Rotations in three-dimensional Euclidean space

Navaza, J.

doi:10.1107/97809553602060000682

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 13.2, pp. 269-270 | 1 | 2 |

Section 13.2.2. Rotations in three-dimensional Euclidean space

J. Navaza^a ^*

^aLaboratoire de Génétique des Virus, CNRS-GIF, 1. Avenue de la Terrasse, 91198 Gif-sur-Yvette, France
Correspondence e-mail: jnavaza@pasteur.fr

13.2.2. Rotations in three-dimensional Euclidean space

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A rotation R is specified by an oriented axis, characterized by the unit vector u, and the spin, χ, about it. Positive spins are defined by the right-hand screw sense and values are given in degrees. An almost one-to-one correspondence between rotations and parameters (χ, u) can be established. If we restrict the spin values to the positive interval $[0 \leq \chi \leq 180]$ , then for each rotation there is a unique vector χu within the sphere of radius 180. However, vectors situated at opposite points on the surface correspond to the same rotation, e.g. (180, u) and (180, −u).

When the unit vector u is specified by the colatitude ω and the longitude $[\varphi]$ with respect to an orthonormal reference frame (see Fig. 13.2.2.1a), we have the spherical polar parameterization of rotations (χ, ω, φ). The range of variation of the parameters is $[0 \leq \chi \leq 180\hbox{;}\ 0 \leq \omega \leq 180\hbox{;}\ 0 \leq \varphi \lt 360.]$ Rotations may also be parameterized with the Euler angles (α, β, γ) associated with an orthonormal frame (x, y, z). Several conventions exist for the names of angles and definitions of the axes involved in this parameterization. We will follow the convention by which (α, β, γ) denotes a rotation of α about the z axis, followed by a rotation of β about the nodal line n, the rotated y axis, and finally a rotation of γ about p, the rotated z axis (see Fig. 13.2.2.1b): $[{\bf R}(\alpha,\beta,\gamma) = {\bf R}(\gamma,{\bf p}){\bf R}(\beta,{\bf n}){\bf R}(\alpha,{\bf z}). \eqno(13.2.2.1)]$ The same rotation may be written in terms of rotations around the fixed orthonormal axes. By using the group property $[{\bf TR}(\chi,{\bf u}){\bf T}^{-1} = {\bf R}(\chi,{\bf Tu}), \eqno(13.2.2.2)]$ which is valid for any rotation T, we obtain (see Appendix 13.2.1) $[{\bf R}(\alpha, \beta, \gamma) = {\bf R}(\alpha, {\bf z}){\bf R}(\beta, {\bf y}){\bf R}(\gamma, {\bf z}). \eqno(13.2.2.3)]$ The parameters (α, β, γ) take values within the parallelepiped $[0 \leq \alpha \lt 360\hbox{;}\ 0 \leq \beta \leq 180\hbox{;}\ 0 \leq \gamma \lt 360.]$ Here again, different values of the parameters may correspond to the same rotation, e.g. (α, 180, γ) and $[(\alpha - \gamma, 180, 0)]$ .

Figure 13.2.2.1| top | pdf |

Illustration of rotations defined by (a) the spherical polar angles (χ, ω, φ); (b) the Euler angles (α, β, γ).

