Shortest path to a geodesic

Question

This question has already been asked here, but has not been answered fully. I really want to know the answer, so I ask again.

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.

Answer 1

It's possible I'm not understanding your question correctly, but let's think of this (more general) approach:

Let $\alpha:I\to M$ be an curve on some Riemannian manifold $(M,g)$ (this could be your geodesic connecting $A$ and $B$). Then under suitable conditions, we can treat the image as a $1$-dimensional submanifold. Then you can consider your length function $L:\Omega\to\mathbf{R}$, where $\Omega$ is the space of of all piecewise smooth curves $\gamma:[a,b]\to M$ with the conditions that $\gamma(a)\in\alpha(I)$, $\gamma'(a)\in T_{\gamma(a)}\alpha(I)^\perp$, and $\gamma(b)=C$. The usual variational analysis of such a functional/space would then apply. That is, your critical points of the functional should be geodesics, and they should minimize if the second variation is positive definite on $T_\gamma\Omega$ (your second variation should contain the shape operator of the submanifold $\alpha(I)\times\{C\}\subset M\times M$).

I'm not sure where exactly your question is going, but hopefully I've helped guide it in some way. There is a good source (though, I don't suggest it if it's you're first attempt at learning Riemannian geometry) with Sakai's "Riemannian Geometry" text that covers such methods.