I have a variety of related questions that all seem to involve deeper math that I'm used to dealing with (undergraduate calculus and linear algebra). My feeling is that these are classical (and solved) questions in (advanced) analytic geometry.
Consider a fixed ball $S$ in $\mathbb{R}^3$ and some axis of rotation $\mathbf{a}$.
I) For an arbitrary plane $\pi$, determine the set of angles $\theta$ for which the rotated ball intersects the plane. A little easier (but still beyond me); determine if that set is empty. This second problem seems to require finding the (smallest) distance from the plane $\pi$ to a point on a circle (the trajectory of the ball centre under the given rotation). That seemingly elementary problem still eludes me.
II) Same as I, but with an arbitrary line $l$ instead of the plane $\pi$.
There are a lot of special cases to consider, so I am happy to focus on the particular case where the ball has radius 1 and is centered at (2,0,0) and the axis of rotation is the positive z-axis.
Thanks.