Implicit equation

The surface of the basic superquadric is given by

${\displaystyle \left|x\right|^{r}+\left|y\right|^{s}+\left|z\right|^{t}=1}$

where r, s, and t are positive real numbers that determine the main features of the superquadric. Namely:

• less than 1: a pointy octahedron modified to have concave faces and sharp edges.
• exactly 1: a regular octahedron.
• between 1 and 2: an octahedron modified to have convex faces, blunt edges and blunt corners.
• exactly 2: a sphere
• greater than 2: a cube modified to have rounded edges and corners.
• infinite (in the limit): a cube

Each exponent can be varied independently to obtain combined shapes. For example, if r=s=2, and t=4, one obtains a solid of revolution which resembles an ellipsoid with round cross-section but flattened ends. This formula is a special case of the superellipsoid's formula if (and only if) r = s.

If any exponent is allowed to be negative, the shape extends to infinity. Such shapes are sometimes called super-hyperboloids.

The basic shape above spans from -1 to +1 along each coordinate axis. The general superquadric is the result of scaling this basic shape by different amounts A, B, C along each axis. Its general equation is

${\displaystyle \left|{\frac {x}{A}}\right|^{r}+\left|{\frac {y}{B}}\right|^{s}+\left|{\frac {z}{C}}\right|^{t}=1.}$

Parametric description

Parametric equations in terms of surface parameters u and v (equivalent to longitude and latitude if m equals 2) are

${\displaystyle {\begin{aligned}x(u,v)&{}=Ag\left(v,{\frac {2}{r}}\right)g\left(u,{\frac {2}{r}}\right)\\y(u,v)&{}=Bg\left(v,{\frac {2}{s}}\right)f\left(u,{\frac {2}{s}}\right)\\z(u,v)&{}=Cf\left(v,{\frac {2}{t}}\right)\\&-{\frac {\pi }{2}}\leq v\leq {\frac {\pi }{2}},\quad -\pi \leq u<\pi ,\end{aligned}}}$

where the auxiliary functions are

${\displaystyle {\begin{aligned}f(\omega ,m)&{}=\operatorname {sgn}(\sin \omega )\left|\sin \omega \right|^{m}\\g(\omega ,m)&{}=\operatorname {sgn}(\cos \omega )\left|\cos \omega \right|^{m}\end{aligned}}}$

and the sign function sgn(x) is

${\displaystyle \operatorname {sgn}(x)={\begin{cases}-1,&x<0\\0,&x=0\\+1,&x>0.\end{cases}}}$

Spherical product

Barr introduces the spherical product which given two plane curves produces a 3D surface. If

$${\displaystyle f(\mu )={\begin{pmatrix}f_{1}(\mu )\\f_{2}(\mu )\end{pmatrix}},\quad g(\nu )={\begin{pmatrix}g_{1}(\nu )\\g_{2}(\nu )\end{pmatrix}}}$$

are two plane curves then the spherical product is

$${\displaystyle h(\mu ,\nu )=f(\mu )\otimes g(\nu )={\begin{pmatrix}g_{1}(\nu )\ f_{1}(\mu )\\g_{1}(\nu )\ f_{2}(\mu )\\g_{2}(\nu )\end{pmatrix}}}$$

This is similar to the typical parametric equation of a sphere:

$${\displaystyle {\begin{aligned}x&=x_{0}+r\sin \theta \;\cos \varphi \\y&=y_{0}+r\sin \theta \;\sin \varphi \qquad (0\leq \theta \leq \pi ,\;0\leq \varphi <2\pi )\\z&=z_{0}+r\cos \theta \end{aligned}}}$$

which give rise to the name spherical product.

Barr uses the spherical product to define quadric surfaces, like ellipsoids, and hyperboloids as well as the torus, superellipsoid, superquadric hyperboloids of one and two sheets, and supertoroids.^[1]