In geometry

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 ....

 and 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....

, a supertoroid or supertorus is usually understood to be a family of doughnut

Doughnut

A doughnut or donut is a fried dough food and is popular in many countries and prepared in various forms as a sweet snack that can be homemade or purchased in bakeries, supermarkets, food stalls, and franchised specialty outlets...

-like surfaces (technically, a topological

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 torus) whose shape is defined by mathematical formulas similar to those that define the superquadrics. The plural of "supertorus" is either supertori or supertoruses.
The family was described and named by Alan Barr in 1994.

Barr's supertoroids have been fairly popular in computer graphics as a convenient model for many objects, such as smooth frames for rectangular things. One quarter of a supertoroid can provide a smooth and seamless 90-degree joint between two superquadric cylinder

Cylinder (geometry)

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...

s. However they are not algebraic surface

Algebraic surface

In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...

s (except in special cases).

Formulas

Alan Barr's supertoroids are defined by parametric equations similar to the trigonometric

Trigonometry

Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

 equations of the torus, except that the sine

Sine

In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....

 and cosine terms are raised to arbitrary powers. Namely, the generic point P(u, v) of the surface is given by
where , , and the parameters u and v range from 0 to 360 degrees (0 to 2π radian

Radian

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

s).

In these formulas, the parameter s>0 controls the "squareness" of the vertical sections, t controls the squareness of the horizontal sections, and a, b ≥ 1 are the major radii in the X and Y directions. With s=t=1 and a=b=R one obtains the ordinary torus with major radius R and minor radius 1, with the center at the origin and rotational symmetry

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

 about the Z axis.

In general, the supertorus defined as above spans the interval

Interval

Interval may refer to:* Interval , a range of numbers * Interval measurements or interval variables in statistics is a level of measurement...

s in X, in Y, and in Z. The whole shape is symmetric about the planes X=0, Y=0, and Z=0. The hole runs in the Z direction and spans the intervals in X and in Y.

A curve of constant u on this surface is an horizontal Lamé curve with exponent 2/t, scaled in X and Y and displaced in Z. A curve of constant v, projected on the plane X=0 or Y=0, is a Lamé curve with exponent 2/s, scaled and horizontally shifted. If v is 0, the curve is planar and spans the interval in X, and in Z; and similarly if v is 90, 180, or 270 degrees. The curve is planar also if a = b.

In general, if a≠b and v is not a multiple of 90 degrees, the curve of constant v will not be planar; and, conversely, a vertical plane section of the supertorus will not be a Lamé curve.

The basic supertoroid shape defined above is often modified by non-uniform scaling to yield supertoroids of specific width, length, and vertical thickness.

Plotting code

The following GNU Octave

GNU Octave

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 plots of a supertorus:


function supertoroid(epsilon,a)
n=50;
d=.1;
etamax=pi;
etamin=-pi;
wmax=pi;
wmin=-pi;
deta=(etamax-etamin)/n;
dw=(wmax-wmin)/n;
k=0;
l=0;
for i=1:n+1
eta(i)=etamin+(i-1)*deta;
for j=1:n+1
w(j)=wmin+(j-1)*dw;
x(i,j)=a(1)*(a(4)+sign(cos(eta(i)))*abs(cos(eta(i)))^epsilon(1))*sign(cos(w(j)))*abs(cos(w(j)))^epsilon(2);
y(i,j)=a(2)*(a(4)+sign(cos(eta(i)))*abs(cos(eta(i)))^epsilon(1))*sign(sin(w(j)))*abs(sin(w(j)))^epsilon(2);
z(i,j)=a(3)*sign(sin(eta(i)))*abs(sin(eta(i)))^epsilon(1);
endfor;
endfor;
mesh(x,y,z);
endfunction;