I need to sketch the image under the stereographical projection of sphere where the spherical cap $A < X < 1,$ with center lying on the equator, for fixed $ A. $ Separating the cases according to various ranges of $A.$
Here $X^2 + Y^2 + Z^2 = 1$ and $ (X,Y,Z) $ are points the unit sphere.
Here is a graphical answer. The stereographic projection is from the North Pole (materialized by a star) onto the equatorial plane. Please note that all the blue disks (images of spherical caps whose limits are in red) have their centers on the $x$ axis, are orthogonal to the equator circle but are not concentrical. Waiting for your comments.
For the computations, see the reference I have given about an identical question.