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 S 3 ⊂ R 4 with a single punctured (removed) point and the three-dimensional space R 3) 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 S 3 onto the hyperplane R 3. For the normalized (Hamiltonian) Rodrigues-Hamilton parameters, we present a method of stereographic projection of a point belonging to the three-dimensional sphere S 3 onto the oriented space R 3. We present a family of local kinematic parameters obtained by the method of mapping four symmetric kinematic parameters of the space R 4 onto the oriented real space R 3.
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 R 4.
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Original Russian Text © S.E. Perelyaev, 2009, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2009, No. 2, pp. 47–58.
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Perelyaev, S.E. On the correspondence between the three- and four-dimensional parameters of the three-dimensional rotation group. Mech. Solids 44, 204–213 (2009). https://doi.org/10.3103/S0025654409020058
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DOI: https://doi.org/10.3103/S0025654409020058