This spherical projection is a map from the sphere to the plane: Imagine straight lines going from above the center of the sphere down onto the plane. Each line hits the sphere in one point and the plane in one point. This projection maps that point on the sphere to the point on the plane.
If you put a light source in the center of this bowl, the rays of light effectively do a stereographic projection! The curves on the sphere cast shadows, mapping them to a straight line grid on the plane.
Inspired by https://www.thingiverse.com/thing:202774 I desined this with OpenSCAD.