# How do I calculate the area of an ellipse projected onto a hemisphere?

I'm trying to figure out how to calculate the area of an elliptic region projected onto the surface of a hemisphere. Explicitly, I'm trying to evaluate the following integral:

$$\\I=\int_{0}^{2\pi} \int_0^{\sqrt{\alpha^2 \sin^2 \phi + \beta^2 \cos^2\phi}} \sin \theta\, d\theta \,d\phi$$

Notice that if $\alpha = \beta$, the problem simplifies to calculating the area of a circle projected onto a hemisphere, and is much easier. I've tried some changes of variables but haven't been able to make any headway. The only thing left I can think of would be to transform to cartesian coordinates, then represent all the square roots as infinite series...Is there some simple trick that I am missing here?

• The inner integral is $\int_0^{f(\phi)} \sin\theta\,\mathrm d\theta$, which simplifies to $1 - \cos f(\phi)$. So you want to evaluate $\int_0^{2\phi}\big(1 - \cos\sqrt{\alpha^2\sin^2\phi+\beta^2\cos^2\phi}\big)\mathrm\,\mathrm d\phi$, right? Yeah, that doesn't look pretty. – Rahul Jan 4 '14 at 5:08