If we have two points A and B on the surface of a sphere, a geodesic between them, and another point C on the same sphere surface, but not on the geodesic, is there any concept of a "perpendicular" geodesic to AB that passes through C ?
Because I'm not able to describe the problem mathematically (because I don't know what is the exact concept I'm searching for, and I don't have the proper mathematical vocabulary), I'm going to describe the practical problem for which I need this.
A and B are two points on the surface of the Earth with a geodesic between them, C is another point on the surface of the Earth, which does not pass through AB, and I need to calculate the coordinates of D, on the AB geodesic, so that the geodesic distance between C and D is minimized. It is sort of a "shortest distance from point to line" problem applied to geodesics. In 2D geometry D would be the perpendicular foot from C to AB.
Consider the plane tangent to the sphere at a point where two geodesics intersect. Within this plane, there are two lines, each tangent to one of the geodesics. If these two lines are perpendicular, then the two corresponding geodesics are "perpendicular."
As for your problem, if there isn't a restriction against this, then you could stereographically project (link to Wikipedia) from the sphere onto the plane. Then, you can work in the plane and, when finished, map back into the sphere. This is a common technique in differential geometry.