superellipse
The superellipse (or LamÃ© curve) is the geometric figure defined in the cartesian coordinate system as the set of all points (x, y) with
Effects of n
When n is a nonzero rational number ^{p}⁄_{q} (in lowest terms), then the superellipse is a plane algebraic curve. For positive n the order is pq; for negative n the order is 2pq. In particular, when a and b are both one and n is an even integer, then it is a Fermat curve of degree n. In that case it is nonsingular, but in general it will be singular. 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 x^{4/3} + y^{4/3}=1, then the curve is an algebraic curve of degree twelve and genus three, given by the implicit equationThe area inside the ellipse can be expressed in terms of the gamma function, Γ(x), as
Generalizations
The superellipse is further generalized as:Superellipsoid
In threedimensions, a superellipsoid or superegg can be made by revolving a superellipse into a surface of revolution and scaling. Following , it is convenient to distinguish a northsouth parameter n, an eastwest parameter e, and length, width, and depth parameters a_{x}, a_{y}, a_{z}. Then an implicit equation for the surface isHistory
Though he is often credited with its invention, the Danish poet and scientist Piet Hein (1905–1996) did not discover the superellipse. The general Cartesian notation of the form comes from the French mathematician Gabriel LamÃ© (1795–1870) who generalized the equation for the ellipse. However, Piet Hein did popularize the use of the superellipse in architecture, urban planning, and furniture making, and he did invent the superegg or superellipsoid by starting with the superellipseCity planners in Stockholm, Sweden needed a solution for a roundabout in their city square Sergels Torg. Piet Hein's superellipse provided the needed aesthetic and practical solution. In 1968, when negotiators in Paris for the Vietnam War could not agree on the shape of the negotiating table, Balinski and Holt suggested a superelliptical table in a letter to the New York Times (Gardner 1977:251). The superellipse was used for the shape of the 1968 Azteca Olympic Stadium [1], in Mexico City.
Hermann Zapf's typeface 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 Knuth built into his Computer Modern type family the ability to choose between true ellipses and superellipses (both approximated by cubic splines).
 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 superellipse solved the problem. It is neither round nor rectangular, but in between. Yet it is fixed, it is definite — it has a unity. —Piet Hein
See also
 Astroid, the superellipse with n = ^{2}⁄_{3} and a = b
 Ellipse
 Ellipsoid, a higherdimensional analogue of an ellipse
 Spheroid, the ellipsoids obtained by rotating an ellipse about its major or minor axis
 Squircle, a special case of the superellipse found by setting
 Superformula, a generalization of the superellipse
 Superquadrics
References

id="CITEREFBarr1983">Barr, Alan H. (1983), Geometric Modeling and Fluid Dynamic Analysis of Swimming Spermatozoa, Rensselaer Polytechnic Institute (Ph.D. dissertation using superellipsoids)

id="CITEREFBarr1992">Barr, Alan H. (1992), "Rigid Physically Based Superquadrics", in Kirk, David, Graphics Gems III, Academic Press, pp. 137–159 (code: 472–477), ISBN 9780124096721

id="CITEREFGardner1977">Gardner, Martin (1977), "Piet Hein’s Superellipse", Mathematical Carnival. A New RoundUp of Tantalizers and Puzzles from Scientific American, New York: Vintage Press, pp. 240–254, ISBN 9780394723495

id="CITEREFGielis2003">Gielis, Johan (2003), Inventing the Circle: The Geometry of Nature, Antwerp: Geniaal Press, ISBN 9789080775619

