When $requals;2$, 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 $tequals;requals;2$, the solid is an ordinary ellipsoid. In particular, if $Aequals;Bequals;C$, the solid is a sphere of radius A.
When $Aequals;Bequals;3comma;Cequals;4comma;tequals;2.5$ and $requals;2$, the superellipsoid is a special solid known as Piet Hein's "superegg".