NDF (normal distribution function) are typically given in terms of microfacet area. To convert to macrosurface area, we need to multiple by . This is because . To get into spherical coordinates, we need to multiple by since .

So before creating sampling routines for an NDF, first multiply by
, the jacobian determinant of the transformation.

Quick reference:

To get a sampling routine given the PDF in spherical coordinate domain, first integrate out to get the marginal density . Then get conditional density .

Then use inverse transform sampling. For each function calculate the CDF by integration:

Finally set the CDF equal to a random variable and solve for the integration variable:

Links:
https://www.reedbeta.com/blog/hows-the-ndf-really-defined/
https://agraphicsguynotes.com/posts/sample_microfacet_brdf/
https://www.tobias-franke.eu/log/2014/03/30/notes_on_importance_sampling.html
https://jcgt.org/published/0007/04/01/