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The elliptic hyperboloid is the generalization of the hyperboloid to three distinct semimajor axes. The elliptic hyperboloid of one sheet is a ruled surface and has Cartesian equation
(1)
and parametric equations
(2)
(3)
(4)
for
, or
(5)
(6)
(7)
or
(8)
(9)
(10)
The two-sheeted elliptic hyperboloid oriented along the z-axis has Cartesian equation
(11)
and parametric equations
(12)
(13)
(14)
The two-sheeted elliptic hyperboloid oriented along the x-axis has Cartesian equation
(15)
and parametric equations
(16)
(17)
(18)
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