Where does the cos(phi) term come from that often shows up in spherical integrals?

Question

When integrating functions defined on the surface of a unit sphere, you often get an integral that looks something like:

$\int{d\theta}\int{d\phi}\cos\phi S(\theta,\phi)$

Where $S(\theta,\phi)$ is your function defined on the surface of the sphere, and $\theta$ and $\phi$ are the azimuth and elevation angles, respectively (or longitude and latitude angles, if you prefer).

I've most often seen this in situations where you're trying to find the amount of total energy falling on a point (i.e. by summing in all incoming directions to that point), so you see it a lot in rendering equations (especially radiosity), computational acoustics simulations using plane waves, flux, etc...

My question though, is where does the $\cos\phi$ or, depending on your coordinate system, sometimes $\sin\phi$, term come from?

My guess is this is accounting for the fact that as you near the poles on the sphere, the elevation angles become closer together, and this accounts for the change in infinitesimal area that you're breaking the integral up into. Is that the correct interpretation?

Answer 1

In performing a change of variables to spherical coordinates, $(\rho, \phi, \theta) \mapsto (\rho \sin\phi \cos\theta, \rho \sin\phi \sin\theta, \rho \cos\phi)$, you need to take into account the Jacobian determinant $$ \left| \begin{matrix} \sin\phi\cos\theta & \rho\cos\phi\cos\theta & -\rho\sin\phi\sin\theta \\ \sin\phi\sin\theta & \rho\cos\phi\sin\theta & \rho\sin\phi\cos\theta \\ \cos\phi & -\rho\sin\phi & 0 \end{matrix} \right| = \rho^2 \sin\phi. $$

Answer 2

In the above spherical system, $r$ is the radius, $\theta$ is the azimuth angle and $\phi$ is the elevation angle.

Here $\theta$ is the angle between $x$ axis (in the Cartesian system ) and $r\sin\phi$.

Considering $r$ as radius, infinitesimal change in arc length for infinitesimal change in $\phi $ is $dl = r \ d\phi$.

Similarly you can get $(r\sin\phi)\ d\theta$.

We can vary any $2$ of these $3$ quantities infinitesimally keeping the other quantity constant and get elemental areas,

$dS_{r = \text{const}} = r\sin\phi \ d\theta \cdot rd\phi = r^2\sin\phi \ d\theta d\phi$
$dS_{\phi = \text{const}} = dr \cdot r\sin\phi \ d\theta = r\sin\phi \ dr d\theta$
$dS_{\theta= \text{const}} = dr \cdot rd\phi = r \ dr d\phi$

If you integrate over the sphere keeping $r = 1$ as constant, you need to consider the 1st case,

$dS_{r = 1} =\sin\phi \ d\theta d\phi$

Based on your definition of the angles, you may get either $\sin\phi$ or $\cos\phi$