Although rotations are abstract objects, there is a one-to-one correspondence with the orthogonal matrices in three-dimensional space. In the following sections, R will denote a $[3 \times 3]$ orthogonal matrix. An explicit expression for the matrix which corresponds to the rotation (χ, u) is $[\left[\matrix{\cos \chi + u_{1} u_{1} (1 - \cos \chi) &u_{1} u_{2} (1 - \cos \chi) - u_{3} \sin \chi &u_{1} u_{3} (1 - \cos \chi) + u_{2} \sin \chi\cr u_{2} u_{1} (1 - \cos \chi) + u_{3} \sin \chi &\cos \chi + u_{2} u_{2} (1 - \cos \chi) &u_{2} u_{3} (1 - \cos \chi) - u_{1} \sin \chi\cr u_{3} u_{1} (1 - \cos \chi) - u_{2} \sin \chi &u_{3} u_{2} (1 - \cos \chi) + u_{1} \sin \chi &\cos \chi + u_{3} u_{3} (1 - \cos \chi)\cr}\right] (13.2.2.4)]$ or, in condensed form, $[{{\bf R}(\chi, {\bf u})_{ij} = \delta_{ij} \cos \chi + u_{i} u_{j} (1 - \cos \chi) + {\textstyle\sum\limits_{k=1}^{3}} \varepsilon_{ikj} u_{k} \sin \chi,} \eqno(13.2.2.5)]$ where $[\delta_{ij}]$ is the Kronecker tensor, $[u_{i}]$ are the components of u, and $[\varepsilon_{ijk}]$ is the Levi–Civita tensor. The rotation matrix in the Euler parameterization is obtained by substituting the matrices in the right-hand side of equation (13.2.2.3) by the corresponding expressions given by equation (13.2.2.4).

13.2.2.1. The metric of the rotation group

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The idea of distance between rotations is necessary for a correct formulation of the problem of sampling and for plotting functions of rotations (Burdina, 1971; Lattman, 1972). It can be demonstrated that the quantity $[\hbox{d}s^{2} = \hbox{Tr} \left(\hbox{d}{\bf R}\ \hbox{d}{\bf R}^{+}\right) = {\textstyle\sum\limits_{i, \, j=1}^{3}} (\hbox{d}R_{ij})^{2} \eqno(13.2.2.6)]$ defines a metric on the rotation group, unique up to a multiplicative constant, which cannot be reduced to a Cartesian metric. This is a topological property of the group, independent of its parameterization. ds is interpreted as the distance between the rotations R and $[{\bf R} + \hbox{d}{\bf R}]$ . With the Euler parameterization, equation (13.2.2.6) becomes $[\hbox{d}s^{2} = \hbox{d}\alpha^{2} + 2\cos (\beta)\ \hbox{d}\alpha\ \hbox{d}\gamma + \hbox{d}\gamma^{2} + \hbox{d}\beta^{2}. \eqno(13.2.2.7)]$ The volume element of integration, $[\sin (\beta)\ \hbox{d}\alpha\ \hbox{d}\beta\ \hbox{d}\gamma]$ , corresponding to this length element guarantees the invariance of integrals over the rotation angles with respect to initial reference orientations.

The distance defined by equation (13.2.2.6) has a simple physical interpretation. Let us consider a molecule with initial atomic coordinates $[\{{\bf x}\}]$ , referred to an orthonormal frame parallel to the molecule's principal moments of inertia, $[I_{i}]$ . Then, the coordinates satisfy the conditions $[\eqalignno{\langle x_{i} \ x_{j} \rangle &= 0 \quad \hbox{ if } i \neq j,\cr \langle x_{i} \ x_{i} \rangle &= I_{i}, &(13.2.2.8)}]$ where $[\langle\ldots \rangle]$ means `average over atoms'. If we move the molecule from a rotated position characterized by the rotation R to a close one characterized by $[{\bf R} + \hbox{d}{\bf R}]$ , the mean-square shift of the atomic coordinates is $[\sigma^{2} = \langle (\hbox{d} {\bf R \ x)}^{2} \rangle = {\textstyle\sum\limits_{i=1}^{3}}\ I_{i} {\textstyle\sum\limits_{j=1}^{3}} (\hbox{d}R_{ij})^{2}. \eqno(13.2.2.9)]$ When all the $[I_{i}]$ are equal, σ becomes proportional to ds.

References

Burdina, V. I. (1971). Symmetry of rotation function. Sov. Phys. Crystallogr. 15, 545–550.Google Scholar

Lattman, E. E. (1972). Optimal sampling of the rotation function. Acta Cryst. B28, 1065–1068.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 13.2, pp. 269-270