A
superellipse is a geometric figure defined in the
Cartesian coordinate systemA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
as the set of all points (
x,
y) with
-
where
n,
a and
b are positive numbers.
This formula defines a closed curve contained in the
rectangleIn Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...
−
a ≤
x ≤ +
a and −
b ≤
y ≤ +
b. The parameters
a and
b are called the
semi-diameters of the curve.
When
n is between 0 and 1, the superellipse looks like a four-armed star with concave (inwards-curved) sides. For
n = 1/2, in particular, the sides are arcs of
parabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
s.
When
n is 1 the curve is a
diamondIn mineralogy, diamond is an allotrope of carbon, where the carbon atoms are arranged in a variation of the face-centered cubic crystal structure called a diamond lattice. Diamond is less stable than graphite, but the conversion rate from diamond to graphite is negligible at ambient conditions...
with corners (±
a, 0) and (0, ±
b). When
n is between 1 and 2, it looks like a diamond with those same corners but with
convexIn Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
(outwards-curved) sides. The
curvatureIn mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
increases without
limitThe limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
as one approaches the corners.
When
n is 2, the curve is an ordinary
ellipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
(in particular, a
circleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
if
a =
b). When
n is greater than 2, it looks superficially like a
rectangleIn Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...
with
chamferA chamfer is a beveled edge connecting two surfaces. If the surfaces are at right angles, the chamfer will typically be symmetrical at 45 degrees. A fillet is the rounding off of an interior corner. A rounding of an exterior corner is called a "round" or a "radius"."Chamfer" is a term commonly...
ed (rounded) corners. The curvature is zero at the points (±
a, 0) and (0, ±
b).
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
ballIn mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
of
R2 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" ).
Algebraic properties
When
n is a nonzero rational number
p /
q (in lowest terms), then the superellipse is a
plane algebraic curveIn algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
. For positive
n the order is
pq; for negative
n the order is 2
pq. In particular, when
a and
b are both one and
n is an even integer, then it is a
Fermat curveIn mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates by the Fermat equationX^n + Y^n = Z^n.\ Therefore in terms of the affine plane its equation is...
of degree
n. In that case it is nonsingular, but in general it will be
singularIn mathematics, a singular point of an algebraic variety V is a point P that is 'special' , in the geometric sense that V is not locally flat there. In the case of an algebraic curve, a plane curve that has a double point, such as the cubic curveexhibits at , cannot simply be parametrized near the...
. If the numerator is not even, then the curve is pasted together from portions of the same algebraic curve in different orientations.
For example, if
x4/3 +
y4/3 = 1, then the curve is an algebraic curve of degree twelve and
genusIn algebraic geometry, the geometric genus is a basic birational invariant pg of algebraic varieties and complex manifolds.-Definition:...
three, given by the implicit equation
or by the parametric equations
or
The area inside the superellipse can be expressed in terms of the
gamma functionIn mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
, Γ(
x), as
Generalizations
The superellipse is further generalized as:
or
(Note that :
is NOT a physical angle of the figure, but just a parameter)
History
The general Cartesian notation of the form comes from the French mathematician
Gabriel LaméGabriel Léon Jean Baptiste Lamé was a French mathematician.-Biography:Lamé was born in Tours, in today's département of Indre-et-Loire....
(1795–1870) who generalized the equation for the ellipse.
Though he is often credited with its invention, the
DanishDenmark is a Scandinavian country in Northern Europe. The countries of Denmark and Greenland, as well as the Faroe Islands, constitute the Kingdom of Denmark . It is the southernmost of the Nordic countries, southwest of Sweden and south of Norway, and bordered to the south by Germany. Denmark...
poet and scientist
Piet HeinPiet Hein was a Danish scientist, mathematician, inventor, designer, author, and poet, often writing under the Old Norse pseudonym "Kumbel" meaning "tombstone"...
(1905–1996) did not discover the super-ellipse. In 1959, city planners in
StockholmStockholm is the capital and the largest city of Sweden and constitutes the most populated urban area in Scandinavia. Stockholm is the most populous city in Sweden, with a population of 851,155 in the municipality , 1.37 million in the urban area , and around 2.1 million in the metropolitan area...
,
SwedenSweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe. Sweden borders with Norway and Finland and is connected to Denmark by a bridge-tunnel across the Öresund....
announced a design challenge for a
roundaboutA roundabout is the name for a road junction in which traffic moves in one direction around a central island. The word dates from the early 20th century. Roundabouts are common in many countries around the world...
in their city square
Sergels TorgSergels torg is the most central public square in Stockholm, Sweden, named after 18th century sculptor Johan Tobias Sergel, whose workshop was once located north of the square.- Overview :...
