<p><span class="math-container">$L_i(p,\omega_i) = L_o(p+t\omega_i,-\omega_i)$</span> where t is some distance to another point, so <span class="math-container">$\omega_i$</span> is &quot;positive&quot; (-) when evaluating <span class="math-container">$L_o$</span> but &quot;negative&quot; (+) for <span class="math-container">$L_i$</span>. So if <span class="math-container">$-\omega_i$</span> is used in <span class="math-container">$L_i$</span>, we get problems in <span class="math-container">$L_o$</span> since <span class="math-container">$\omega_i$</span> points into geometry.</p>

CC BY-SA 4.0
2023-09-13T09:23:35.507Z
2023-09-13T09:58:06.670Z
2023-09-13T09:58:06.670Z
Mathis
<p>I am studying from pbrt-v3 and the rendering equation has just been introduced as
<span class="math-container">$$
\underbrace{L_\mathrm{o}(\mathrm{p},\omega_\mathrm{o})}_\text{Outgoing radiance }=\underbrace{L_\mathrm{e}(\mathrm{p},\omega_\mathrm{o})}_\text{Emitted Radiance}+\underbrace{\int_{\mathrm{S}^2}f(\mathrm{p},\omega_\mathrm{o},\omega_\mathrm{i})L_\mathrm{i}(\mathrm{p},\omega_\mathrm{i})|\cos(\theta_\mathrm{i})|\mathrm{d}\omega_\mathrm{i}}_\text{Incident radiance from all directions}.
$$</span>
Please bear with me as I have some confusion regarding the integration variable <span class="math-container">$\omega_\mathrm{i}$</span>. It represents a point on the sphere <span class="math-container">$\mathrm{S}^2$</span>, which is centred at <span class="math-container">$\mathrm{p}$</span>, so wouldn't this mean the direction of the incoming light points <em>away</em> from <span class="math-container">$\mathrm{p}$</span>? I took a look at the <a href="https://en.wikipedia.org/wiki/Rendering_equation" rel="nofollow noreferrer">Wikipedia page</a> for the rendering equation and indeed, <span class="math-container">$\omega_\mathrm{i}$</span> is stated as the <em>negative</em> direction of incoming light.</p>
<p>I'm wondering why the direction is stated as negative. Perhaps it makes the equation simpler? How would the incident radiance function <span class="math-container">$L_\mathrm{i}(\mathrm{p},\omega_\mathrm{i})$</span> handle a negative direction?</p>

CC BY-SA 4.0
2023-09-12T20:21:00.017Z
$L_i$ is just defined wrt an incident direction, i.e. pointing towards where the radiance is coming from.
lightxbulb
<rendering><mathematics><pbrt>
Confusion regarding incident direction in render equation
Scene
# Confusion regarding incident direction in render equation

(Asked by: username_0 on 2023-09-12)

I am studying from pbrt-v3 and the rendering equation has just been introduced as
$$
\underbrace{L\_\mathrm{o}(\mathrm{p},\omega\_\mathrm{o})}\_\text{Outgoing radiance }=\underbrace{L\_\mathrm{e}(\mathrm{p},\omega\_\mathrm{o})}\_\text{Emitted Radiance}+\underbrace{\int\_{\mathrm{S}^2}f(\mathrm{p},\omega\_\mathrm{o},\omega\_\mathrm{i})L\_\mathrm{i}(\mathrm{p},\omega\_\mathrm{i})|\cos(\theta\_\mathrm{i})|\mathrm{d}\omega\_\mathrm{i}}\_\text{Incident radiance from all directions}.
$$
Please bear with me as I have some confusion regarding the integration variable $\omega\_\mathrm{i}$. It represents a point on the sphere $\mathrm{S}^2$, which is centred at $\mathrm{p}$, so wouldn't this mean the direction of the incoming light points *away* from $\mathrm{p}$? I took a look at the [Wikipedia page](https://en.wikipedia.org/wiki/Rendering_equation) for the rendering equation and indeed, $\omega\_\mathrm{i}$ is stated as the *negative* direction of incoming light.

I'm wondering why the direction is stated as negative. Perhaps it makes the equation simpler? How would the incident radiance function $L\_\mathrm{i}(\mathrm{p},\omega\_\mathrm{i})$ handle a negative direction?

**Tags:** rendering, mathematics, pbrt

> **Comments:**
> **username_1**: $L_i$ is just defined wrt an incident direction, i.e. pointing towards where the radiance is coming from.  

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## 1 answers

### Answer by username_2 on 2023-09-13. Score: 1 | Accepted answer

$L\_i(p,\omega\_i) = L\_o(p+t\omega\_i,-\omega\_i)$ where t is some distance to another point, so $\omega\_i$ is "positive" (-) when evaluating $L\_o$ but "negative" (+) for $L\_i$. So if $-\omega\_i$ is used in $L\_i$, we get problems in $L\_o$ since $\omega\_i$ points into geometry.


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