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# Eight Surface

The surface of revolution given by the parametric equations

 (1) (2) (3)

for and .

It is a quartic surface with equation

 (4)

An essentially equivalent surface called by Hauser the octdong surface follows by making the transformation in the above, leading to

 (5)

Setting , , and (i.e., scaling by half and relabeling the -axis as the -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

 (6) (7) (8)

and of the second fundamental form are

 (9) (10) (11)

The Gaussian and mean curvatures are given by

 (12) (13)

The Gaussian curvature can be given implicitly as

 (14)

The eight surface has surface area and volume given by

 (15) (16)

Its centroid is at and its moment of inertia tensor is

 (17)

for a solid with uniform density and mass .

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