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Pseudosphere


Pseudosphere

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

x=sechucosv
(1)
y=sechusinv
(2)
z=u-tanhu
(3)

for u in (-infty,infty) and v in [0,2pi).

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.
(4)

Other parametrizations include

x=cosusinv
(5)
y=sinusinv
(6)
z=cosv+ln[tan(1/2v)]
(7)

for u in [0,2pi) and v in (0,pi) (Gray et al. 2006, p. 480) and

x=phi(v)cosu
(8)
y=phi(v)sinu
(9)
z=psi(v)
(10)

for u in [0,2pi) and v in (-infty,infty), where

phi(v)={e^v for v<0; e^(-v) for v>=0
(11)
psi(v)={sqrt(1-e^(2v))-tanh^(-1)(sqrt(1-e^(2v))) for v<0; ln(e^v+sqrt(e^(2v)-1))-e^(-v)sqrt(e^(2v)-1) for v>=0
(12)

(Gray et al. 2006, p. 477).

In the first parametrization, the coefficients of the first fundamental form are

E=tanh^2u
(13)
F=0
(14)
G=sech^2u,
(15)

the second fundamental form coefficients are

e=-sechutanhu
(16)
f=0
(17)
g=sechutanhu,
(18)

and the surface area element is

 dS=sechutanhu.
(19)

The surface area is

 S=2int_0^(2pi)int_0^inftysechutanhududv=4pi,
(20)

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 z=u-tanhu, giving dz=tanh^2udu, and substituting into the equation for a solid of revolution, giving

 V=piint_(-infty)^inftysech^2utanh^2udu=2/3pi,
(21)

which is exactly half that of the usual sphere.

The Gaussian and mean curvatures are

K=-1
(22)
H=1/2(sinhu-cschu).
(23)

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

 cosh^2u+(v+c)^2=k^2.
(24)

See also

Funnel, Gabriel's Horn, Hyperbolic Geometry, Tractrix

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References

Fischer, G. (Ed.). Plate 82 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 77, 1986.Geometry Center. "The Pseudosphere." http://www.geom.umn.edu/zoo/diffgeom/pseudosphere/.Gray, A.; Abbena, E.; and Salamon, S. Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. Boca Raton, FL: CRC Press, pp. 477 and 480, 2006.JavaView. "Classic Surfaces from Differential Geometry: Pseudo Sphere." http://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_PseudoSphere.html.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 251, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 199-200, 1991.

Cite this as:

Weisstein, Eric W. "Pseudosphere." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Pseudosphere.html

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