I've seen the attached images describing surfaces of negative curvature. I was wondering if there exist such surfaces with constant Gaussian negative curvature.
To this end, I attempted to model the surfaces by: $$f(r,\theta)=A(r)\sin(n\theta)$$ And attempted to find $A(r)$ such that the surface would have a constant negative curvature, $K =const < 0$.
Unfortunately, the resulting differential equation is rather complex, and I not even sure it has a solution. So:
- Is there a surface of constant negative Gaussian curvature, that can be described by $A(r)\sin(n\theta)$, for some finite range of $r$?
- If not, are there other surfaces of constant negative $K$ that are similar to the images, and if so, what would be there parameterization?
Addendum:
It looks like theses surfaces are similar to Periodic Amsler surfaces, that are claimed to have constant Gaussian curvature. The paper in the link contains the following image, and it would be nice to understand how to parameterize them.
