Help with limits of integration in spherical coordinates

Question

$$\int_\Omega F(\theta,\phi) \sin(\phi) \, d\Omega$$
where $\Omega$ represents the (outer) surface of a spherical cap on a unit sphere.

Specifically, let the center of the spherical cap be in some direction $(\phi_s,\theta_s)$, and let the apex angle of the spherical sector corresponding to the cap be $2\sigma$.

The problem I'm having is finding the limits of integration. If this was a cap symmetric about the z axis, it'd be easy because of symmetry. It'd just be:
$$\int_{\theta=0}^{2\pi}\int_{\phi=0}^{\sigma} F(\theta,\phi)\sin(\phi) \, d\phi \, d\theta$$.

I can't figure out how to take into consideration the rotation of the spherical cap. I don't see a way to rotate F , and I can't describe the portion of the sphere's surface in terms of just $\theta$ and $\phi$.

I saw this post, which suggests I should convert the function into Cartesian Coordinates, multiply a rotation matrix, and convert it back into Spherical coordinates, but I don't think that is reasonable with my particular F.

(For additional information, F is a second-order polynomial in $\theta$ and $\phi$. I think that if I were to transform,rotate,and transform it back, I would get a ton of nested sinusoidal and inverse sinusoidal functions that would not be able to be integrated properly/handily)

Any insights or pointers would be greatly appreciated!

Answer 1

set up a spherical triangle with one point at the pole ( $\theta =0$) one point at $(\theta_S, \phi_S)$ and the third point being $(\theta, \phi)$ subject to the condition that the length of the arc from $(\theta_S, \phi_S)$ to $(\theta, \phi)$ is $\sigma$ .

You have all three sides of this triangle $\theta, \theta_S \text{ and } \sigma$ , now you can use the cosine law for spherical triangles to solve for the dihedral angle subtended at the pole, which equals $\phi - \phi_S$

$$\cos \sigma = \cos \theta \cos \theta_S + \sin \theta \sin \theta_S \cos(\phi - \phi_S)$$

so that

$$\cos(\phi - \phi_S) = \frac{ \cos \sigma - \cos \theta \cos \theta_S }{ \sin \theta \sin \theta_S} $$

having the solution

$$ \phi = \phi_S \pm g(\theta) \text { where } g(\theta) \equiv \cos^{-1}\left( \frac{ \cos \sigma - \cos \theta \cos \theta_S }{ \sin \theta \sin \theta_S} \right)$$

If $\sigma < \theta_S$ you can set up the limits as ...

$$\int_{\theta_S - \sigma}^{\theta_S + \sigma}\int_{\phi_S-g(\theta)}^{\phi_S+g(\theta)} F(\theta,\phi)\sin(\phi) \, d\phi \, d\theta $$

If $\sigma > \theta_S$ you should split into two integrals ...

$$\int_\Omega F(\theta,\phi) \sin(\phi) \, d\Omega = $$

$$\int_{0}^{ \sigma - \theta_S }\int_{0}^{2 \pi} F(\theta,\phi)\sin(\phi) \, d\phi \, d\theta +\int_{\sigma - \theta_S }^{\sigma + \theta_S }\int_{\phi_S-g(\theta)}^{\phi_S+g(\theta)} F(\theta,\phi)\sin(\phi) \, d\phi \, d\theta $$