A pseudosphere (or antisphere or tractricoid) has a surface of constant negative Gaussian curvature, in contrast to a sphere, which has constant positive Gaussian curvature.
Handwaving the model of physics and spacetime that would allow the formation of bodies closely approximating this top-like shape—pseudosphere planets, pseudosphere stars, etc—how would you optimally (i.e. with least distortion) display the surface as a 2D map?
My instinct says it would be comparable to the Sinusoidal projection, but with the maximum distortion at the equator instead of poles. But I have a hard time resolving the equator as the widest and largest portion of the surface having the most distortion.