Dini's Surface
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 = [画像:-(a^2cosvsinv)/(sqrt(a^2+b^2))]
(7)
f = [画像:(abcosv)/(sqrt(a^2+b^2))]
(8)
g = [画像:(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 = [画像:-1/(a^2+b^2)]
(11)
H = [画像:-(cot(2v))/(sqrt(a^2+b^2)).]
(12)
See also
PseudosphereExplore with Wolfram|Alpha
WolframAlpha
More things to try:
References
Geometry Center. "Dini's Surface." http://www.geom.uiuc.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.htm.Referenced on Wolfram|Alpha
Dini's SurfaceCite this as:
Weisstein, Eric W. "Dini's Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DinisSurface.html