Should I use th = acos(r) or th = acos(sqrt(r)) where r is a random number between 0 and 1. 'th' is the angle of the ray from the normal to the surface.
Dear Sudarshan.
I have deleted my previous post because the answer I gave was wrong.
If your purpose is to obtain a photon propagation distributed uniformly in all directions of the half plane, then neither of your proposed procedures would work correctly.
Could you give more details about how you would like the photons be distributed ?
Surface emission should satisfy Lambert's cosine law.
http://en.wikipedia.org/wiki/Lambert's_cosine_law
So, when emitting photons in 3D, you would have to choose two angles : one in the xy plane( azimuthal angle) and the angle between z axis and the ray : polar angle.
The azimuthal angle can be uniformly distributed. A book on this subject suggests that the polar angle must be generated using th = acos( sqrt(r)) to satisfy lambert;s law ( attaching proof ) .
I wrote a code to calculate view factor between 2 inclined plates . My numerical solution does not get validated when I use th = acos(sqrt(r)) but rather works when I use th = acos(r) to a high degree of accuracy. What mistake am I making ?
In order to account for Lambert's cosine law, the azimuth angle A and the polar angle P must satisfy the following
A/(2pi) = r1
sin²(P) = r2
where r1 and r2 are random numbers uniformly distributed in [0, 1].
see for example
Siegel R., Howell J. R., Thermal radiation heat transfer, (1992).
In other words, A will be uniformly distributed whereas P will be calculated as
P = acos( sqrt(1 - r2) )
which is the same as Eq. (3.35) in your second file.
Dear Sudarshan, "What mistake am I making ?"
I don't know the answer to this question. But if you are actually using A/(2pi) = r1 and cos(P) = r2 then you are simply generating photons directions that are uniformly distributed over the upper half sphere (z > 0). Obviously, such a sampling of photon directions would be inconsistent with Lambert's cosine law.
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