## Microfacet Multiple Scattering Simplifications

### GGX Lambda

First, we can write $\alpha_i$ in terms of $\omega_i = (\sin\theta_i \cos\phi_i, \sin\theta_i \sin\phi_i, \cos\theta_i) = (x_i, y_i, z_i)$:

Next, we can substitute this form into $a$ and simplify:

We can then do the same for $\Lambda$:

### Projected Area

### Free Path Length

### New Height

The change in height is $\Delta h = \ell \cos\theta_r$, therefore the new height is