The projection of a disk, with an arbitrary orientation, onto an sphere is an spherical ellipse(right image), I believe.
Now, what is the projection of this shperical ellipse on to the base plane of the hemisphere? I cannot figure it out what geometry shap it is? Is it just compsed of two arc segments of two different ellipses? or an ellipse and a circle?
My motivation is to find the projected area of the disk on the hemisphere base plane and then to sample it.
Like this: (left is projection of the spherical ellipse on the base plane of the hemisphere, and right image is 3d view of the disk and it's projection on to the sphere)
Or this:
