Lambertian surfaces and Monte-Carlo

Question

I'm writing a ray tracer (actually to learn rust).
During this project I'm learning a lot about optics/physics.

Here is a problem for which I have no nice solution:

Take a plane surface and assume it reflects lambertian ("diffuse", AOI independent).
Question: What is the probability distribution function (psd) of the reflected light? How does it integrate?

Example:

Assume that the surface is the x-y-plane and that light incidences from above.
Then light is reflected by the cosine-law, and hence the scattered intensity is given by cos(theta)*sin(theta) (where the cosine is the lambertian reflectance and the sin is due to spherical coordinates).
So a reflected ray is generate as follows:
phi is uniform distributed between 0 and 2pi.
theta is given by asin(sqrt(a)), where a is a uniform random number between 0 and 1
(integrate to get the cfd (cumulative distribution function), then invert)

If I try to do the same thing for an inclined surface - even for easy examples like with normal (1/sqrt(2),0,1/sqrt(2)) - then I fail at integrating the psd to get the cfd.
The psd is given by the inner product between a sample ray and the surface normal (set to 0 if it's negative - the plane reflects only to one side) - of course multiplied with sin(theta) (to correct for spherical coordinates.

Thanks for your help.

Answer 1

Here are some pointers to formulas used for diffuse ray tracing:

C.M. Goral et al., Modeling the interaction of light between diffuse surfaces, ACM SIGGRAPH Computer Graphics 18 (1984). http://www.cs.rpi.edu/~cutler/classes/advancedgraphics/S10/papers/goral.pdf

G.J. Ward, F.M. Rubinstein, and R.D. Clear. "A ray tracing solution for diffuse interreflection." ACM SIGGRAPH Computer Graphics 22 (1988):,85-92. https://eetd.lbl.gov/sites/all/files/publications/22789.pdf

T. Whitted, An improved illumination model for shaded display, ACM Siggraph 2005 Courses. ACM, 2005. https://www.cs.drexel.edu/~david/Classes/CS586/Papers/p343-whitted.pdf

X.D. He, et al., A comprehensive physical model for light reflection, ACM SIGGRAPH computer graphics 22 (1991). https://hal.archives-ouvertes.fr/file/index/docid/510144/filename/HTSG91.pdf

Comments

thanks for your answer

embarrassingly, the upshot is:
generate the ray in a spherical coordinates relative to the surface normal instead of the z-axis

(this is also explained nicely in a book by Modest on heat transfer)