On a surface of revolution using cylindrical coordinates $r,z $ primed with respect to $ \theta $ how to trace geodesics on a cone $ r= z \tan \gamma \; $ with metric
$$ \; ds= \sqrt{r^2-r^{'2}+z^{'2}}\; d\theta\; ?\; \tag1 $$
Using Euler-Lagrange equation on above metric with cone $z'= r'\cot\gamma$ we obtain a new geodesic definition:
$$ r. \frac{r}{\sqrt{r^2-r^{'2}+z^{'2}}}=c \tag2 $$
To solve this
$$ \frac{r^4}{c^2}= r^2-r^{'2}+z^{'2} $$
$$ \int \dfrac{dr}{r\sqrt{1-\dfrac{r^2}{c^2}}} =\dfrac{\theta}{ (1-\cot^2\gamma) } \tag3 $$
With $\theta$ parameter integrates to :
$$\tanh\frac{r(\theta)}{2c}=\tan\frac{\theta}{2\sqrt{1-\cot^2 \gamma}}; z(\theta ) = r(\theta)\cot\gamma \tag4 $$
Background how it was obtained:
With Euclidean metric $ds= \sqrt{r^2+r^{'2}+z^{'2}}\; $ and Euler-Lagrange with Beltrami special relation ( not dependent on $\theta$)
$$ \sqrt{r^2+r^{'2}+z^{'2}}- r'\frac{r'}{\sqrt{r^2+r^{'2}+z^{'2}}}-z'\frac{z'}{\sqrt{r^2+r^{'2}+z^{'2}}}= r. \frac{r}{\sqrt{r^2+r^{'2}+z^{'2}}}=c $$
that simplifies to:
$$ r\cdot \sin \psi = c$$
which is the Clairaut relation.
Your comments will be most appreciated in finding a geometric interpretation of this length invariant $c$ in this research effort. Shown blue rings have radius $c$.
As is known in elliptic geometry the invariant $c$ represents the Clairaut minimum radial distance of geodesic to axis of symmetry.