I am having trouble deriving Crofton's formula for rectifiable curves in $\mathbb{R}^4$. My approach so far has been as follows. Hyperplanes in $\mathbb{R}^4$ may be given by the data of a real number $r$ and a point $v$ on $S^3$ via $(r,v)\mapsto\{x\,|\,\langle x,v\rangle=r\}$. There will be a formula of the form $$l(\gamma)=C\int_{S^3} \int_{\mathbb{R}}|\gamma\cap\Pi(r,v)|K(r,v)d\lambda(r)dS(v)$$ for some universal constant $C>0$, where $K(r,v)$ is a function which ensures the right hand side is invariant under rigid motions, $\Pi(r,v)$ is the aforementioned hyperplane associated to the pair $(r,v)$, $d\lambda$ is the Lebesgue measure and $dS$ is the surface measure of $S^3$. We need to calculate $K$ and $C$.
For $K$, we observe that the action of the isometry group of $\mathbb{R}^4$ doesn't distort space (that is, the Jacobian of the left-action of any element in this group is 1). Therefore by the change of variable formula $K$ should be identically 1.
For $C$, it suffices to compute the integral for a simple choice of $\gamma$. I choose $\{0\leq x\leq1\}\cap\{y=z=w=0\}$, a line segment of length 1 on the $x$-axis. Using spherical coordinates $$x=\cos(\theta),\;y=\sin(\theta)\cos(\phi),\;z=\sin(\theta)\sin(\phi)\cos(\psi),\;w=\sin(\theta)\sin(\phi)\sin(\psi)$$ where $0\leq\theta,\phi\leq\pi$ and $0\leq\psi<2\pi$, I think the hyperplane $\Pi(r,\theta,\phi,\psi)$ intersects $\gamma$ if and only if $0<|r|<\cos\theta\leq1$, and in this case there is precisely 1 intersection point. Then \begin{align} 1&=C\int_{S^3} \int_{-\infty}^{\infty}|\gamma\cap\Pi(r,v)|drdS(v)\\&=2C\int_0^{2\pi}\int_0^{\pi}\int_0^{\pi} \int_0^{\infty}|\gamma\cap\Pi(r,v)|r^3\sin^2\theta\sin\phi drd\theta d\phi d\psi\\&=2C\int_0^{2\pi}\int_0^{\pi}\int_0^{\pi/2} \int_{0}^{\cos\theta}r^3\sin^2\theta\sin\phi drd\theta d\phi d\psi\\&=8C\pi\int_0^{\pi/2} \int_{0}^{\cos\theta}r^3\sin^2\theta drd\theta\\ &=2C\pi\int_0^{\pi/2} \cos^4\theta\sin^2\theta d\theta\\ &=\frac{C\pi^2}{16} \end{align} implying that $C=16/\pi^2$. Now I believe that Crofton's formula in this context is actually given by $$l(\gamma)=\frac{1}{\pi^2}\int_{S^3\times\mathbb{R}}|\gamma\cap\Pi(r,v)|d\lambda(r)dS(v)$$ as can be seen here on page 89 for example. I have made an error therefore, but I can't see where it is. I am not even totally sure if the logic in this approach is correct. Any help in identifying either of these things would be much appreciated.
