Am elliptic torus is a surface of revolution which is a generalization of the ring torus. It is produced by rotating an ellipse embedded in the -plane having horizontal semi-axis , vertical semi-axis , and located a distance away from the -axis about the -axis. It is given by the parametric equations
(1)
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(2)
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(3)
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for .
This gives first fundamental form coefficients of
(4)
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(5)
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(6)
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second fundamental form coefficients of
(7)
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(8)
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(9)
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The Gaussian curvature and mean curvature are
(10)
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(11)
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By Pappus's centroid theorems, the surface area and volume are
(12)
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(13)
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(14)
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(15)
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where is a complete elliptic integral of the first kind and
(16)
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is the eccentricity of the ellipse cross section.