Eight Surface

The surface of revolution given by the parametric equations

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

It is a quartic surface with equation

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

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

and of the second fundamental form are

The Gaussian and mean curvatures are given by

The Gaussian curvature can be given implicitly as

The eight surface has surface area and volume given by

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

for a solid with uniform density and mass M.

See also

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References

Referenced on Wolfram|Alpha

Cite this as:

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

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