Given two points, $A$ and $B$ that define a great circle on a unit hypersphere, how do I find the point $D$ on that great circle that is closest to the point $C$. (Closest in the sense of having the shortest distance along a great circle)
I have the points as vectors of numbers in Euclidian space. If all the points are near each other, I can approximate an answer by projecting C onto a straight line though AB in Euclidean space, then normalising the result to put it on the sphere. I'm after a more general, more elegant solution.
This answer to a similar question makes sense:
Let be the target point and the arc. Consider an arc through that is perpendicular to . Then the point in closest to lies in the intersection of with .
Except I don't know how to find an arc that goes through C and that's perpendicular to the AB arc.
I don't know what the thing I'm trying to do is called, and may be using terms incorrectly - I welcome suggestions for better terminology.