Geodesics in a surface of revolution

Consider a surface of revolution with respect to $z$-axis and suppose that the generating curve $s\mapsto (f(t),0,g(t))$ is parametrized by the arc-length. Let $\alpha(s)=X(u(s),v(s))$ be a geodesic where $X$ is the parametrization of the surface given by

$$X(u,v)=(f(u)\cos(v),f(u)\sin(v),g(u)).$$Suppose that $\alpha$ is a geodesic. Then we know that the functions $u(s)$, and $v(s)$ satisfy  two equations involving the Christoffel symbols. We pay attention in the second one, which is $$v''+\Gamma_{11}^2 u'^2+2\Gamma_{12}^2u'v'+\Gamma_{22}^2v'^2=0.$$

The computation of these symbols are: $\Gamma_{11}^2=\Gamma_{22}^2=0$ and $\Gamma_{12}^2=f'/f$. Thus the equation writes as

$$v''+\frac{f'}{2f}u'v'=0.$$ We deduce that $$(f^2v')'=2ff'v'+f^2v''=0.$$

If we compute the angle $\theta(s)$ that makes $\alpha$ with each parallel that meets, we have

$$\cos\theta=\frac{\langle u'X_u+v' X_v,X_v\rangle}{1}=v'$$ because $\alpha$ is parametrized by the arc-lengt since it is a geodesic. As a conclusion

Theorem: In a surface of revolution, we have that the angle that makes a geodesic with every parallel that intersects satisfies

$$f(u(s))^2\cos\theta(s)=\mbox{constant}.\quad (*)$$

Application. A sailor what to find/keep a specific path along a trip on the Earth. He needs to know at every time what is its position. The knowledge of the parallel is easy measuring with respect to the Polar star and this gives the value of $f$ by the radius of the Earth. Thus if the sailor want to follow a way doing a given angle $\theta$ with the parallel, he only has to maintain the value of the constant at (*): the value of $v(s)$ indicates which is the meridian where he is positioned.