Physically based rendering is a catch all term for any technique that tries to achieve photorealism via physical simulation of light. Currently the best model to simulate light is captured by an equation known as the rendering equation. The rendering equation tries to describe how a "unit" of light is obtained given all the incoming light that interacts with a specific point of a given scene. Image intensities = f ( normal, surface reflectance, illumination direction, viewing direction)
The following is a simplified version:
$$ L_{o}\left(p, \omega_{o}\right)=\int_{\Omega} f_{r}\left(p, \omega_{i}, \omega_{o}\right) L_{i}\left(p, \omega_{i}\right) \mathbf{n} \cdot \omega_{i} d \omega_{i} $$
<aside> 🌟 The equation is simply representing the outgoing radiance given the incoming radiance weighted by the cosine of the angle between every incoming ray and the normal to the surface.
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The equation describes the outgoing radiance from a point Lo(p,ωo) , which is all we need to colour a pixel on screen. To calculate it we need the normal of the surface where our pixel lies on (nn), and the irradiance of the scene, which is given by Li(p,ωi)∀ωi. To obtain the irradiance we sum them all the incoming radiance, hence the integral sign in front of the equation.
The dot product n⋅ωi is there to take into account the angle of incidence angle of the light ray. If the ray is perpendicular to the surface it will be more localized on the lit area, while if the angle is shallow it will be spread across a bigger area, eventually spreading across too much to actually being visible.
We can use radiance to describe the amount of flux, passing through an infinitely small solid angle, hitting an infinitely small area, and that describes the behaviour of a light ray. So when we talk about radiance we talk about some amount of light going in some direction to some area. Irradiance and radiance are our main physical quantities.