Pseudosphere: A Sphere Of Negative Constant Curvature

A surface of constant negative curvature (contrary to a sphere, which has a constant positive curvature) which resembles a funnel. This surface when endowed with the Riemannian structure induced from the 3-dimensional Euclidean space $\mathbb R^3$ , is the same as the hyperbolic plane $\mathbb H^2$ . However, the hyperbolic plane cannot be fully embedded in the Euclidean 3-space isometrically. What we can do at best is to truncate the pseudosphere at some base circle. Then, that surface with a circular base or boundary will be isometrically isomorphic to $\mathbb H^2$ truncated at some $y=\text{constant}(\geq 0)$ line in the upper-half-plane model.
If one does not truncate the pseudosphere at some base circle and keeps extending it, then eventually one sees the curled patterns developing, and these curls eventually become too crowded making it impossible to accommodate the entire thing in the 3-dimensional space isometrically, i.e. without introducing any distortion in lengths and angles. This surface is in fact the surface of revolution of the curve called the Tractrix.