. 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. By rotating a superellipse around the longest axis, he created the
supereggIn 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....
, a solid egg-like shape that could stand upright on a flat surface, and was marketed as a novelty toy.
In 1968, when negotiators in
ParisParis is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...
for the
Vietnam WarThe Vietnam War was a Cold War-era military conflict that occurred in Vietnam, Laos, and Cambodia from 1 November 1955 to the fall of Saigon on 30 April 1975. This war followed the First Indochina War and was fought between North Vietnam, supported by its communist allies, and the government of...
could not agree on the shape of the negotiating table, Balinski, Kieron Underwood and Holt suggested a superelliptical table in a letter to the New York Times. The superellipse was used for the shape of the 1968
Azteca Olympic StadiumEstadio Azteca is a stadium in Santa Ursula, Mexico City, Mexico. It is the official home stadium of the Mexico national football team and the Mexican team Club América.The stadium was the venue for football soccer in the 1968 Summer Olympics....
, in
Mexico CityMexico City is the Federal District , capital of Mexico and seat of the federal powers of the Mexican Union. It is a federal entity within Mexico which is not part of any one of the 31 Mexican states but belongs to the federation as a whole...
.
Waldo R. ToblerWaldo Tobler is an American-Swiss geographer and cartographer. Tobler's idea that "Everything is related to everything else, but near things are more related to each other" is referred to as the "first law of geography." Tobler is a Professor Emeritus at the University of California, Santa Barbara...
developed a
map projectionA map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...
, the
Tobler hyperelliptical projectionThe Tobler hyperelliptical projection is a family of equal-area pseudocylindrical projections used for mapping the earth. It is named for Waldo R...
, published in 1973, in which the
meridiansA meridian is an imaginary line on the Earth's surface from the North Pole to the South Pole that connects all locations along it with a given longitude. The position of a point along the meridian is given by its latitude. Each meridian is perpendicular to all circles of latitude...
are arcs of superellipses.
Hermann ZapfHermann Zapf is a German typeface designer who lives in Darmstadt, Germany. He is married to calligrapher and typeface designer Gudrun Zapf von Hesse....
's
typefaceIn typography, a typeface is the artistic representation or interpretation of characters; it is the way the type looks. Each type is designed and there are thousands of different typefaces in existence, with new ones being developed constantly....
Melior, published in 1952, uses superellipses for letters such as
o. Many web sites say Zapf actually drew the shapes of Melior by hand without knowing the mathematical concept of the superellipse, and only later did Piet Hein point out to Zapf that his curves were extremely similar to the mathematical construct, but these web sites do not cite any primary source of this account. Thirty years later
Donald KnuthDonald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...
built into his
Computer ModernComputer Modern is the family of typefaces used by default by the typesetting program TeX. It was created by Donald Knuth with his METAFONT program, and was most recently updated in 1992. However, the family font was superseded by CM-Super , the latest release dating 2008...
type family the ability to choose between true ellipses and superellipses (both approximated by cubic splines).
Three connected superellipses are used in the logo of the
Pittsburgh SteelersThe Pittsburgh Steelers are a professional football team based in Pittsburgh, Pennsylvania. The team currently belongs to the North Division of the American Football Conference in the National Football League . Founded in , the Steelers are the oldest franchise in the AFC...
.
See also
- Astroid, the superellipse with n = and a = b is a hypocycloid with four cusps.
- Deltoid curve
In geometry, a deltoid, also known as a tricuspoid or Steiner curve, is a hypocycloid of three cusps. In other words, it is the roulette created by a point on the circumference of a circle as it rolls without slipping along the inside of a circle with three times its radius...
, the three cusps hypocycloid.
- Squircle
A squircle is a mathematical shape with properties between those of a square and those of a circle. It is a special case of superellipse. The word "squircle" is a portmanteau of the words, "square" and "circle".-Equation:...
, the superellipse with n = 4 and a = b looks like a "The Four-Cornered Wheel."
- Reuleaux triangle
A Reuleaux triangle is, apart from the trivial case of the circle, the simplest and best known Reuleaux polygon, a curve of constant width. The separation of two parallel lines tangent to the curve is independent of their orientation...
, "The Three-Cornered Wheel."
- Superformula
The superformula is a generalization of the superellipse and was first proposed by Johan Gielis.Gielis suggested that the formula can be used to describe many complex shapes and curves that are found in nature...
, a generalization of the superellipse
- Superquadrics
In mathematics, the superquadrics or super-quadrics are a family of geometric shapes defined by formulas that resemble those of elipsoids and other quadrics, except that the squaring operations are replaced by arbitrary powers...
, the three-dimensional "relatives" of superellipses
External links