In
mathematics, the superquadrics or super-quadrics (also superquadratics) are a family of
geometric shapes defined by formulas that resemble those of
ellipsoids and other
quadrics, except that the
squaring operations are replaced by arbitrary powers. They can be seen as the three-dimensional relatives of the
superellipses. The term may refer to the solid object or to its
surface, depending on the context. The equations below specify the surface; the solid is specified by replacing the equality signs by less-than-or-equal signs.
The superquadrics include many shapes that resemble
cubes,
octahedra,
cylinders,
lozenges and
spindles, with rounded or sharp corners.[1] Because of their flexibility and relative simplicity, they are popular
geometric modeling tools, especially in
computer graphics. It becomes an important geometric primitive widely used in computer vision,[2][3] robotics,[4] and physical simulation.[5]
Some authors, such as
Alan Barr, define "superquadrics" as including both the
superellipsoids and the
supertoroids.[1][6] In modern computer vision literatures, superquadrics and
superellipsoids are used interchangeably, since
superellipsoids are the most representative and widely utilized shape among all the superquadrics.[2][3] Comprehensive coverage of geometrical properties of superquadrics and methods of their recovery from
range images and
point clouds are covered in several computer vision literatures.[1][3][7][8]
Formulas
Implicit equation
The surface of the basic superquadric is given by
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
concavefaces and sharp
edges.
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
Parametric description
Parametric equations in terms of surface parameters u and v (equivalent to longitude and latitude if m equals 2) are
Barr introduces the spherical product which given two plane curves produces a 3D surface. If
are two plane curves then the spherical product is
This is similar to the typical parametric equation of a
sphere:
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]
Plotting code
The following
GNU Octave code generates a mesh approximation of a superquadric: