Pilou, you have created an interesting surface.

I did 3 versions with circular array 4, mirror, and network, and want to try one with edge of two half tractrixes, connected tail to tail.

Re: pseudosphere, which has negative curvature, a few of the many search results:
http://www.rudyrucker.com/blog/2009/08/28/pseudospheres/

https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-pseudosphere/

http://www.cs.unm.edu/~joel/NonEuclid/pseudosphere.html

Wikipedia results for tractrix were limited IMHO, but had good information, including loudspeaker horns.

- Brian