The term Supershape is used to refer to a 3d-parametric shape based on the so called superformula popularized by Johan Gielis. This formula is an extension of the superellipse and has been promoted using the term Superformula™.
Johan Gielis' superformula is a generalization of the Superellipsoid by Piet Hein.
Superdupershapes are based on the superduperformula which can be applied to arbitrary parametric functions (including the superformula)