Superquadrics

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

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

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

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

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]

The following GNU Octave code generates a mesh approximation of a superquadric:

function superquadric(epsilon,a)  n = 50;  etamax = pi/2;  etamin = -pi/2;  wmax = pi;  wmin = -pi;  deta = (etamax-etamin)/n;  dw = (wmax-wmin)/n;  [i,j] = meshgrid(1:n+1,1:n+1)  eta = etamin + (i-1) * deta;  w   = wmin + (j-1) * dw;  x = a(1) .* sign(cos(eta)) .* abs(cos(eta)).^epsilon(1) .* sign(cos(w)) .* abs(cos(w)).^epsilon(1);  y = a(2) .* sign(cos(eta)) .* abs(cos(eta)).^epsilon(2) .* sign(sin(w)) .* abs(sin(w)).^epsilon(2);  z = a(3) .* sign(sin(eta)) .* abs(sin(eta)).^epsilon(3);  mesh(x,y,z);end

• Superegg
• Superellipsoid
• Ellipsoid

• Bibliography: SuperQuadric Representations
• Superquadric Tensor Glyphs
• SuperQuadric Ellipsoids and Toroids, OpenGL Lighting, and Timing
• Superquadrics by Robert Kragler, The Wolfram Demonstrations Project.
• Superquadrics in Python
• Superquadrics recovery algorithm in Python and MATLAB