I am trying to write a code that calculates the cap area of prolate and oblate spheroids, while avoiding integrals. Through this online calculator I got the equation for a prolate spheroid (i.e. c > a) cap area:
example geometry from the calculator here - sorry, not allowed to embed yet...
$$ e = \sqrt{1 - \frac{a^2}{c^2}}, $$ $$ e_{1} = e\left(1-\frac{h}{c} \right) $$ $$ A = \pi a c \left(\frac{\sin^{-1}e - \sin^{-1}e_1}{e} + \frac{a}{c} - \left( 1-\frac{h}{c}\right)\sqrt{1-e_{1}^{2}} \right) $$
However, it does not hold/work for oblate spheroids (i.e. a > c), where the first square root would be of a negative value. Although the online calculator still works for oblate spheroids, it unfortunately does not provide the required equation to calculate the cap area for that type. Also the original source or derivation for the equation given is not provided.
What is the correct equation to calculate the cap area at a specific height h for an oblate spheroid?
I found an equation in this publication that seems to produce correct results:
$$ e = \sqrt{\frac{a^2}{c^2}-1}, $$ $$ e_{1} = e\left(1-\frac{h}{c} \right) $$ $$ A = \pi a c \left(\frac{\sinh^{-1}e - \sinh^{-1}e_1}{e} + \frac{a}{c} - \left( 1-\frac{h}{c}\right)\sqrt{1+e_{1}^{2}} \right) $$