TITLE: Practical drawing of geodesics
QUESTION [5 upvotes]: I want to use a computer to draw geodesics on a known parameterized surface of revolution, starting from a known point and at a known angle to the meridian.
What would be the easiest way of doing this? 
Some methods I've considered:

Using Clairaut's relation ($r \cos \theta = C$), and using
small increments, but I'm afraid that this is not a general solution, as this will cause the geodesic to
"get stuck" on parallels whenever the angle crosses zero.
Use the relation that for a surface given by:
$$\left(\varphi(v)\cos u, \varphi(v)\sin u, \psi(v)\right)$$ The geodesics are given by: $$\frac{du}{dv}=\frac{\sqrt{\varphi'^2+\psi'^2}}{\varphi\sqrt{\varphi^2-c^2}}\ \longrightarrow\ u(v_1) = u_0 + \int_{v_0}^{v_1} \frac{\sqrt{\varphi'^2+\psi'^2}}{\varphi\sqrt{\varphi^2-c^2}} dv$$ This presents the problem of finding the correct value of $c$ given the starting angle, and as user levap pointed out, has the same "getting stuck" problem, since $u(v)$ is no longer a function.
Various numerical methods which work for any triangulated surface, but I feel it would be a shame to use these when I know the exact form of the surface.

Are there any easier methods or modifications to the above?

REPLY [2 votes]: Triangulation need not any more be adopted once you know Clairaut's Law as operative and geodesic trajectory is found by quadrature for the surfaces of revolution.
EDIT2:
The parametrization you gave is not full, valid for all meridians.
Wlog you could consider 
$$ X(u,v) = (v \cos u, v \sin u, \psi (v) ) $$
$C$ is determined by initial condition:  $ C_{initial}=  v_0 \sin \theta_{0}. $
It is beneficial with respect to easily crossing  the parallels in numerical computations to work on arc length basis (primes) using Liouville's theorem for zero geodesic curvature lines aka  geodesics.
$$ \theta^{'}(s) = -\sin \theta \sin \phi / v $$
where the meridian has slope: 
$$ \dfrac{dv}{dz} = \tan \phi $$
and polar coordinate rotation:
$$ v\,u^{'} = \sin \theta $$ 
EDIT1:
Say you want to practically compute and construct a great circle on a sphere. Using Runge-Kutta_4 numerically integrate the set of equations to define the circle in space and supply boundary conditions.
$$ \theta ^{'} = -\sin \theta \sin \phi / v ;\;  \phi^{'} = 1/ (a \cos \phi ) ;\;   v^{'} = \sin \phi \cos \theta;\;  z^{'} = \cos \phi \cos \theta\; ; u{'}=  \sin \theta / v ;  $$
(The following is more advanced, after some experience gained in geodesic tracing:)
Geodesics appear to cross all parallels before the one corresponding to
$$ r_{min} = C, \theta = \pi/2 $$
There are however, three ways geodesics move during turn-around at above extreme parallel: 
The returning behavior types are commonly traced for:

Positive/Negative Gauss curvatures: Geodesics are returning after meeting parallel circle $ r_{min}< C_{initial} $ at turn-around.

Non-returning types are for Negative Gauss curvature:

Geodesics shoot through after meeting $ r_{min}= C$ if $ r_{min} > C_{initial} $
Geodesics are asymptotic (never reach)  $ r_{min}= C$ if $ r_{min} = C_{initial} $

The latter are demonstrated in WolframAlpha Demo:
ONE_sheet_Hyperboloid_geodesics