<p>In PBRT 4ed</p>
<p><a href="https://www.pbr-book.org/4ed/Volume_Scattering/Transmittance#eq:volume-attenuation-differential" rel="nofollow noreferrer">https://www.pbr-book.org/4ed/Volume_Scattering/Transmittance#eq:volume-attenuation-differential</a></p>
<p>How is this transmittance equation transformed from 11.9 to 11.10?</p>
<p><span class="math-container">\begin{equation}
\int_0^d \frac{dL(p + t\omega)}{dt} dt = L(p') - L(p) = \int_0^d -\sigma_t(p + t\omega) L(p + t\omega) dt \tag{11.9}
\end{equation}</span></p>
<p><span class="math-container">\begin{equation}
T_r(p \to p') = 1 - \int_0^d \sigma_t(p + t\omega) T_r(p + t\omega \to p') dt \tag{11.10}
\end{equation}</span></p>
<p>The book says that dividing 11.9 by L(p) gives 11.10, but the question is why deos Tr convert like that?</p>
<p>Since,
<span class="math-container">\begin{equation}
T(p \rightarrow p') = \frac{L(p')}{L(p)}
\end{equation}</span></p>
<p>shouldn't the Tr inside the right integral(11.10) be Tr(p -&gt; p +tw)?</p>

CC BY-SA 4.0
2024-07-02T14:38:42.400Z
I agree with you. However, 11.10 somewhat relates to the latter derivation, such as equation 11.13. The book is rather unclear about the derivation, and I am only able to get an exponential formulation, starting from 11.11. Yet I think 11.13 is correct while 11.10 is not, and if you care about this, I can post an answer about why this is so (intuitively). I recommend you **file an issue on the PBR-book website github page** if this is confirmed to be wrong.
Enigmatisms
CC BY-SA 4.0
2024-07-04T23:41:26.223Z
Thank you! @Enigmatisms. I found the original paper that derived the equation, and the difference comes from the change of variable part, which expresses the equation in the opposite direction of the light ray.

https://cs.dartmouth.edu/~wjarosz/publications/georgiev19integral.html
Sopiro
<raytracing><mathematics><volume-rendering>
How is this transmittance derived?
Sopiro
# How is this transmittance derived?

(Asked by: username_0 on 2024-07-01)

In PBRT 4ed

<https://www.pbr-book.org/4ed/Volume_Scattering/Transmittance#eq:volume-attenuation-differential>

How is this transmittance equation transformed from 11.9 to 11.10?

\begin{equation}
\int\_0^d \frac{dL(p + t\omega)}{dt} dt = L(p') - L(p) = \int\_0^d -\sigma\_t(p + t\omega) L(p + t\omega) dt \tag{11.9}
\end{equation}

\begin{equation}
T\_r(p \to p') = 1 - \int\_0^d \sigma\_t(p + t\omega) T\_r(p + t\omega \to p') dt \tag{11.10}
\end{equation}

The book says that dividing 11.9 by L(p) gives 11.10, but the question is why deos Tr convert like that?

Since,
\begin{equation}
T(p \rightarrow p') = \frac{L(p')}{L(p)}
\end{equation}

shouldn't the Tr inside the right integral(11.10) be Tr(p -> p +tw)?

**Tags:** raytracing, mathematics, volume-rendering

> **Comments:**
> **username_1**: I agree with you. However, 11.10 somewhat relates to the latter derivation, such as equation 11.13. The book is rather unclear about the derivation, and I am only able to get an exponential formulation, starting from 11.11. Yet I think 11.13 is correct while 11.10 is not, and if you care about this, I can post an answer about why this is so (intuitively). I recommend you **file an issue on the PBR-book website github page** if this is confirmed to be wrong.  
> **username_0**: Thank you! @username_1. I found the original paper that derived the equation, and the difference comes from the change of variable part, which expresses the equation in the opposite direction of the light ray.  https://cs.dartmouth.edu/~wjarosz/publications/georgiev19integral.html  

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## 0 answers

