<p>In appendix <strong>A</strong> of the paper <a href="https://www.cs.cornell.edu/~srm/publications/EGSR07-btdf.pdf" rel="nofollow noreferrer">Microfacet Models for Refraction through Rough Surfaces</a> there is a derivation provided for $\Lambda(w)$ but the mathematics is very confusing. Could somebody help me by explaining it?</p>

<hr>

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

<p>$G(w_o, w_i)$ $=$ $G_1(w_o)G_1(w_i)$</p>

<p>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.</p>

<p>$G_1(w_o,w_m)$ $=$ $1\over1+\Lambda(w_o)$</p>

<p>For the Trowbridge-Reitz distribution it is quite simple:</p>

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

<p>How is this $\Lambda(w)$ function derived?</p>

CC BY-SA 3.0
2017-11-08T15:53:58.183Z
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.

https://twvideo01.ubm-us.net/o1/vault/gdc2017/Presentations/Hammon_Earl_PBR_Diffuse_Lighting.pdf
Tare
CC BY-SA 3.0
2017-11-08T18:26:06.650Z
@Tare Thanks so much, this is exactly what I needed! I'm gonna post an answer with the derivation soon.
Arjan Singh
<pbr><ggx><ndf>
Explanation of math of the Smith G1 term
Arjan Singh
# Explanation of math of the Smith G1 term

(Asked by: username_0 on 2017-10-27)

In appendix **A** of the paper [Microfacet Models for Refraction through Rough Surfaces](https://www.cs.cornell.edu/~srm/publications/EGSR07-btdf.pdf) 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?

**Tags:** pbr, ggx, ndf

> **Comments:**
> **username_1**: 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.  https://twvideo01.ubm-us.net/o1/vault/gdc2017/Presentations/Hammon_Earl_PBR_Diffuse_Lighting.pdf  
> **username_0**: @username_1 Thanks so much, this is exactly what I needed! I'm gonna post an answer with the derivation soon.  

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## 0 answers

