard vector and scalar products, the formulas obtained allow for effective software implementation.

\emph {Предпосылки} В предыдущей статье авторов был подробно рассмотрен вопрос использования бикватернионов для задания точек, прямых и плоскостей и решения стандартных геометрических задач. Данная статья является логическим продолжением и раскрывает применение бикватернионов для описания изометрий трёхмерного пространства. \emph {Цель} Вывод всех необходимых формул для винтового движения точек, прямых и плоскостей, а также зеркальной симметрии (отражения) относительно плоскости. Доработка обозначений и формализма. \emph {Методы} Используется алгебра дуальных чисел, кватернионов и бикватернионов, а также элементы теории винтов и скользящих векторов. \emph {Результаты} Получены и систематизированы формулы для вращения, трансляции, отражения, винтового движения и зеркального вращения. \emph {Выводы} Бикватернионы могут служить полноценным инструментом для описания винтового движения в пространстве. Благодаря возможности выражения бикватернионных операций через стандартное векторное и скалярное произведения, полученные формулы допускают эффективную программную реализацию.

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