# Why integrate over a hemisphere (and not a sphere) to solve the rendering equation?

In most text books that I have seen, this is how the rendering equation is written:

$$L_0( \omega_0)= L_e(\omega_0)+\int_{\Omega}{f(\omega_i, \omega_0)L_i(\omega_i)\,\mathrm{d}\omega_i}$$

Where $\Omega$ is defined to be a hemisphere (and all those functions depend on more variables, omitted here for simplicity's sake).

Now suppose the surface being rendered is some kind of glass, or some transparent plastic. Why would it make sense to only integrate over a hemisphere? I would imagine that there can be incoming light from any direction, and thus the integration domain should be the entire sphere. How is the light coming from behind the glass accounted for?

• note that the subscript is not a 0 (zero), but an O (oh). it reads like... "Light out at the out angle equations light emitted towards the out angle plus ... ". o and i are complements, meaning out and in respectively (: Sep 4, 2016 at 21:57

The form of the rendering equation that uses only the BRDF ($f$ in your example, often called $f_r$) and integrates over one hemisphere does not account for transmission.