Take the 2-minute tour ×  I am programming an algorithm where I
have broken up the surface of a sphere into grid points (for
simplicity I have the grid lines are parallel and perpendicular
to the meridians). Given a point A on the sphere, I would like
to be able to efficiently take any grid "square" and determine
the point B in the square with the least spherical coordinate
distance AB.    • Near the poles, the "squares" degenerate into
"triangles".   • As noted by Trithematician, this is a special
case of to finding the shortest distance from a point to an arc
on a sphere. For the longitudinal lines, these are arcs of great
circles, but for the latitudinal they are not.   • TonyK
provides a method below that solves the longitudinal case.   •
Minimising the distance in 3d also minimises the distance on the
sphere  I actually asked this question here:
http://stackoverflow.com/questions/1463606/closest-grid-square-
to-a-point-in-spherical-coordinates and decided to use an
approximation instead, but I am still cu