Sine Surface
SineSurface
The surface given by the parametric equations
x = asinu
(1)
y = asinv
(2)
z = asin(u+v).
(3)
It is a sextic surface with algebraic equation
| 4x^2y^2z^2+a^2(x-y-z)(x+y-z)(x-y+z)(x+y+z)=0. |
(4)
|
The coefficients of the first fundamental form are
E = a^2[cos^2u+cos^2(u+v)]
(5)
F = a^2cos^2(u+v)
(6)
G = a^2[cos^2v+cos^2(u+v)],
(7)
the second fundamental form coefficients are
the area element is
| dS=a^2sqrt(cos^2ucos^2v+(cos^2u+cos^2v)cos^2(u+v)), |
(11)
|
the Gaussian curvature is
| [画像: K=(cosucosv[sinusinv-cosucosvsin^2(u+v)])/([acos^2ucos^2v+a(cos^2u+cos^2v)cos^2(u+v)]^2), ] |
(12)
|
![画像: K=(cosucosv[sinusinv-cosucosvsin^2(u+v)])/([acos^2ucos^2v+a(cos^2u+cos^2v)cos^2(u+v)]^2),](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/SineSurface/NumberedEquation3.svg){kind=link}
and the mean curvature is a complicated expression.
It has volume
| V=pia^3. |
(13)
|
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References
Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 315-316, 1997.Referenced on Wolfram|Alpha
Sine SurfaceCite this as:
Weisstein, Eric W. "Sine Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SineSurface.html