The Euler-angle pole: a degenerate parametrization of $SO(3)$ while the physical orientation remains regular

Pole values: by the composition law and $R_y(0)=I_3$ of [L1], $\mathrm{\Phi }(\alpha ,0,\gamma )=R_z(\alpha )R_z(\gamma )=R_z(\alpha +\gamma )$, so the value depends only on $\alpha +\gamma$. For every real $s$, $\mathrm{\Phi }(\alpha +s,0,\gamma -s)=R_z(\alpha +s+\gamma -s)=R_z(\alpha +\gamma )=\mathrm{\Phi }(\alpha ,0,\gamma )$, and the points $(\alpha +s,0,\gamma -s)$ approach $(\alpha ,0,\gamma )$ as $s\to 0$ while remaining distinct for $s\mathrm{\ne }0$. Hence $\mathrm{\Phi }$ is not injective on any neighbourhood of the pole point; this is clause 1.

Partial derivatives at the pole: differentiating with respect to $\alpha$ with $\beta ,\gamma$ fixed and using the product rule and [L1], ${\mathrm{\partial }}_{\alpha }\mathrm{\Phi }(\alpha ,\beta ,\gamma )=[e_3{]}_{\times }{R}_z(\alpha )R_y(\beta )R_z(\gamma )$, and at $\beta =0$ this is $[e_3{]}_{\times }{R}_z(\alpha +\gamma )$ by clause 1. Differentiating with respect to $\gamma$ gives ${\mathrm{\partial }}_{\gamma }\mathrm{\Phi }(\alpha ,\beta ,\gamma )=R_z(\alpha )R_y(\beta )R_z(\gamma )[e_3{]}_{\times }$; at $\beta =0$ the middle factor is $I_3$ and the factors $R_z(\gamma )$ and $[e_3{]}_{\times }$ commute by the identity $R_z(a)[e_3{]}_{\times }=[e_3{]}_{\times }{R}_z(a)$ of [L1], so ${\mathrm{\partial }}_{\gamma }\mathrm{\Phi }(\alpha ,0,\gamma )=R_z(\alpha )[e_3{]}_{\times }{R}_z(\gamma )=[e_3{]}_{\times }{R}_z(\alpha )R_z(\gamma )=[e_3{]}_{\times }{R}_z(\alpha +\gamma )$, the same value as ${\mathrm{\partial }}_{\alpha }\mathrm{\Phi }$.

Rank and absence of a local inverse: the Jacobian matrix $J\mathrm{\Phi }(\alpha ,0,\gamma )$ has columns ${\mathrm{\partial }}_{\alpha }\mathrm{\Phi },{\mathrm{\partial }}_{\beta }\mathrm{\Phi },{\mathrm{\partial }}_{\gamma }\mathrm{\Phi }$ [L3], and the first and third columns coincide by step 1.2. Thus the linear map $D\mathrm{\Phi }:{\mathbb{R}}^3\to M_3(\mathbb{R})\cong {\mathbb{R}}^9$ annihilates the nonzero vector $(1,0,-1)$ and has rank at most $2$, so it is not injective. Moreover, step 1.1 gives distinct points with equal images in every neighbourhood of the pole point, ruling out any local inverse, including a differentiable inverse onto a neighbourhood in $SO(3)$. This proves clause 2.

Angular velocity along a curve through the pole: let ${\mathrm{\Phi }}_t:=\mathrm{\Phi }(\alpha (t),\beta (t),\gamma (t))$; by the product rule and [L1] at $t=0$, where $\beta (0)=0$ and $R_y(0)=I_3$, ${\mathrm{\Phi }}_0^{\mathrm{\prime }}=({\alpha }^{\mathrm{\prime }}(0)+{\gamma }^{\mathrm{\prime }}(0))[e_3{]}_{\times }{R}_z(\alpha (0)+\gamma (0))+{\beta }^{\mathrm{\prime }}(0)R_z(\alpha (0))[e_2{]}_{\times }{R}_z(\gamma (0)),$ using the two computations of step 1.2 for the $\alpha$- and $\gamma$-directions and the middle derivative ${\mathrm{\partial }}_{\beta }\mathrm{\Phi }=R_z(\alpha )[e_2{]}_{\times }{R}_z(\gamma )$ at the pole. Multiplying on the left by ${\mathrm{\Phi }}_0^{\mathsf{T}}=R_z(-\gamma (0))R_z(-\alpha (0))={\mathrm{\Phi }}_0^{-1}$ (the matrix is orthogonal [L1]) and using the conjugation law $R[\omega {]}_{\times }{R}^{\mathsf{T}}=[R\omega {]}_{\times }$ of [L2] together with $R_z(-\gamma (0))e_2$ for the second term gives ${\mathrm{\Phi }}_0^{\mathsf{T}}{\mathrm{\Phi }}_0^{\mathrm{\prime }}=[({\alpha }^{\mathrm{\prime }}(0)+{\gamma }^{\mathrm{\prime }}(0))e_3+{\beta }^{\mathrm{\prime }}(0)R_z(-\gamma (0))e_2{]}_{\times },$ so by the injectivity clause of [L2] and the characterisation of the body angular velocity [L4], $\omega (0)=({\alpha }^{\mathrm{\prime }}(0)+{\gamma }^{\mathrm{\prime }}(0))e_3+{\beta }^{\mathrm{\prime }}(0)R_z(-\gamma (0))e_2$. The rates ${\alpha }^{\mathrm{\prime }}$ and ${\gamma }^{\mathrm{\prime }}$ enter only through the sum, so $({\alpha }^{\mathrm{\prime }}+h,{\beta }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }}-h)$ and $({\alpha }^{\mathrm{\prime }},{\beta }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})$ give the same angular velocity: the rate map is not injective at the pole. This is clause 3.

The regular physical path: $R(t)=R_y(\kappa t)$ has $R(0)=I_3$ and is the composition of the smooth coordinate-rotation curve with the smooth map $t\mapsto \kappa t$, hence of class $C^{\mathrm{\infty }}$ [L1]; by the derivative clause of [L1], $R^{\mathrm{\prime }}(t)=\kappa R_y(\kappa t)[e_2{]}_{\times }$, so $R(t{)}^{\mathsf{T}}{R}^{\mathrm{\prime }}(t)=[\kappa e_2{]}_{\times }$ for every $t$ and the body angular velocity is the constant vector $\kappa e_2$ by [L2] and [L4]; in particular the orientation path passes through the pole orientation $R(0)=I_3=\mathrm{\Phi }(0,0,0)$ with a well-defined, constant angular velocity. The failure of clause 2 is thus a failure of the parameter triple to be a chart at the pole, not a singularity of the orientation path or of its angular velocity. This proves clause 4.

Summary: the parametrization $\mathrm{\Phi }$ is smooth as a map, but at every pole point it is non-injective (clause 1), its derivative has rank at most $2$ and it admits no differentiable local inverse (clause 2), and the angular velocity does not uniquely determine the Euler-angle rates at a fixed pole triple (clause 3), while a physical orientation path through the same pole orientation is regular with a perfectly well-defined angular velocity (clause 4).