# Confusion regarding incident direction in render equation

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 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?

• $L_i$ is just defined wrt an incident direction, i.e. pointing towards where the radiance is coming from. Sep 12 at 20:21

$$L_i(p,\omega_i) = L_o(p+t\omega_i,-\omega_i)$$ 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.