Besides the tracioid are there other surfaces of revolution that have a constant negative curvature?

Question

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?

Answer 1

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):

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/

Answer 2

The topic is quite old.

There are three and only three types of rotationally symmetric surfaces for constant $K = -1/a^2$ where $a$ is the cuspidal radius of the central pseudosphere. These are the central pseudosphere (parameter $m =1$) or tractricoid, Conic type ($m < 1$) and Ring type ($m > 1$). The descriptions Ring, Conic type etc. are given by Klein in:

Felix Klein," Vorlesungen über nicht-euklidische Geometrie" 3rd ed. (Berlin, 1928). with a reference/reprint iirc from Crelle's Journal.

Asymptotic lines of a Chebyshev Net on these three surfaces are given by the Sine-Gordon Equation.

Present day English translations of Vorlesungen may be available on googling, else one can contact German newsgroups e.g., de.sci.mathematik.

Although the central pseudosphere is often referred to as "the" Beltrami surface, the physical paper model he made is isometrically equivalent to the Conic type ( m > 1) that you can readily verify in Daina's blog. I am discussing it the concurrent thread here

In 1868 when making his hyperbolic plane model Beltrami was a professor of mathematics in Bologna. It had to be important to him to take his paper models with him when Beltrami returned to University of Pavia 1876.