Pseudospherical Surfaces
These surfaces have constant negative Gaussian curvature. There are no complete
pseudospherical surfaces immersed in 3-space, by Hilbert's theorem. They all have singularities.
Each of these surfaces is the unique pseudospherical surface containing a particular space curve
as a cuspidal edge singular curve. The surfaces here are colored by mean curvature.
The theory used to produce them
is described
here