A minimal atlas for the rotation group SO (3)
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A minimal atlas for the rotation group S O(3) Erik W. Grafarend · Wolfgang Kühnel
Received: 22 December 2010 / Accepted: 26 April 2011 / Published online: 15 May 2011 © Springer-Verlag 2011
Abstract We describe explicitly an atlas for the rotation group S O(3) consisting of four charts where each chart is defined by Euler angles or each chart is defined by Cardan angles. This is best possible since it is well known that three charts do not suffice. Keywords
Euler angles · Cardan angles · Quaternions · Chart · Gimbal lock
Mathematics Subject Classification (2000) 20H15 · 53Z05 · 57R55 · 86A30
Primary: 22E70; Secondary:
It is our daily experience that the Earth rotates, and it is our yearly experience that the Earth revolves around the Sun. The rotation of the Earth is nowadays described by a rotation matrix, an element of the three-dimensional rotation group S O(3). The rotation group is presented in various monographs, for instance in Carmeli and Malin (1976) and Gel’fand et al (1963). The rotation matrix of the Earth is given by the International Earth Rotation and Reference Service (IERS) in terms of daily, monthly and yearly data, namely for precession/nutation versus polar motion/length of day variations (POM/LOD). The problem we are discussing here originates in the various parameter systems of the rotation of rigid or deformable bodies. The characteristic equations are the
E. W. Grafarend (B) Institute of Geodesy, University of Stuttgart, Stuttgart, Germany e-mail: [email protected]; [email protected]; [email protected] W. Kühnel Institute of Geometry and Topology, University of Stuttgart, Stuttgart, Germany e-mail: [email protected]
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Int J Geomath (2011) 2:113–122
kinematic Euler equations and the dynamic Euler equations parameterized in terms of Euler or Cardan angles. For a deformable body the dynamic Euler equations are generalized into Euler-Liouville equations. As another parameter system Hamilton’s unit quaternions are used. General references are Burša (1979), Carmeli (1968), Hamilton (1981), Richter et al (2010) and, in particular, the previous article Grafarend (2009) by the first author on the same problem and the references quoted there. The Earth has to be considered as a gyroscope with exotic movements like precession and nutation in an inertial frame of reference or polar motion and length of day variation in an Earth-fixed frame of reference. These movements are described by elements of the rotation group S O(3) which is defined as the set of all real (3×3)-matrices with det A = 1 and with three orthonormal rows and columns. It is a compact threedimensional subgroup of the 9-dimensional group G L(3, R). Compare Stuelpnagel (1964) for the reduction of nine parameters to three parameters. In particular S O(3) is a connected Lie group and an analytic 3-manifold. Its universal 2-sheeted covering is the group Spin(3) which can be identified with the set of all unit quaternions H1 = Sp(1) = {q ∈ H ||q|| = 1} which, as a ma
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