How to find the coordinates of the nearest point on the segment of the arc of the great circle to the given one?

Question

Mathematics

Given 3 points on a sphere using longitude and latitude. Two of these points form a segment of a large arc, the third is a random point on the surface of the same sphere.
It is necessary to find the coordinates of the nearest point on the obtained segment of the arc from the given third point.
And of course, I would like to choose a method that gives the smallest error in the calculations.
During the study of this issue, I managed to find a useful article in which there are formulas for calculating the distance from a point to a large arc, but I just can’t figure out how to derive coordinates from it.
https://www.movable-type.co.uk/scripts/latlong.htm...

Answer 1

Let the points be called A, B, C. An arc through A, B. The center of the sphere O.
One can construct a normal n for the plane of the great circle AOB. Find the intersection point M of the line passing through C parallel to n. then the point can be found by stretching this vector OM to the radius of the sphere.
Well, we need to consider cases when the nearest point is the end of the arc.