If we are only sampling from the surface of the hemisphere, wouldn't just randomly selecting x,y in [0,1] and z in [0.5,1] and plugging them into the equation for the hemisphere do the random sampling?

To get from sampling the sphere to sampling the hemisphere, can't you just sample from the sphere and negate the samples outside the hemisphere?

For the "warp" approach, how do we know the samples are still spatially uniform after warping?

If we want a sample of the whole sphere, how do we modify the z-component content to achieve this?

I think the sampling of a whole sphere would be: (2sqrt(e1(1-e1)cos(2pie2), 2sqrt(e1(1-e1)sin(2pie2), 1-2e1). I found this website really helpful in proving the correctness of the warp: https://pbr-book.org/3ed-2018/Monte_Carlo_Integration/2D_Sampling_with_Multidimensional_Transformations.

Why is this sampling uniform on the z axis?

Why is z value constant in the wrap function?

Is this similar to the method we use for uniform sampling of a circle in the previous lecture?