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S. E. Perelyaev, "On the correspondence between the three- and four-dimensional parameters of the three-dimensional rotation group," Mech. Solids. 44 (2), 204-213 (2009)
Year 2009 Volume 44 Number 2 Pages 204-213
DOI 10.3103/S0025654409020058
Title On the correspondence between the three- and four-dimensional parameters of the three-dimensional rotation group
Author(s) S. E. Perelyaev (Moscow Institute of Electromechanics and Automatics, Aviatsionnyy per. 5, Moscow, 125319, Russia, pers2030@yandex.ru)
Abstract A fundamental kinematic theorem due to Euler permits synthesizing a series of three- and four-dimensional orientation parameters that correspond to each other in spaces of the same dimension.

We use the theorem about the homeomorphism of two topological spaces (the three-dimensional sphere S3⊂R4 with a single punctured (removed) point and the three-dimensional space R3) to establish a one-to-one mutually continuous correspondence between the four- and three-dimensional kinematic parameters prescribed in these spaces. The latter can be proved using the stereographic projection of points of the sphere S3 onto the hyperplane R3. For the normalized (Hamiltonian) Rodrigues-Hamilton parameters, we present a method of stereographic projection of a point belonging to the three-dimensional sphere S3 onto the oriented space R3. We present a family of local kinematic parameters obtained by the method of mapping four symmetric kinematic parameters of the space R4 onto the oriented real space R3.

In contrast to the well-known four symmetric global parameters of the Rodrigues-Hamilton orientation, the synthesized three-dimensional orientation parameters are local (have two singular points ±360°). The differential equations of rotation in the three-dimensional orientation parameters are obtained by the projection method.

We present the three-dimensional parameters corresponding to the classical Hamiltonian quaternions defined in the four-dimensional vector space R4.
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Received 21 September 2007
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