I am trying to calculate a partial surface of a prolate spheroid in terms of the polar and azimuthal angles. I have been searching the internet for quite some time, but without finding a solution or an idea how to solve it that I understood.
So, here is what I think is the closest I got to a solution.
The general form of a spheroid:
$$\frac{x^2+y^2}{a^2}+\frac{z^2}{c^2}=1$$ with $c>a$ making it prolate.
It can be parameterised in the following form
$$x=a \> \cos(\phi) \> \sin(\theta)$$
$$y=a \> \sin(\phi) \> \sin(\theta)$$
$$z=c \> \cos(\theta)$$
with $\phi \in [0, 2\pi]$ and $\theta \in [0, \pi]$.
If I determine the coefficients of the first fundamental form $S = \int_{\phi_0}^{\phi_1}d\phi\int_{\theta_0}^{\theta_1}\sqrt{EG-F^2} \> d\theta$, they turn out to be
$$E = a^2 \> \sin^2(\theta)$$
$$F = 0$$
$$G = \frac{a^2+c^2+(a^2-c^2) \> \cos(2 \> \theta)}{2}$$
This was actually the approach as described on Wolfram Mathworld for a general spheroid. If the coefficients are plugged into the above equation for $S$, it results in $$S = \frac{a}{\sqrt{2}}\int_{\phi_0}^{\phi_1}d\phi\int_{\theta_0}^{\theta_1}\sqrt{a^2+c^2+(a^2-c^2) \> \cos(2 \> \theta)}\>\sin(\theta) \> d\theta$$
This equation can also be found on Wolfram, but is said to be the conventional form for the oblate spheroid ($a>c$). Apparently, the integral $\int_{\theta_0}^{\theta_1}\sqrt{a^2+c^2+(a^2-c^2) \> \cos(2 \> \theta)}\>\sin(\theta) \> d\theta$ can be solved to $$\dfrac{-c^2\ln\left(\left|c\sqrt{\left(a^2-c^2\right)\cos^2\left(x\right)+c^2}+\sqrt{a^2-c^2}\left|c\right|\cos\left(x\right)\right|\right)-c\sqrt{a^2-c^2}\cos\left(x\right)\sqrt{\frac{\left(a^2-c^2\right)\cos^2\left(x\right)}{c^2}+1}}{\sqrt{2}\sqrt{a-c}\sqrt{c+a}}$$
This works well indeed for oblate spheroids. I used the ke!san Online Calculator to confirm.
Now, what do I have to do differently to find a similar function $S(\phi, \theta)$ for a prolate spheroid? Simply flipping $c$ and $a$ did not seem to do the trick. The parameterisation will stay the same as it is independent of the ratio of $c$ and $a$. Therefore, the fundamental coefficients should change neither.
I hope you can push me in the right direction. Any help is much appreciated!