If two points $P,Q$ which are not umbilical are given, then there are in general several geodesics but only one that is shortest among them all. Their  $ u^{'}, v^{'}$ are different, but unique for any one of the geodesic choices.

If $P,u^{'}, v^{'}$ are given there is no guarantee that the line so defined passes through $Q.$ 

These can be physically verified between two points on a cylinder/cone using a taut thread for example. 

EDIT1:

It can be compared to the dynamic geodesic trajectory situation when a gun is fired with a given velocity and inclination from start point $P$. The initial velocity and direction should be adjusted to make trajectory pass through another desired point $Q$ in the vertical plane, as it cannot meet a desired point with arbitrary setting of dynamic parameters like $ u^{'}, v^{'} $ in case of geometrical geodesics.

EDIT2:

The solution also reflects in numerical procedures adopted. We have both initial value and boundary value problems. The former is straight forward and for the latter position/ derivative values at $P$ and $Q$ are given as input and numerical iterative (again!) shoot-through technique is employed.