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, 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):
http://math.stackexchange.com/questions/77396/revolution-surfaces-of-constant-gaussian-curvaturehttps://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/