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 of revolution 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 218 219 and 220, Klein gives three surfaces for hyperbolic surfaces:


figure 218 looks like an hill 


figure 219 the tracioid


and figure 220 looks like an single sheet hyperboloid or catenoid 


Unfortunedly Klein doesn't give 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, but maybe he refers to surfaces that are not surfaces of revolution.


In <https://math.stackexchange.com/a/666101/88985> there is a link to 
<http://www.dm.unibo.it/~arcozzi/beltrami_sent1.pdf>


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.
> 
> 
> 


Again sadly there is no reference to the publication of Minding.


I did ask a similar question at the mathematics stack exchange site <https://math.stackexchange.com/q/930847/88985>


and one answer refered to the virtual math museum, the gallery of famous surfaces


<http://virtualmathmuseum.org/Surface/gallery_o.html>


And under Pseudospherical Surfaces (K = -1) 


It mentions three surfaces of revolution with a constant negative curvature:


The tracioid : <http://virtualmathmuseum.org/Surface/pseudosphere/pseudosphere.html>


The Conic K =-1 Surface of revolution :
<http://virtualmathmuseum.org/Surface/conic_k-1_sor/conic_k-1_sor.html>
 looks like Klein's hill


and 


The hyperbolic K =-1 Surface of revolution :
<http://virtualmathmuseum.org/Surface/hyperbolic_k1_sor/hyperbolic_k1_sor.html>
 looks like Klein's hyperbolioid


These pages sadly do not give a lot of information. 


Now I am stuck: What are those other surfaces of revolution that have a constant negative curvature? and what are their (parametric) equations?


Or don't they have a constant negative curvature and can it be proved that the tracioid is the only surface of revolution with an constant negative curvature?



Besides the tracioid are there other surfaces of revolution that have a constant negative curvature?
There are many examples of surfaces in $\mathbb{R}^3$ with constant negative curvature. They can be described by using the so-called *parametrization by Chebyshev nets.* Have a look at the paper by Robert McLachlan


[*A gallery of constant-negative-curvature surfaces*](http://link.springer.com/article/10.1007%2FBF03024701), 
The Mathematical Intelligencer **16** (1994), 31-37. 


However (and this answers your question) there are precisely *three* types of *revolution* surfaces with constant curvature $-1$. The reason is explained here (it boils down to solving a second order ODE): 


<https://math.stackexchange.com/questions/77396/revolution-surfaces-of-constant-gaussian-curvature>


One of them of course is the tracioid. Pictures of the three surfaces can be found here:


<http://demonstrations.wolfram.com/SurfacesOfRevolutionWithConstantGaussianCurvature/>


