Minor correction: The line "Then, what is that PDF $$F_\Phi(\phi) = P[\phi \leq \Phi]$$" should say:
"Then what is that CDF $$F_\Phi(\phi) = P[\phi \leq \Phi]$$"?
The equation for $$f_\Phi(\phi)$$ (lowercase f) refers to the PDF, not the CDF (uppercase F). The provided fraction refers to the PDF, not the CDF.
Additionally, the provided diagram defines phi with respect to the plane, but in our parameterization it is defined with respect to the normal of the plane. This doesn't break our parameterization however -- if you change z to sin(phi), which is how it would look if you defined it as in the picture -- x and y would similarly be defined differently and the parameterization would still work out.
Minor correction: The line "Then, what is that PDF $$F_\Phi(\phi) = P[\phi \leq \Phi]$$" should say:
"Then what is that CDF $$F_\Phi(\phi) = P[\phi \leq \Phi]$$"?
The equation for $$f_\Phi(\phi)$$ (lowercase f) refers to the PDF, not the CDF (uppercase F). The provided fraction refers to the PDF, not the CDF.
Additionally, the provided diagram defines phi with respect to the plane, but in our parameterization it is defined with respect to the normal of the plane. This doesn't break our parameterization however -- if you change z to sin(phi), which is how it would look if you defined it as in the picture -- x and y would similarly be defined differently and the parameterization would still work out.
Link in comments copypasted
https://math.stackexchange.com/questions/35500/parameterizing-the-upper-hemisphere-of-a-sphere-with-an-upward-pointing-normal