I have a question about radiance and irradiance in a very simple scene.

Let's have a scene with just 2 infinite planes: ground and sky. The camera is
somewhere between them so that both are visible.

The ground is pure lambertian surface (ie. totally matte surface, like a white
piece of paper).

The sky emits light evenly and let's say the outgoing radiance is 1.0.
(Actually, I'm not sure what is the correct unit to define this light source,
radiant exitance is probably correct?) And the camera registers that value as
pure white pixel.

The question is: what is the irradiance at each point in the ground? I guess it
should also be 1.0. So the sky should look identical to ground: pure white.
Right? This is the most important question as I need a reference scene to verify
my renderer.

But how do I calculate it?

The way to do it is to integrate over hemisphere at that point acording to light
transport equation. In 2D scene it would be I(1.0 * cos x  dx) and we have to do
it from -pi/2 to pi/2. Right?

The value of that is sin (pi/2) - sin(-pi/2) = 2. Hmm. Not right. Or is it? Most
likely I'm integrating incorrectly or the wrong thing.

Yes, I'm lost and my maths suck...but the correct result would help a lot.

Thanks,
Severi

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