Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the superquadrics or super-quadrics (also superquadratics) are a family of geometric shapes
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
defined by formulas that resemble those of elipsoids and other quadric
Quadric
In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
s, except that the squaring operations are replaced by arbitrary powers. They can be seen as the three-dimensional relatives of the Lamé curves ("superellipses").
The superquadrics include many shapes that resemble cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
A lozenge , often referred to as a diamond, is a form of rhombus. The definition of lozenge is not strictly fixed, and it is sometimes used simply as a synonym for rhombus. Most often, though, lozenge refers to a thin rhombus—a rhombus with acute angles of 45°...
s and spindles, with rounded or sharp corners. Because of their flexibility and relative simplicity, they are popular geometric modeling tools, especially in computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
.
Some authors, such as Alan Barr, define "superquadrics" as including both the superellipsoid
Superellipsoid
In mathematics, a super-ellipsoid or superellipsoid is a solid whose horizontal sections are super-ellipses with the same exponent r, and whose vertical sections through the center are super-ellipses with the same exponent t....
In geometry and computer graphics, a supertoroid or supertorus is usually understood to be a family of doughnut-like surfaces whose shape is defined by mathematical formulas similar to those that define the superquadrics...
s. However, the (proper) supertoroids are not superquadrics as defined above; and, while some superquadrics are superellipsoids, neither family is contained in the other.
Implicit equation
The basic superquadric has the formula
where r, s, and t are positive real numbers that determine the main features of the superquadric. Namely:
less than 1: a pointy octahedron with concave faces
Face (geometry)
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
.
exactly 1: a regular octahedron.
between 1 and 2: an octahedron with convex faces, blunt edges and blunt corners.
exactly 2: a sphere
greater than 2: a cube with rounded edges and corners.
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
): 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
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...
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 (longitude and latitude) are
where the auxiliary functions are
and the sign function
Sign function
In mathematics, the sign function is an odd mathematical function that extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function ....
GNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command-line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB...
code generates a mesh approximation of a superquadric:
function retval=superquadric(epsilon,a)
n=50;
etamax=pi/2;
etamin=-pi/2;
wmax=pi;
wmin=-pi;
deta=(etamax-etamin)/n;
dw=(wmax-wmin)/n;
k=0;
l=0;
[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);
In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
In geometry and computer graphics, a supertoroid or supertorus is usually understood to be a family of doughnut-like surfaces whose shape is defined by mathematical formulas similar to those that define the superquadrics...
In mathematics, a super-ellipsoid or superellipsoid is a solid whose horizontal sections are super-ellipses with the same exponent r, and whose vertical sections through the center are super-ellipses with the same exponent t....
In geometry, a superegg is a solid of revolution obtained by rotating an elongated super-ellipse with exponent greater than 2 around its longest axis. It is a special case of super-ellipsoid....
