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

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.

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?


3 Answers 3


Take a look here for pictures. Also take a look here for the references.

  • 1
    $\begingroup$ Yes, I was just going to comment about the Bäcklund transform that takes one surface of constant negative curvature in $\Bbb R^3$ to another. $\endgroup$ Sep 14, 2014 at 12:29
  • $\begingroup$ In addition, she has a lot of examples, one of them (a surface of revolution) might be the surface Klein alludes to. $\endgroup$ Sep 14, 2014 at 12:31
  • $\begingroup$ thanks but i could not find the equations of the surfaces $\endgroup$
    – Willemien
    Sep 14, 2014 at 14:26
  • $\begingroup$ @Willemien: How hard did you try? Did you click on the links under the pictures? Did you look at Terng's homepage I linked to? $\endgroup$ Sep 14, 2014 at 16:56
  • $\begingroup$ @studious i had a look at the pictures (especially virtualmathmuseum.org/Surface/hyperbolic_k1_sor/… and virtualmathmuseum.org/Surface/conic_k-1_sor/conic_k-1_sor.html ) i read the lecture notes of Chuu-Lian Terng , but they were a bit to advanced for me , what i am looking for are formulas like $ t \mapsto \left( t - \tanh{t}, \operatorname{sech}\,{t} \right) $ (this is for a tractrix) or $ (t, \alpha)\mapsto \left( (t - \tanh{t}) \cos \alpha, (t - \tanh{t}) \sin \alpha, \operatorname{sech}\,{t} \right) $ (tracioid) and they are not given. $\endgroup$
    – Willemien
    Sep 14, 2014 at 17:52

You can find three space surfaces carrying constant negative curvature at p.14-16 of http://www.dm.unibo.it/~arcozzi/beltrami_sent1.pdf, in the subsection "Imbedding the psudosphere in Euclidean space" (they are in fact Beltrami's three surfaces). The parametrization of the surfaces is not explicit, but all ingredients are there.


The meridians of revolved surfaces with constant Gauss curvatutre $ \pm 1$ are given by ( arc length is independent variable, $r$ radius):

$ r'' (s) \pm r (s) = 0 $ . You can integrate this and take it further.


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