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A spheroid is an ellipsoid having two axes of equal length, making it a surface of revolution. By convention, the two distinct axis lengths are denoted and , and the spheroid is oriented so that its axis of rotational symmetric is along the -axis, giving it the parametric representation
(1)
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(2)
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(3)
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with , and .
The Cartesian equation of the spheroid is
(4)
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If , the spheroid is called oblate (left figure). If , the spheroid is prolate (right figure). If , the spheroid degenerates to a sphere.
In the above parametrization, the coefficients of the first fundamental form are
(5)
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(6)
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(7)
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and of the second fundamental form are
(8)
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(9)
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(10)
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The Gaussian curvature is given by
(11)
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the implicit Gaussian curvature by
(12)
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and the mean curvature by
(13)
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The surface area of a spheroid can be variously written as
(14)
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(15)
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(16)
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(17)
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where
(18)
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(19)
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and is a hypergeometric function.
The volume of a spheroid can be computed from the formula for a general ellipsoid with ,
(20)
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(21)
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(Beyer 1987, p. 131).
The moment of inertia tensor of a spheroid with -axis along the axis of symmetry is given by
(22)
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