Map Two Polar Disks To Four Dimensional Spherical Coordinates ($\left(\mathbb{R}^2\right)^2\to\mathbb{R}^4$) We have two points $(x_1, y_1), (x_2, y_2) \in \mathbb{R}^2$ with polar parameterizations $(r_1, \theta_1), (r_2, \theta_2)$ (i.e., $x_1=r_1 cos(\theta)$ and so on.) How would the point $(x_1, y_1, x_2, y_2) \in \mathbb{R}^4$ be represented with four-dimensional spherical coordinates $(r, \phi_1, \phi_2, \phi_3)$?
Please correct my proposed answer if I have made a mistake or upvote it if you concur.
Finally, how would one inductively generalize equivalency to $\mathbb{R}^{2^n}$ or $\mathbb{R}^{2n}$?

While the other answer would take you to the standard spherical coordinates, the formula gets very unwieldy especially for higher dimensions. An alternative will get you the same radial coordinate but different angles
$$\phi_1=\theta_1$$
$$\phi_2=\theta_2$$
$$r_3\cos\phi = r_1$$
$$r_3\sin\phi = r_2$$
The interesting thing about these coordinates is instead of having the angles be maps $[0,2\pi]\times[0,\pi]^{2n-2}\to S^{2n-1}$, these are maps $[0,2\pi]^{n}\times\left[0,\frac{\pi}{2}\right]^{n-1}\to S^{2n-1}$ (embedded in $\Bbb{R}^{2n}$). The generalization to $\Bbb{R}^{2n}$ is simply setting all of the radial coordinates to the standard spherical coordinates in $\Bbb{R}^n$ but restricted to the first quadrant/octant/etc..
