The surface of revolution given by the parametric equations
(1)
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(2)
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(3)
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for and .
It is a quartic surface with equation
(4)
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An essentially equivalent surface called by Hauser the octdong surface follows by making the transformation in the above, leading to
(5)
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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)
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(7)
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(8)
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and of the second fundamental form are
(9)
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(10)
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(11)
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The Gaussian and mean curvatures are given by
(12)
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(13)
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The Gaussian curvature can be given implicitly as
(14)
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The eight surface has surface area and volume given by
(15)
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(16)
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Its centroid is at and its moment of inertia tensor is
(17)
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for a solid with uniform density and mass .