¥¯ ¥¯ ¥¯ ¥ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥¯ ¥ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥¯ ¥¯ ¥ ¥ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥¯ ¥ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥¯ ¥ ¥ ¥ ¥¯ ¥¯ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥ ¥¯ ¥ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥¯ ¥¯ ¥¯ ¥¯ ¥¯ ¥¯ ¥¯ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¨ ¥¯ ¯ Universit¥ at Fachbereich ¥¯ ¥¯ ¥ ¥ ¥¯ ¥ ¥¯ ¥ ¥¯ ¥ ¥¯ ¥ ¥ Stuttgart ¥Mathematik ¥¯ ¥¯ ¥¯ ¥ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥¯ ¥ ¥ ¥ ¥¯ ¥¯ ¥ A minimal atlas for¥ the rotation group SO(3) ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ Erik W. Grafarend, Wolfgang K¨uhnel ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥¯ ¥¯ ¥ ¥ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥¯ ¥ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥ ¥ ¥¯ ¥¯ ¥¯ ¥ ¥ ¥ Preprint 2010/015 Universitat¨ Stuttgart Fachbereich Mathematik A minimal atlas for the rotation group SO(3) Erik W. Grafarend, Wolfgang K¨uhnel Preprint 2010/015 Fachbereich Mathematik Fakultat¨ Mathematik und Physik Universitat¨ Stuttgart Pfaffenwaldring 57 D-70 569 Stuttgart E-Mail: [email protected] WWW: http://www.mathematik/uni-stuttgart.de/preprints ISSN 1613-8309 c Alle Rechte vorbehalten. Nachdruck nur mit Genehmigung des Autors. LATEX-Style: Winfried Geis, Thomas Merkle A minimal atlas for the rotation group SO(3) Erik W. Grafarend & Wolfgang Kuhnel¨ We describe explicitly an atlas for the rotation group SO(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. 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 descibed by a rotation matrix, an element of the three-dimensional rotation group SO(3). The rotation group is presented in various monographs, for instance in [5] and [7]. 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 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 [3], [4], [10], [18] and, in particular, the previous article [8] 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 SO(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 three-dimensional× subgroup of the 9-dimensional group GL(3, R). Compare [22] for the reduction of 9 parameters to 3 parameters. In particular SO(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 {3 ∈ 4 || || } manifold, is diffeomorphic with the 3-dimensional standard sphere S R . It is also isomorphic with the group SU(2). See [2] for the details. ⊂ It is necessary to have a parameterization of the rotation group by three independent parameters. In particular it is a very natural goal to define an atlas for the rotation group by charts. A chart is an injective and differentiable map Φ: U SO(3) of maximal rank where U is an open subset of R3 diffeomorphic with an open 3-dimensional→ ball. In other words: A chart Φ is a diffeomorphic map from U R3 onto its image Φ(U) SO(3). An atlas is a set of charts covering the entire group SO(3)⊂ such that all coordinate changes⊂ are differentiable maps. For a Lie group one may require that all coordinate changes are real analytic. Our contribution deals with the problem how to find such an atlas explicitly, especially one with a minimum number of maps. Here we can take advantage of the Lusternik-Schnirelmann category cat [12], [13], [14], [6]. The category of a manifold is defined as the minimum number of subsets of a covering such that each subset has a contractible neighborhood in the manifold. The number of critical points of a real function on M cannot be smaller than cat(M), see [23]. In particular the well known equation cat(SO(3)) = 4 indicates that at least four charts should be needed to cover the group SO(3) completely. This holds for any type of charts, not only for Euler charts or Cardan charts. From the practical point of view, the construction of a minimum atlas on SO(3) runs into the same problem as the construction of a minimum atlas on the unit sphere S2. For the unit sphere