In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, a supertoroid or supertorus is usually understood to be a family of
doughnut
A doughnut or donut () is a type of food made from leavened fried dough. It is popular in many countries and is prepared in various forms as a sweet snack that can be homemade or purchased in bakeries, supermarkets, food stalls, and franc ...
-like
surfaces (technically, a
topological torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
) whose shape is defined by mathematical formulas similar to those that define the
superquadric
In mathematics, the superquadrics or super-quadrics (also superquadratics) are a family of geometry, geometric shapes defined by formulas that resemble those of ellipsoids and other quadrics, except that the square (algebra), squaring operations a ...
s. The plural of "supertorus" is either supertori or supertoruses.
The family was described and named by
Alan Barr in 1994.
[Alan H. Barr (1981) ''Superquadrics and Angle-Preserving Transformations''. IEEE Computer Graphics and Applications, volume 1 issue 1. pp. 11-23.]
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
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...
s. However, they are not
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 (as a complex manifold, when it is non-singular) and so of di ...
s (except in special cases).
Formulas
Alan Barr's supertoroids are defined by parametric equations similar to the
trigonometric equations of the torus, except that the
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
and
cosine terms are raised to arbitrary
powers
Powers may refer to:
Arts and media
* ''Powers'' (comics), a comic book series by Brian Michael Bendis and Michael Avon Oeming
** ''Powers'' (American TV series), a 2015–2016 series based on the comics
* ''Powers'' (British TV series), a 200 ...
. 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 2''π''
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s).
In these formulas, the parameter ''s'' > 0 controls the "squareness" of the vertical sections, ''t'' > 0 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
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
about the ''Z'' axis.
In general, the supertorus defined as above spans the
intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets
* A statistical level of measurement
* Interval est ...