A superellipsoid is a 3-dimensional solid whose horizontal cross-sections are superellipses with the same exponent r, and whose vertical cross-sections through the centre are superellipses with the same exponent t. The general implicit equation for a superellipsoid is
The parameters t and r are positive real numbers which control the amount of flattening at the tips and equator of the solid. Factors A, B, and C scale the basic shape along each axis and are called the semi-diameters of the solid.
If t = r, the equation for a superellipsoid becomes a special case of the superquadric equation.
When r = 2, the horizontal cross sections of the solid are circles, so the superellipsoid is a solid of revolution: it can be obtained by rotating a superellipse of exponent t around the vertical axis.
When t = r =2, the solid is an ordinary ellipsoid. In particular, if A = B = C, the solid is a sphere of radius A.
When A = B = 3, C = 4, t = 2.5 and r = 2, the superellipsoid is a special solid known as Piet Hein's "superegg".
The following graph shows a superellipsoid. Use the sliders to adjust the semi-diameters and parameters to see what solids you can make.
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