Anyway, because of the negative curvature, we get Pseudosphere in Euclidean World when we express a plane surface of Hyperbolic NonEuclidean World. We have not been aware of it. It is hard to see that Pseudosphere has constant curvature by visual observation. You get a curved line by cutting a surface in the normal direction, and curvature K . It depends on how you cut.
Gaussian curvature is product of the maximum curvature and the minimum curvature .
The maximum curvature is getting smaller and the minimum curvature is getting larger toward the tail of Pseudosphere. That is to say, Gaussian curvature is kind of an average curvature in consideration of curvature in all direction. Now we change curvature of Pseudosphere. When you enlarge K , ratio of curving goes down. So Pseudosphere gets fat.