Supershape

Supershape is a spectral surface, given by a parametrisation, immersed.

Open Supershape in the interactive viewer →

aperiodic def-parametric immersed implemented spectral

Formula

x(u, v)

\left(\operatorname{abs}\left(\cos\left(\frac{5 u}{4}\right)\right)^{7} + \operatorname{abs}\left(\sin\left(\frac{5 u}{4}\right)\right)^{7}\right)^{-0.5} \cos\left(u\right) \left(\operatorname{abs}\left(\cos\left(\frac{5 v}{4}\right)\right)^{7} + \operatorname{abs}\left(\sin\left(\frac{5 v}{4}\right)\right)^{7}\right)^{-0.5} \cos\left(v\right)

y(u, v)

\left(\operatorname{abs}\left(\cos\left(\frac{5 u}{4}\right)\right)^{7} + \operatorname{abs}\left(\sin\left(\frac{5 u}{4}\right)\right)^{7}\right)^{-0.5} \sin\left(u\right) \left(\operatorname{abs}\left(\cos\left(\frac{5 v}{4}\right)\right)^{7} + \operatorname{abs}\left(\sin\left(\frac{5 v}{4}\right)\right)^{7}\right)^{-0.5} \cos\left(v\right)

z(u, v)

\left(\operatorname{abs}\left(\cos\left(\frac{5 v}{4}\right)\right)^{7} + \operatorname{abs}\left(\sin\left(\frac{5 v}{4}\right)\right)^{7}\right)^{-0.5} \sin\left(v\right)

Properties

Family: spectral
Given by: a parametrisation
Curvature: no condition imposed
Periodicity: not periodic
Ends: 0
Embedding: immersed
Fidelity: exact
Exactness: elementary

Definition

spherical product of two Gielis superformulas at the operator's default preset STARFISH, (m, n1, n2, n3, a, b) = (5, 2, 7, 7, 1, 1) for both; the longitude seam closes because the superformula radius is even in its angle

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.36 of tol, 0.1% of oracle beyond tol (tol floor 2.5 x 0.08894).

Gielis, J. (2003). "A generic geometric transformation that unifies a wide range of natural and abstract shapes." American Journal of Botany 90(3), 333-338. doi:10.3732/ajb.90.3.333 -- the paper that introduced the superformula.
Bourke, P. "Supershapes / SuperFormula." https://paulbourke.net/geometry/supershape/ -- the 3D spherical product, shell and toroidal mappings this module follows.
Barr, A. H. (1981). "Superquadrics and Angle-Preserving Transformations." IEEE Computer Graphics & Applications 1(1), 11-23 -- the superellipsoid (SUPERELLIPSOID mode).