The pseudosphere is the constant negative-Gaussian curvature surface of revolution generated by a tractrix about its asymptote. It is sometimes also called the tractroid, tractricoid, antisphere, or tractrisoid (Steinhaus 1999, p. 251). The Cartesian parametric equations are
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(1)
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(2)
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(3)
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for 
and 
.
It can be written in implicit Cartesian form as
![z^2=[asech^(-1)(sqrt((x^2+y^2)/a))-sqrt(a^2-x^2-y^2)]^2.](/images/equations/Pseudosphere/NumberedEquation1.svg) |
(4)
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Other parametrizations include
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(5)
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(6)
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 |  | ![cosv+ln[tan(1/2v)]](/images/equations/Pseudosphere/Inline20.svg) |
(7)
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for 
and 
(Gray et al. 2006, p. 480) and
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(8)
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(9)
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(10)
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for 
and 
,
where
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(11)
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(12)
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(Gray et al. 2006, p. 477).
In the first parametrization, the coefficients of the first fundamental form are
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(13)
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(14)
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(15)
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the second fundamental form coefficients are
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(16)
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(17)
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(18)
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and the surface area element is
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(19)
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The surface area is
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(20)
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which is exactly that of the usual sphere.
Even though the pseudosphere has infinite extent, it has finite volume. The volume can be found by making the change of variables 
, giving 
, and substituting into the equation for a solid
of revolution, giving
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(21)
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which is exactly half that of the usual sphere.
The Gaussian and mean curvatures are
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(22)
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(23)
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The pseudosphere therefore has the same volume as the sphere while having constant negative Gaussian curvature (rather than the constant positive curvature of the sphere), leading to the name "pseudo-sphere." The pseudosphere's constant negative curvature also makes it a local partial model of hyperbolic geometry, just as a cone or cylinder is a local partial model of Euclidean geometry of the plane.
An equation for the geodesics on a pseudosphere is given by
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(24)
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