the equation CAT (S2) = 2 holds because the sphere is not contractible itself, and because we can cover it by two overlapping 2-discs around north and south pole. So in any case we need at least two distinct charts to cover the sphere completely. The standard spherical coordinates (spherical longitude and spherical latitude as a parameter system) become singular at the two poles. A well defined transverse coordinate system, also called meta-longitude and meta-latitude, 5 as a parameter system has to be constructed. The union of the charts longitude/latitude and meta-longitude/meta-latitude cover the unit sphere completely in the sense of a minimum atlas, see [9, Sect.3-3]. For the three-dimensional rotation group the singularities of the parameter system Euler angles, Cardan angles and Hamilton unit quaternions are well known. They have been already analysed in the monographs [16], [24] and [19]. For spaceborne gyroscopes, deforming in time, we refer to [20], [21]. For Euler angles and Cardan angles Rimrott [19] found six different charts which constitute a complete atlas. Thus it remained an open question to find four charts of Euler or Cardan type which constitute a minimum atlas for SO(3). Definition (Euler angles, Cardan angles) We say that two charts F1: U1 SO(3), F2: U2 SO(3) are of the same type if there is a → 3 → translation T in the parameter domain R and a rotation R SO(3) such that F2 = R F1 T . ∈ ◦ ◦ 1. By the Cayley parameters we mean the Cayley map C: so(3) SO(3) defined by → C(A)=(1+ A)(1 A)−1. − The exceptional locus SO(3) C(so(3)) consists of all rotation matrices with an angle π. The inverse map is explicitly given\ by C−1(B) = (B + 1)−1(B 1). − 2. The Geodesic polar coordinates are given by the exponential map exp: A so(3) A < π SO(3) { ∈ | || || } → with 1 exp(tA)= (tA)n = 1 + (sin t)A + (1 cos t)A2 X n! − n≥0 whenever A = 1. Here denotes the operator norm of a matrix. Again the exceptional locus consists of|| all|| rotation matrices|| || with an angle π. If B SO(3) is a matrix an angle distinct from π then we have tr(B) = 1, and B is not symmetric unless∈ it is the identity matrix 1 = exp(0). Hence we can recover6 the− corresponding A from the skew-symmetric part of B and its operator norm, where the scalar factor sin t uniquely determines the parameter t in the interval 0 <t<π. 3. The standard Euler chart is the following map F : U SO(3) for a suitable domain U R3 (for the precise choice of U see below) → ⊂ ∗ ∗ F (γ,α,γ )= Rz(γ) Rx(α) Rz(γ ) · · cos γ sin γ 0 1 0 0 cos γ∗ sin γ∗ 0 = sin γ −cos γ 0 0 cos α sin α sin γ∗ −cos γ∗ 0 · − · 0 01 0 sin α cos α 0 01 cos γ cos γ∗ sin γ cos α sin γ∗ cos γ sin γ∗ sin γ cos α cos γ∗ sin γ sin α = sin γ cos γ∗ +− cos γ cos α sin γ∗ − sin γ sin γ∗ +− cos γ cos α cos γ∗ cos γ sin α − − sin α sin γ∗ sin α cos γ∗ cos α The angles γ,α,γ∗ are called the Euler angles1 associated with the matrix on the right hand side. It is well known that the map F : R3 SO(3) is surjective, compare [17] for a theoretical foundation in terms of generators and relations→ of groups. 1named after Leonhard Euler (1707–1783) 6 4. The standard Cardan chart G: V SO(3) is the following map for a suitable domain V R3 (for the precise choice of V see below)→ ⊂ G(α,β,γ)= Rx(α) Ry(β) Rz(γ) · · 1 0 0 cos β 0 sin β cos γ sin γ 0 = 0 cos α sin α 01− 0 sin γ −cos γ 0 − · · 0 sin α cos α sin β 0 cos β 0 01 cos β cos γ cos β sin γ sin β = cos α sin γ sin α sin β cos γ cos α cos−γ + sin α sin β sin γ sin− α cos β − − sin α sin γ + cos α sin β cos γ sin α cos γ cos α sin β sin γ cos α cos β − Accordingly, the angles α,β,γ are called the Cardan angles2 associated with the matrix on the right hand side, sometimes also called Tait-Bryan angles or yaw, pitch and roll. The map G is surjective since the three standard rotations generated the entire rotation group. The exceptional sets of matrices where the Euler angles or the Cardan angles fail to be unique will be determined in the proof of our theorem below.