Fun thing to think about: Parametrize a sphere from the outside

Question

Suppose I have a sphere of radius R centered at $(x_0,0,0)$ where $x_0 > R$. I would like to parametrize the "face" of the sphere closest to the origin, using the angles $\theta, \phi$, where $\theta$ is the angle between the x-axis and the ray extending from the origin to a point on the sphere, and $\phi$ is the parameter that allows for "rotation" about the x-axis. I've been thinking about this for some time, and I'm struggling to come up with a solution. Is this even possible given the parameters listed? I would think yes.

Answer 1

Ignoring the $\phi$ angle, which is easily dealt with, and focusing on the $xy$-plane (say), simply solve for $$t(\cos(\theta),\sin(\theta))=(x_0,0)+R(\cos(\varphi),\sin(\varphi)),$$ where $t$ is the irrelevant parameter (distance to origin), and $\varphi$ becomes the angle relative to the sphere.

The maximum angle that's allowed is when the ray becomes tangent to the sphere at intersection, which occurs at $\theta=\arcsin(R/x_0)$.