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Eight Surface


EightSurface

The surface of revolution given by the parametric equations

x(u,v)=cosusin(2v)
(1)
y(u,v)=sinusin(2v)
(2)
z(u,v)=sinv
(3)

for u in [0,2pi) and v in [-pi/2,pi/2].

It is a quartic surface with equation

 4z^4+a^2(x^2+y^2-4z^2)=0.
(4)

An essentially equivalent surface called by Hauser the octdong surface follows by making the transformation z->z/2 in the above, leading to

 z^4+4a^2(x^2+y^2-z^2)=0.
(5)

Setting x=0, z=x/2, and a^'=a/2 (i.e., scaling by half and relabeling the z-axis as the x-axis) gives the eight curve, of which the eight surface is therefore "almost" a surface of revolution.

The coefficients of the first fundamental form are

E=a^2sin^2(2v)
(6)
F=0
(7)
G=1/2a^2[5+cos(2v)+4cos(4v)]
(8)

and of the second fundamental form are

e=-(4sqrt(2)cos^3vsin^2v)/(|sin(2v)|sqrt(5+cos(2v)+4cos(4v)))
(9)
f=0
(10)
g=-(2sqrt(2)[5cosv+cos(3v)]sin^2v)/(|sin(2v)|sqrt(5+cos(2v)+4cos(4v))).
(11)

The Gaussian and mean curvatures are given by

K=(4[2+cos(2v)])/([5+cos(2v)+4cos(4v)]^2)
(12)
H=(cos(v)[-11+3cos(2v)-2cos(4v)])/(sqrt(2)|sin(2v)|[5+cos(2v)+4cos(4v)]^(3/2)).
(13)

The Gaussian curvature can be given implicitly as

 K(x,y,z)=(3a^6-2a^4z^2)/((5a^4-17a^2z^2+16z^4)^2).
(14)

The eight surface has surface area and volume given by

S=(pia^2[240+136sqrt(5)+31ln(17+8sqrt(5))])/(128)
(15)
V=(16)/(15)pia^3.
(16)

Its centroid is at (0,0,0) and its moment of inertia tensor is

 I=[(13)/(21)Ma^2 0 0; 0 (13)/(21)Ma^2 0; 0 0 8/(21)Ma^2]
(17)

for a solid with uniform density and mass M.


See also

Eight Curve

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References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 310, 1997.Hauser, H. "Gallery of Singular Algebraic Surfaces: Octdong." https://homepage.univie.ac.at/herwig.hauser/gallery.html.

Cite this as:

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

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