Why is radiance defined per projected area normal to the beam direction?

[brightlint]

Radiance is defined as radiant flux per solid angle per projected area normal to the beam direction: $L = \frac{d^2 \Phi}{d \vec\omega \cdot d A_\perp}$ where $A_\perp = A \cos \theta$ and $\theta$ is the angle between the beam direction $\vec\omega$ and the surface normal. I kind of understand that radiance is simply the infinitesimal flux $d\Phi$ contained in the infinitesimal cone/ray which is described by the infinitesimal solid angle and the surface segment $d A$. However I don't understand why it's necessary to project the surface segment normal to the beam. Why would $L = \frac{d^2 \Phi}{d \vec\omega \cdot d A}$ be a bad definition of radiance?

 

[mfb]

Then L would depend on the definition of your area and its orientation relative to the flux. The number alone would become meaningless.

 

[brightlint]

Then L would depend on the definition of your area and its orientation relative to the flux. The number alone would become meaningless.

I'm still having trouble to see why that would be a problem. I would be glad if you or someone else could illustrate it with an example like this:

Suppose there is a surface segment $d A$ inside a sphere and the sphere emits light on the inside like a Lambertian radiator. If I measure the incident radiance at the surface segment $d A$ coming from a certain direction $d \vec\omega$ without the projection of $dA$ normal to the beam, then the measured value would be small for directions near the horizon because of the Tilting principle. However, if I project the surface segment normal to the beam, then the radiance would be constant across the whole hemisphere.

Is this correct so far? Why would it be meaningless if the radiance would change depending on the direction?

 

[mfb]

Why would it be meaningless if the radiance would change depending on the direction?

Radiance is supposed to be a property of the light, not a property of the interaction of light with some (not necessarily real!) surface with some specific orientation.

 

[brightlint]

Why don’t we use a differential area normal to the beam in the first place, instead of projecting a non-normal one?

Besides that, radiance is often described as an measure for how bright an object appears, wouldn't that be a property of the interaction of light with a surface?