Explanation of math of the Smith G1 term

In appendix A of the paper Microfacet Models for Refraction through Rough Surfaces there is a derivation provided for $\Lambda(w)$ but the mathematics is very confusing. Could somebody help me by explaining it?

Smith's shadow masking function $G1$ gives the fraction of microfacets with normal $w_h$ that are visible from direction $w$.

$G(w_o, w_i)$ $=$ $G_1(w_o)G_1(w_i)$

In the literature, the Smith masking function is often expressed as a fraction involving a function $\Lambda(w_o)$. This function is expressed as an integral over the slopes of the microsurface and its form is derived with a raytracing formulation of the masking probability.

$G_1(w_o,w_m)$ $=$ $1\over1+\Lambda(w_o)$

For the Trowbridge-Reitz distribution it is quite simple:

$\Lambda(w)$ $=$ $-1+ \sqrt{1+\alpha^2tan^2\theta}\over 2$

How is this $\Lambda(w)$ function derived?

• There was a derivation done by Earl Hammon Jr in his talk, I hope that helps. He starts a derivation via Ray Tracing on slide 71. There is more information about it in the Appendix (slide 148ff.) I think. twvideo01.ubm-us.net/o1/vault/gdc2017/Presentations/…
– Tare
Nov 8 '17 at 15:53
• @Tare Thanks so much, this is exactly what I needed! I'm gonna post an answer with the derivation soon. Nov 8 '17 at 18:26