A superellipse, also known as a Lamé curve after
Gabriel Lamé
Gabriel Lamé (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equations by the use of curvilinear coordinates, and the mathematical theory of elasticity (for which linear elasticity ...
, is a closed curve resembling the
ellipse, retaining the geometric features of
semi-major axis and
semi-minor axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lo ...
, and symmetry about them, but a different overall shape.
In the
Cartesian coordinate system, the set of all points
on the curve satisfy the equation
:
where
and
are positive numbers, and the vertical bars around a number indicate the
absolute value of the number.
Specific cases
This formula defines a
closed curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
contained in the
rectangle −''a'' ≤ ''x'' ≤ +''a'' and −''b'' ≤ ''y'' ≤ +''b''. The parameters ''a'' and ''b'' are called the ''semi-diameters'' of the curve.
The overall shape of the curve is determined by the value of the exponent ''n'', as shown in the following table:
If ''n'' < 2, the figure is also called a hypoellipse; if ''n'' > 2, a hyperellipse.
When ''n'' ≥ 1 and ''a'' = ''b'', the superellipse is the boundary of a
ball of R
2 in the
''n''-norm.
The extreme points of the superellipse are (±''a'', 0) and (0, ±''b''), and its four "corners" are (±''sa, ±sb''), where
(sometimes called the "superness").
Mathematical properties
When ''n'' is a positive
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
''p''/''q'' (in lowest terms), then each quadrant of the superellipse is a
plane algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
of order ''pq''. In particular, when ''a'' = ''b'' = 1 and ''n'' is an even integer, then it is a
Fermat curve
In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (''X'':''Y'':''Z'') by the Fermat equation
:X^n + Y^n = Z^n.\
Therefore, in terms of the affine plane its equation is
:x^ ...
of degree ''n''. In that case it is non-singular, but in general it will be
singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, ...
. If the numerator is not even, then the curve is pieced together from portions of the same algebraic curve in different orientations.
The curve is given by the
parametric equations (with parameter
having no elementary geometric interpretation)
:
where each ± can be chosen separately so that each value of
gives four points on the curve. Equivalently, letting
range over
:
where the
sign function is
:
Here
is not the angle between the positive horizontal axis and the ray from the origin to the point, since the tangent of this angle equals ''y/x'' while in the parametric expressions
The area inside the superellipse can be expressed in terms of the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
as
:
or in terms of the
beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^( ...
as
:
The
pedal curve
A pedal (from the Latin '' pes'' ''pedis'', "foot") is a lever designed to be operated by foot and may refer to:
Computers and other equipment
* Footmouse, a foot-operated computer mouse
* In medical transcription, a pedal is used to control ...
is relatively straightforward to compute. Specifically, the pedal of
:
is given in
polar coordinate
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the ...
s by
:
Generalizations
The superellipse is further generalized as:
:
or
:
Note that
is a parameter which is not linked to the physical angle through elementary functions.
History
The general Cartesian notation of the form comes from the French mathematician
Gabriel Lamé
Gabriel Lamé (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equations by the use of curvilinear coordinates, and the mathematical theory of elasticity (for which linear elasticity ...
(1795–1870), who generalized the equation for the ellipse.
Hermann Zapf's
typeface
A typeface (or font family) is the design of lettering that can include variations in size, weight (e.g. bold), slope (e.g. italic), width (e.g. condensed), and so on. Each of these variations of the typeface is a font.
There are thousands o ...
Melior, published in 1952, uses superellipses for letters such as ''o''. Thirty years later
Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer sc ...
would build the ability to choose between true ellipses and superellipses (both approximated by
cubic spline
In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is, by its values and first derivatives at the end points of the correspondin ...
s) into his
Computer Modern
Computer Modern is the original family of typefaces used by the typesetting program TeX. It was created by Donald Knuth with his Metafont program, and was most recently updated in 1992. Computer Modern, or variants of it, remains very widely u ...
type family.
The superellipse was named by the
Danish
Danish may refer to:
* Something of, from, or related to the country of Denmark
People
* A national or citizen of Denmark, also called a "Dane," see Demographics of Denmark
* Culture of Denmark
* Danish people or Danes, people with a Danish a ...
poet and scientist
Piet Hein (1905–1996) though he did not discover it as it is sometimes claimed. In 1959, city planners in
Stockholm,
Sweden announced a design challenge for a
roundabout in their city square
Sergels Torg
Sergels torg ("Sergel's Square") is a major public square in Stockholm, Sweden, constructed in the 1960s and named after 18th-century sculptor Johan Tobias Sergel, whose workshop was once located north of the square.
Overview
Sergels torg ha ...
. Piet Hein's winning proposal was based on a superellipse with ''n'' = 2.5 and ''a''/''b'' = 6/5.
As he explained it:
:''Man is the animal that draws lines which he himself then stumbles over. In the whole pattern of civilization there have been two tendencies, one toward straight lines and rectangular patterns and one toward circular lines. There are reasons, mechanical and psychological, for both tendencies. Things made with straight lines fit well together and save space. And we can move easily — physically or mentally — around things made with round lines. But we are in a straitjacket, having to accept one or the other, when often some intermediate form would be better. To draw something freehand — such as the patchwork traffic circle they tried in Stockholm — will not do. It isn't fixed, isn't definite like a circle or square. You don't know what it is. It isn't esthetically satisfying. The super-ellipse solved the problem. It is neither round nor rectangular, but in between. Yet it is fixed, it is definite — it has a unity.''
Sergels Torg was completed in 1967. Meanwhile, Piet Hein went on to use the superellipse in other artifacts, such as beds, dishes, tables, etc.
[''The Superellipse'']
in ''The Guide to Life, The Universe and Everything'' by BBC #REDIRECT BBC #REDIRECT BBC
Here i going to introduce about the best teacher of my life b BALAJI sir. He is the precious gift that I got befor 2yrs . How has helped and thought all the concept and made my success in the 10th board exam. ...
...
(27 June 2003) By rotating a superellipse around the longest axis, he created the
, a solid egg-like shape that could stand upright on a flat surface, and was marketed as a
.