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A pseudosphere is a Riemannian manifold with constant negative intrinsic curvature. It can be defined with any number of dimensions >=2; the rest of the article will discuss the two-dimensional one.

The pseudosphere is the space described by hyperbolic geometry. It has a unit of length, the anti-radian (is that right?), which is the radius r of a circle whose circumference is 2πsinh(1)r.

It is impossible to embed a complete pseudosphere smoothly in R3. It is possible to embed part of it as a tractricoid.