Equidistant projection of points on spherical cap

Given a sphere of radius $R$, centered in the origin of an euclidean reference frame, I need to calculate the position of $N$ points on the spherical cap of heigth $h$ with $h\lt R$ so that the projections of the points on the $x,y$ plane are equidistant. The center of the spherical cap is on the $z$ axis. The projection is normal to the $x,y$ plane. How can I choose the coordinates of the points on the spherical cap in order to satisfy this condition? Thanks in advance.

I think you can use the chordal distance $d(z,w)$ between two points $z,w \in \Bbb C^*$ to be the length of the straight line segment joining the points $P$ and $Q$ on the unit sphere whose stereographic projections are $z$ and $w$, respectively.
Here $d(z,w)= \frac {2|z-w|}{\sqrt{1+|z|^2} \sqrt{1+|w|^2}}$, $z,w \in \Bbb C$.