<p>The book defines operator <span class="math-container">$\mathcal{T}$</span> for the propagation of an exitant function as:</p>
<p><a href="https://i.sstatic.net/e68RC.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/e68RC.jpg" alt="enter image description here" /></a></p>
<p>if I'm assuming that the position of the point <span class="math-container">$x$</span> is on the top-left surface, and that <span class="math-container">$\Psi$</span> is considered as the direction originating from point <span class="math-container">$y = r(x, \Theta)$</span> (<span class="math-container">$r$</span> being the raycast function) then what's the interpretation of <span class="math-container">$cos(N_{x}, \Psi)$</span> ? Why would we use the normal at point <span class="math-container">$x$</span> (on another surface!) instead of <span class="math-container">$cos(N_{r(x, \Theta)}, \Psi)$</span> ?</p>
<p>on the other hand, if I'm assuming that <span class="math-container">$x$</span> is the point on the surface at bottom-right, then where's the point <span class="math-container">$y = r(x, \Theta)$</span> ? And what's the physical meaning of <span class="math-container">$L(y \rightarrow -\Psi)$</span> ?</p>

CC BY-SA 4.0
2022-06-11T06:27:11.890Z
Their picture is misleading/wrong. See this instead: https://viclw17.github.io/2018/06/30/raytracing-rendering-equation
lightxbulb
CC BY-SA 4.0
2022-06-11T15:53:38.197Z
@lightxbulb thank you for the resource, I fully understand the principles behind the standard rendering equation, however in the book they're doing a somewhat long derivation to show 4 different ways of calculating the flux of a set (with exitant/incident radiance/importance) part of this derivation involves the calculation of a function for exitant radiance, which unfortunately as you mentioned seems to be misleading
Domenico
CC BY-SA 4.0
2022-06-11T16:12:59.903Z
Their image is misleading/wrong, but the math is ok. Here's the translation of the usual notation: $L(x, \omega_o) = L_e(x, \omega_o) + \int_{\Omega_x}f(x,\omega_o,\omega_i) L(r(x,\omega_i), -\omega_i) (N_x \cdot \omega_i) d\omega_i$. Set $\omega_o = \Theta$, $\omega_i = \Psi$ and you have their notation. Note that $(N_x \cdot \omega_i) = \cos\angle(N_x, \omega_i)$ and $L_i(x, \omega_i) = L(r(x,\omega_i), -\omega_i)$. The point $x$ is a point on some surface, and $\omega_o$ is the direction you want to compute the radiance exiting $x$ in: $L(x,\omega_o)$.
lightxbulb
CC BY-SA 4.0
2022-06-11T20:48:22.300Z
@lightxbulb just to make sure I understood, you mentioned that the formula itself is ok, but in the comment you just wrote $L(r(x, \omega_{i}), - \omega_{i})$  while the formula in the book is using: $L(r(x, \omega_{o}), - \omega_{i})$
Am I missing something?
Domenico
CC BY-SA 4.0
2022-06-11T21:04:34.623Z
I hadn't noticed that, $L(r(x, \Theta), -\Psi)$ ought to be $L(r(x,\Psi), -\Psi)$ instead.
lightxbulb
<raytracing><physically-based>
Difficulty in understanding the interpretation of propagation of an exitant function from the book Advanced Global Illumination (2nd ed)
Domenico
# Difficulty in understanding the interpretation of propagation of an exitant function from the book Advanced Global Illumination (2nd ed)

(Asked by: username_0 on 2022-06-11)

The book defines operator $\mathcal{T}$ for the propagation of an exitant function as:

[![enter image description here](https://i.sstatic.net/e68RC.jpg)](https://i.sstatic.net/e68RC.jpg)

if I'm assuming that the position of the point $x$ is on the top-left surface, and that $\Psi$ is considered as the direction originating from point $y = r(x, \Theta)$ ($r$ being the raycast function) then what's the interpretation of $cos(N\_{x}, \Psi)$ ? Why would we use the normal at point $x$ (on another surface!) instead of $cos(N\_{r(x, \Theta)}, \Psi)$ ?

on the other hand, if I'm assuming that $x$ is the point on the surface at bottom-right, then where's the point $y = r(x, \Theta)$ ? And what's the physical meaning of $L(y \rightarrow -\Psi)$ ?

**Tags:** raytracing, physically-based

> **Comments:**
> **username_1**: Their picture is misleading/wrong. See this instead: https://viclw17.github.io/2018/06/30/raytracing-rendering-equation  
> **username_0**: @username_1 thank you for the resource, I fully understand the principles behind the standard rendering equation, however in the book they're doing a somewhat long derivation to show 4 different ways of calculating the flux of a set (with exitant/incident radiance/importance) part of this derivation involves the calculation of a function for exitant radiance, which unfortunately as you mentioned seems to be misleading  
> **username_1**: Their image is misleading/wrong, but the math is ok. Here's the translation of the usual notation: $L(x, \omega_o) = L_e(x, \omega_o) + \int_{\Omega_x}f(x,\omega_o,\omega_i) L(r(x,\omega_i), -\omega_i) (N_x \cdot \omega_i) d\omega_i$. Set $\omega_o = \Theta$, $\omega_i = \Psi$ and you have their notation. Note that $(N_x \cdot \omega_i) = \cos\angle(N_x, \omega_i)$ and $L_i(x, \omega_i) = L(r(x,\omega_i), -\omega_i)$. The point $x$ is a point on some surface, and $\omega_o$ is the direction you want to compute the radiance exiting $x$ in: $L(x,\omega_o)$.  
> **username_0**: @username_1 just to make sure I understood, you mentioned that the formula itself is ok, but in the comment you just wrote $L(r(x, \omega_{i}), - \omega_{i})$  while the formula in the book is using: $L(r(x, \omega_{o}), - \omega_{i})$ Am I missing something?  
> **username_1**: I hadn't noticed that, $L(r(x, \Theta), -\Psi)$ ought to be $L(r(x,\Psi), -\Psi)$ instead.  

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## 0 answers

