To generate a sample from a one dimensional PDF, I know that you can invert the cdf (assuming it's invertible) and use a uniformly distributed random number plugged into that function to generate a number from the original PDF.
Is it possible to invert the CDF of a two dimensional PDF in the form $z=f(x,y)$?
I'm a little stumped how you'd turn such a PDF into a CDF, but am more stumped how you would invert it since it isn't just a simple matter of flipping the input and output variable - there being two inputs and one output.