fractional calculus – Solid angle through a circle

I am trying to calculate the proportion of light emitted by a fluorophore that would be transmitted through a small hole. This is similar to calculating the solid angle defined by a circle of radius r and seen by a point.

I can calculate it for the case where the point is on top of the center of the hole, then solid angle is the solid angle defined by a circular cone : $$displaystyle Omega =2pi (1-cos {theta }$$ .
In the case where the point is misaligned it ‘sees’ the circle as en ellipse. i.e there are two different angles on the major and minor axis of the ellipse. I don’t really know how to handle this… i guess the solution lies in formulating some differential surface integration…

Thanks for your help/advice !

Scheme of the point and ‘hole’