The global orientation and its charts: Euler angles, the quaternion double cover and singularities of parametrizations

For the model of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope, the orientation of the body at time $t$ is the element $R(t)\in SO(3)$ of the configuration (Configuration, configuration space and orientation of a rigid-material model in an inertial frame): a global object attached to the placement, unique for a nondegenerate labeling. As a subset of $M_3(\mathbb{R})\cong {\mathbb{R}}^9$, the group $SO(3)$ is a compact three-dimensional embedded submanifold and a matrix group ($SO(3)$ is a compact three-dimensional embedded submanifold of $M_3(\mathbb{R})$, with tangent space the matrices $R\mathrm{\Omega }$ for skew $\mathrm{\Omega }$ clauses 1 and 3, $SO(3)$ is a subgroup of the general linear group, with $R^{-1}=R^{\mathsf{T}}$). Three descriptions of it are used in the literature and on the pages that follow, and they must not be conflated.

Euler angles are local coordinates. The Euler-angle parametrization of The Euler-angle parametrization and the heavy symmetric top system is $\mathrm{\Theta }:\mathbb{R}\times (0,\pi )\times \mathbb{R}⟶SO(3),\mathrm{\Theta }(\phi ,\theta ,\psi )=R_z(\phi )R_y(\theta )R_z(\psi ),$ with the coordinate rotation matrices of The coordinate rotation matrices $R_z(a)$ and $R_y(a)$: values in $SO(3)$, composition law, and derivatives, and its domain is deliberately restricted to $\theta \in (0,\pi )$ so that $\sin \theta >0$. On that domain it is a local parametrisation with smooth local inverse. Indeed the angular-velocity formula in Angular velocity in Euler angles: ${\omega }_3={\psi }^{\mathrm{\prime }}+{\phi }^{\mathrm{\prime }}\cos \theta$ and ${\omega }_1^2+{\omega }_2^2={\theta }^{\mathrm{\prime }2}+{\phi }^{\mathrm{\prime }2}{\sin }^2\theta$ identifies ${\mathrm{\Theta }}^{\mathsf{T}}d\mathrm{\Theta }$ with the skew matrix of $M(\phi ,\theta ,\psi )(d\phi ,d\theta ,d\psi {)}^{\mathsf{T}}$, where $M=\left(\begin{array}{ccc}-\sin \theta \cos \psi & \sin \psi & 0\\ \sin \theta \sin \psi & \cos \psi & 0\\ \cos \theta & 0& 1\end{array}\right),\det M=-\sin \theta \mathrm{\ne }0.$ Thus $d\mathrm{\Theta }$ has zero kernel. Its domain and $SO(3)$ both have dimension three, so it is a tangent-space isomorphism. The map is smooth because its matrix entries are finite products of smooth sines and cosines, and The smooth inverse function theorem on manifolds gives a smooth local inverse; strictly speaking that inverse is the coordinate chart. The boundary values $\theta =0$ and $\theta =\pi$ (the poles) are excluded because there the dependence on the angles collapses: by the composition law $R_z(a)R_z(b)=R_z(a+b)$ the value at $\theta =0$ depends on $\phi +\psi$ alone. A smooth orientation path of the model can pass through a pole orientation, including $I_3$. The restricted chart has no coordinate value there, and the extended formula has no local inverse at a pole: at $\theta =0$, the distinct triples $(\phi +s,0,\psi -s)$ give the same rotation for arbitrarily small nonzero $s$. This does not exclude a continuous pathwise choice of unrestricted parameters; for example, $(\phi ,\theta ,\psi )=(0,t,0)$ represents the smooth path $R_y(t)$ through $t=0$. The companion example computes the deficient differential at a pole. Euler angles are auxiliary local coordinates for the global orientation. The angular-velocity components are not the coordinate derivatives of these angles (Angular velocity in Euler angles: ${\omega }_3={\psi }^{\mathrm{\prime }}+{\phi }^{\mathrm{\prime }}\cos \theta$ and ${\omega }_1^2+{\omega }_2^2={\theta }^{\mathrm{\prime }2}+{\phi }^{\mathrm{\prime }2}{\sin }^2\theta$ is the separate relation, and the model's angular velocity is defined intrinsically in Body and spatial angular velocity of a rigid-material motion, and the material velocity field).

Quaternion lifts are two-valued, not a single-valued chart. Under the countable choice hypothesis recorded in The quaternion rotation map $\rho$: conjugation on the imaginary quaternions, the unit sphere $S^3$, and quaternion lifts, the unit quaternions $S^3$ double-cover $SO(3)$ through the rotation map $\rho$: every orientation has exactly two preimages $q$ and $-q$. Given a $C^1$ rotation path $R$ and an initial unit quaternion $q_0$ with $\rho (q_0)=R(t_0)$, there is exactly one $C^1$ lift $q$ with $q(t_0)=q_0$, and every $C^1$ lift is either $q$ or $-q$ (Existence, uniqueness and sign ambiguity of the quaternion lift of a $C^1$ rotation path). The sign of a lift is thus an initial choice that the orientation path itself does not determine, and no construction on these pages chooses a preferred sign. A quaternion path is a global description of a rotation path in the sense that it avoids the Euler pole, but it represents each orientation twice; it is not a chart of $SO(3)$ and does not select a canonical sign.

• Charts are auxiliary. A consumer that differentiates with respect to Euler angles must restrict to $\theta \in (0,\pi )$, state the convention of the three ordered rotations, and not divide by quantities that vanish at the poles without an explicit limiting argument. The model's orientation $R(t)$ and its angular velocity $\omega (t)$ are defined without any chart (Body and spatial angular velocity of a rigid-material motion, and the material velocity field), and chart-based formulas are read as expressions of these objects on the chart domain.

• The double cover is a mathematical statement, not spin. The two-to-one correspondence between unit quaternions and orientations, and the sign ambiguity of a lift, are mathematical facts about $SO(3)$ and its covering (The quaternion rotation map $\rho$: conjugation on the imaginary quaternions, the unit sphere $S^3$, and quaternion lifts, Existence, uniqueness and sign ambiguity of the quaternion lift of a $C^1$ rotation path). They carry no physical claim about quantum spin, identical particles, relativistic observers or any measurement; such statements belong to other frameworks and are not imported here.

• Degenerate labelings. For a labeling supported on a line the placement does not determine $R$ (Configuration, configuration space and orientation of a rigid-material model in an inertial frame clause (c)), so the orientation is not observable from the placed material points alone. A consumer treating such a body must state which additional data fix the orientation, or restrict to the nondegenerate case.

• Empirical status. The orientation of any concrete body is estimated by instruments with their own calibration and uncertainty; this item records no measured orientation and no measurement procedure. The model quantities are conditional data of the model, exactly as in Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope.