The wheel rotates rigidly, and at constant angular velocity about the hub. Every instance of the wheel is governed by r, the radius of the wheel, u, the velocity of the rim, and t, the time at the hub.

In the non-rotating inertial frame associated with the position and movement of the hub, every point on the wheel is described by its angular position, θ, and its distance from the hub, l. For points on the rim, l = r. For points on the k-th spoke, $$\theta = \frac{2 \pi k}{n}$$.

Putting this all together, we have the following family of worldlines:

$$\textrm{Wheel} \left( r, u; \theta, \ell, t \right) = \begin{pmatrix} \ell \cos \left( \theta + \frac{ u t }{ r } \right) \\ \ell \sin \left( \theta + \frac{ u t }{ r } \right) \\ 0 \\ t \end{pmatrix}, \quad \textrm{Rim}\left( r, u ; \theta, t \right)= \textrm{Wheel} \left( r, u ; \theta, r, t \right), \quad \textrm{Spoke}\left( r, u, n ; k, \ell, t \right) = \textrm{Wheel} \left( r, u ; \frac{ 2 \pi k }{n}, \ell, t \right)$$

We can speak of relative coordinates of θ, l and t:

$$\Delta \textrm{Wheel} \left( r, u; \theta_1, \ell_1, t_1 ; \theta_0, \ell_0, t_0\right) = \begin{pmatrix} \ell_1 \cos \left( \theta_1 + \frac{ u t_1 }{ r } \right) - \ell_0 \cos \left( \theta_0 + \frac{ u t_0 }{ r } \right) \\ \ell_1 \sin \left( \theta_1 + \frac{ u t_1 }{ r } \right) - \ell_0 \sin \left( \theta_0 + \frac{ u t_0 }{ r } \right) \\ 0 \\ t_1 - t_0 \end{pmatrix}$$ with associated Lorentz invariant: $$c^2 (\Delta \tau)^2 = c^2 ( t_1 - t_0)^2 - \ell_0^2 - \ell_1^2 + 2 \ell_0 \ell_1 \cos \left( \theta_1 - \theta_0 + \frac{u}{r} \left( t_1 - t_0 \right) \right) = c^2(\Delta t)^2 - (\Delta \ell)^2 \cos^2 \frac{\Delta \theta + \frac{u}{r} \Delta t}{2} - (\Sigma \ell)^2 \sin^2 \frac{\Delta \theta + \frac{u}{r} \Delta t}{2}$$

Lorentz boosting in the x-direction by the amount v, we get $$ \begin{pmatrix} x' \\ y' \\ z' \\ t' \end{pmatrix} = \textrm{Wheel}' \left( r, u; \theta, \ell, t \right) = \Lambda \textrm{Wheel}\left( r, u; \theta, \ell, t \right) = \begin{pmatrix} \left(\cosh \tanh^{-1} \frac{v}{c} \right) \ell \cos \left( \theta + \frac{ u t }{ r } \right) + \left(\sinh \tanh^{-1} \frac{v}{c} \right) c t \\ \ell \sin \left( \theta + \frac{ u t }{ r } \right) \\ 0 \\ \left(\cosh \tanh^{-1} \frac{v}{c} \right) t + \left(\sinh \tanh^{-1} \frac{v}{c} \right) \frac{\ell}{c} \cos \left( \theta + \frac{ u t }{ r } \right) \end{pmatrix}$$ but this does not let you talk about the shape yet, since x', y' and z' are written in terms of planes which are slices of constant t not constant t'.

Solving $$t' = \left(\cosh \tanh^{-1} \frac{v}{c} \right) t + \left(\sinh \tanh^{-1} \frac{v}{c} \right) \frac{\ell}{c} \cos \left( \theta + \frac{ u t }{ r } \right) = \frac{c^2 t + \ell v \cos \left( \theta + \frac{u t }{r} \right) }{c \sqrt{c^2-v^2}} $$ for a general expression for t in terms of t' looks hopeless.

/// Ran out of time