This sampling approach is necessary because if you choose theta and phi both uniformly at random, the generated points will get 'clustered' at the poles. This is why the correct pdf is p(theta, phi) = sin(theta)/2pi, the probability of a theta,phi point should increase as theta (angle off vertical) increases.
The full derivation of this slide is here:
http://www.pbr-book.org/3ed-2018/Monte_Carlo_Integration/2D_Sampling_with_Multidimensional_Transformations.html#UniformlySamplingaHemisphere

Although I don't quite understand all the details of it, it conceptually makes sense.

zehric

Yes, I think some of the intuition is similar to the previous slide where we need to weight the sampling of a circle by area instead of by radius

xiaoyankang

"Pure diffuse surfaces are only theoretical, but they makes a good approximations of what we can find in the real world." This blog offers a detailed explanation of hemisphere uniform sampling, the part discussing rejection sampling is particularly interesting.
https://blog.thomaspoulet.fr/uniform-sampling-on-unit-hemisphere/

This sampling approach is necessary because if you choose theta and phi both uniformly at random, the generated points will get 'clustered' at the poles. This is why the correct pdf is p(theta, phi) = sin(theta)/2pi, the probability of a theta,phi point should increase as theta (angle off vertical) increases. The full derivation of this slide is here: http://www.pbr-book.org/3ed-2018/Monte_Carlo_Integration/2D_Sampling_with_Multidimensional_Transformations.html#UniformlySamplingaHemisphere

Although I don't quite understand all the details of it, it conceptually makes sense.

Yes, I think some of the intuition is similar to the previous slide where we need to weight the sampling of a circle by area instead of by radius

"Pure diffuse surfaces are only theoretical, but they makes a good approximations of what we can find in the real world." This blog offers a detailed explanation of hemisphere uniform sampling, the part discussing rejection sampling is particularly interesting. https://blog.thomaspoulet.fr/uniform-sampling-on-unit-hemisphere/