A pseudosphere is an surface wth a constant negative curvature.
In most publications, it is almost given that the tracioid (rotated tractrix) is the surface that has a constant negative curvature, and "tracioid" "pseudosphere" are used interchangable.
This made me wonder are there other pseudospheres?
In Klein's "Vorlesungen uber Nicht-Euclidische Geometrie" (1928) $4, page 286, figure 220, KLein mentions an "single sheet hyperboloid-like plane" as hyperbolic plane, and even gives its represetation in the Klein disk model.
I tried to find more recent information on this surface but could not find much more.
(everything I found refers to the hyperboloid model, but this is different, this is not a model of the hyperbolic plane but a surface with a contant negative curvature.)
Still can this made me wonder, what is the equation of this surface. (is it an hyperboloid or doe it only look like one) how does it look in the Poincare disk model.
Or is there a proof that Klein was wrong, such a surface cannot have a constant negative curvature?
I am at a dead end, is there more?