The pseudosphere is the constant negative-Gaussian curvature surface of revolution generated by rotating a tractrix about its asymptote. It is sometimes also called the tractroid, tractricoid, antisphere, or tractrisoid (Steinhaus 1999, p. 251). Unless otherwise specified, the rim radius is 1. The parametric equations in Cartesian coordinates are
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(1)
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(2)
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(3)
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for 
and 
.
This parameter range includes two congruent halves
meeting at the circular rim 
, where the parameterization
is singular. Restricting to 
or 
selects one regular half.
For a pseudosphere with rim radius 
, the implicit form in Cartesian
coordinates is
![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 parameterizations include
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(5)
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(6)
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 |  | ![cosv+ln[tan(1/2v)]](/images/equations/Pseudosphere/Inline24.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 parameterization, 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 surface area and half the volume of 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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