Suppose I have a point $P$ on a unit sphere whose spherical coordinates are $(\theta, \varphi)$, and a great arc from point $Q$ to point $R$, also specified in spherical coordinates. I want to find the minimum distance (great arc distance, preferably) between the point and the arc. Can somebody help me on that?
My best guess would be to take the formula for the Great arc distance between two points on a unit sphere, plug in a point $S$ parameterized by $t$ on the arc between $Q$ and $R$, and find the minimum distance between $P$ and $S$. But I would definitely need help with that.