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Dini's Surface

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DinisSurface

A surface of constant negative curvature obtained by twisting a pseudosphere and given by the parametric equations

x=acosusinv
(1)
y=asinusinv
(2)
z=a{cosv+ln[tan(1/2v)]}+bu.
(3)

The above figure corresponds to a=1, b=0.2, u in [0,4pi], and v in (0,2].

Dini's surface is pictured in the upper right-hand corner of Gray (1997; left figure), as well as on the cover of volume 2, number 3 of La Gaceta de la Real Sociedad Matemática Española (1999; right figure).

The coefficients of the first fundamental form are

E=1/2[a^2+2b^2-a^2cos(2v)]
(4)
F=abcosvcotv
(5)
G=a^2cot^2v,
(6)

the coefficients of the second fundamental form are

e=-(a^2cosvsinv)/(sqrt(a^2+b^2))
(7)
f=(abcosv)/(sqrt(a^2+b^2))
(8)
g=(a^2cotv)/(sqrt(a^2+b^2)),
(9)

and the area element is

dA=asqrt(a^2+b^2)cosvdu ^ dv.
(10)

The Gaussian and mean curvatures are given by

K=-1/(a^2+b^2)
(11)
H=-(cot(2v))/(sqrt(a^2+b^2)).
(12)

See also

Pseudosphere

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References

Geometry Center. "Dini's Surface." http://www.geom.umn.edu/zoo/diffgeom/surfspace/dini/.Gray, A. "Dini's Surface." §21.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 493-495, 1997.Nordstrand, T. "Dini's Surface." http://jalape.no/math/dintxt.

Cite this as:

Weisstein, Eric W. "Dini's Surface." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DinisSurface.html

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