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Radiance is defined as:

$L=\frac{d \Phi}{\mathop{d \omega} \mathop{d A} \mathop{\cos\theta}}.$

I’m wondering why there is a $\cos \theta$ factor in the denominator of the radiance ($\theta$ is the angle between the normal and the light direction vector). Why would $L=\frac{d \Phi}{\mathop{d \omega} \mathop{d A}}$ be a bad definition of radiance?

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It is really the dot product between the area and the solid angle - so the radiance "in this direction" that you are looking at. Thus the cosine term.

Expanding...

There is a nice diagram at http://wtlab.iis.u-tokyo.ac.jp/~wataru/lecture/rsgis/rsnote/cp1/1-6-1.gif of which I reproduce part below:

enter image description here

This shows quite clearly that when you are interested in the radiance in a particular direction, the apparent area of the object emitting radiation (area dS in this diagram) will seem to be smaller by $\cos\theta$. You would expect, then, that the flux per unit apparent area is larger, which is where the $\cos$ term comes from.

For a Lambertian source, the flux term $\Phi$ actually has a $\cos\theta$ dependence as well, and the two cosines will in fact cancel out. This is not guaranteed to be true for "any" source which is why the equation you quote makes the $\cos$ term explicit, and leaves it up to you to reflect the actual angular dependence of the source in the numerator term.

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  • $\begingroup$ Could you elaborate on what you mean exactly by the dot product and why it is needed? $\endgroup$
    – Lenar Hoyt
    Commented Dec 19, 2014 at 18:18
  • $\begingroup$ When you have two vectors, the dot product is the product of their lengths multiplied by the cosine of the angle between them. This is handy for things like work done (force times distance times cosine between them etc). Both solid angle and area can be thought of as having magnitude and direction so the use of the dot product makes sense. And yes the Lambert law follows from the same principle. $\endgroup$
    – Floris
    Commented Dec 19, 2014 at 18:21
  • $\begingroup$ Or is there maybe a geometric interpretation of this? One interpretation I know of is that the dot product is the orthogonal projection of one vector onto the other one, but it seems this does not fit well in this case as the magnitude denotes surface area. $\endgroup$
    – Lenar Hoyt
    Commented Dec 19, 2014 at 19:26
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    $\begingroup$ Would the downvoter care to say why? OK, so the dot product doesn't appear in the numerator, but I think we know that Floris means. A comment asking for better precision and why is much more constructive than a downvote without comment. No one learns anything from the latter. $\endgroup$ Commented Dec 20, 2014 at 3:04
  • $\begingroup$ @mcb Think of working out the fluid flux in a pipe if your measurement surface is tilted relative to the pipe's (and flow's axis). The flow rate, i.e. volume of flux crossing the tilted plane is $A\,\cos\theta$ where $\theta$ is the tilt angle. $\endgroup$ Commented Dec 20, 2014 at 3:06

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