Given a sphere with radius $r$ about a point $c$, what's the apparent angular radius $\alpha$ of that sphere from point $P$? In other words, if $\vec{o} = c - P$, what's the maximum angle another vector $\vec{v}$ may make with $\vec{o}$ to intersect the sphere?
At first I thought of a simple right triangle, with sides:
- $o = |\vec{o}|$
- $m = r$, perpendicular to $o$, from the sphere's center to its boundary
- $l$, closing the triangle.
Then, as long as $m \leq o$, $\alpha = \arctan(m / o) = \arctan(r / |c - P|)$. However, this answer states it's $\arcsin$ instead of $\arctan$. Now I'm lost.