There is no surface in $ R^3 $ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but in most publications on hyperbolic geometry, it is almost given that the tracioid (tractrix rotated about its asymptope) is a surface that has a constant negative curvature, and in many publications "tracioid" and "pseudosphere" are used interchangable.
But I am wondering are there other surfaces that have a constant negative curvature?
I did some searching and did find:
In Klein's "Vorlesungen uber Nicht-Euclidische Geometrie" (1928) $4, page 286, figure 218 - 220, Klein gives three surfaces for hyperbolic surfaces:
- one that looks like an hill
- one that looks like an single sheet hyperboloid
- and then the well known tracioid
Unfortunedly Klein doesn't give the equations of these surfaces.
In Sommerville "The elements of non euclidean geometry" it says (Dover edition page 167)
Furtunedly we do not require to take the imaginary circle a the type of surfaces of constant negative curvature. There are different forms of such surfaces, even of revolution, but the simplest is the surface called pseudosphere, which is formed by revolving a tractrix about its asymptope.
Again a hint that other surfaces exist but no equations
and this publication says at page 6
Gauss published his Theorema egregium in 1827 and it was already clear that, if figures could be moved isometrically, cuvature had to be constant. Minding observed that the converse was true in the 30's, and he found various surfaces of constant negative curvature in Euclidean space, the tractroid among them.
sadly there is no reference to the publication of Minding.
So i am stuck: What are those other surfaces of a constant negative curvature? and what are their equations?