Confusion regarding incident direction in render equation

Question

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?

Answer 1

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