id="CITEREFSokolov2001">Sokolov, D. D. (2001), "LamÃ© curve", Springer Encyclopaedia of Mathematics, <[2]
External links
 Superellipse (MathWorld)
 LamÃ©'s Super Ellipse (JavaApplet)
 Super Ellipsoid (JavaApplet)
 Johan Gielis' and Bert Beirinckx' "Superformula".
Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the xcoordinate and the ycoordinate of the point.
..... Click the link for more information.ellipse (from the Greek ἔλλειψις, literally absence) is the locus of points on a plane where the sum of the distances from any point on the curve to two fixed points is constant.
..... Click the link for more information.rectangle is defined as a quadrilateral where all four of its angles are right angles.
From this definition, it follows that a rectangle has two pairs of parallel sides; that is, a rectangle is a parallelogram.
..... Click the link for more information.rhombus (or homb; plural rhombi) is a quadrilateral in which all of the sides are of equal length, i.e., it is awith two pairs of equal adjacent sides. The opposite sides of a kite are not parallel unless the kite is also a rhombus.
..... Click the link for more information.In 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.
..... Click the link for more information.
..... Click the link for more information.In mathematics, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that V is not locally flat there.
..... Click the link for more information.In algebraic geometry, the geometric genus is a basic birational invariant p_{g} of algebraic varieties, defined for nonsingular complex projective varieties (and more generally for complex manifolds) as the Hodge number h^{n,0}
..... Click the link for more information.Gamma function (represented by the capitalized Greek letter Γ) is an extension of the factorial function to real and complex numbers. For a complex number z with positive real part it is defined by
..... Click the link for more information.surface of revolution is a surface created by rotating a curve lying on some plane (the generatrix) around a straight line (the axis of rotation) that lies on the same plane.
Examples of surfaces generated by a straight line are the cylindrical and conical surfaces.
..... Click the link for more information.In mathematics, an implicit function is a generalization for the concept of a function in which the dependent variable may not be given explicitly in terms of the independent variable.
..... Click the link for more information.sign function is a logical function which extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function (after the Latin form of "sign").
..... Click the link for more information.beta function, also called the Euler integral of the first kind, is a special function defined by
for Re(x), Re(y) > 0.
The beta function was studied by Euler and Legendre and was given its name by Jacques Binet.
..... Click the link for more information.Piet Hein (December 16, 1905  April 17, 1996) was a Danish scientist, mathematician, inventor, author, and poet, often writing under the Old Norse pseudonym "Kumbel" meaning "tombstone".
..... Click the link for more information.Gabriel LamÃ© (July 22, 1795  May 1, 1870) was a French mathematician.Biography
LamÃ© was born in Tours, in today's dÃ©partement of IndreetLoire.
..... Click the link for more information.ellipsoid is a type of quadric surface that is a higher dimensional analogue of an ellipse. The equation of a standard ellipsoid body in an xyz Cartesian coordinate system is
..... Click the link for more information.City of Stockholm
Stockholms stad
Coat of arms
Location of Stockholm in northern Europe
Coordinates:
Country Sweden
Municipality
..... Click the link for more information.Motto
(Royal) "FÃ¶r Sverige  I tiden" ^{1}
"For Sweden – With the Times" Â²
Anthem
Du gamla, Du fria
..... Click the link for more information.roundabout is a type of road junction at which traffic enters a oneway stream around a central island. In the United States it is technically called a modern roundabout, to emphasize the distinction from the older, larger type of traffic circle.
..... Click the link for more information.Sergels torg (translated "Sergel's Square") is a wellknown public square in the centre of Stockholm, Sweden. The square is named after 18th century sculptor Johan Tobias Sergel, whose workshop was located in the area north of the square.
..... Click the link for more information.19th century  20th century  21st century
1930s 1940s 1950s  1960s  1970s 1980s 1990s
1965 1966 1967  1968  1969 1970 1971
Year 1968 (MCMLXVIII
..... Click the link for more information.Ville de Paris
City flag City coat of arms
Motto: Fluctuat nec mergitur
(Latin: "Tossed by the waves, she does not sink")
The Eiffel Tower in Paris, as seen from the esplanade du TrocadÃ©ro.
..... Click the link for more information.Total dead: ~314,000
Total wounded: ~1,490,000 North Vietnam and NLF
dead and missing: ~1,100,000 [1] [2] [3] [4]
wounded: ~600,000+ [5]
People's Republic of China
dead: 1,446
wounded: 4,200
..... Click the link for more information.
The May 8, 2007 front page of
The New York Times
Type Daily newspaper
Format Broadsheet
Owner The New York Times Company
Publisher Arthur Ochs Sulzberger, Jr.
Staff Writers 350
Founded 1851
Price USD 1.
..... Click the link for more information.19th century  20th century  21st century
1930s 1940s 1950s  1960s  1970s 1980s 1990s
1965 1966 1967  1968  1969 1970 1971
Year 1968 (MCMLXVIII
..... Click the link for more information.Estadio Azteca
Location Mexico DF, Mexico
Broke ground 1961
Opened May 29, 1966
Renovated 1985
Owner Televisa
Operator Club America
Surface Grass
Construction cost MXN260 Million
Architect Pedro RamÃrez VÃ¡zquez
..... Click the link for more information.Mexico City
Ciudad de MÃ©xico
Skyline of Mexico City at night
Seal
Nickname: Ciudad de los palacios (City of Palaces)
Motto: Capital en movimiento
..... Click the link for more information.Hermann Zapf (born November 8, 1918) is a German typeface designer who lives in Darmstadt, Germany. He is married to calligrapher and typeface designer Gudrun Zapf von Hesse.
..... Click the link for more information.This article or section is in need of attention from an expert on the subject.
Please help recruit one or [ improve this article] yourself. See the talk page for details.
..... Click the link for more information.Donald Ervin Knuth
Photographed by Jacob Appelbaum, 25 October 2005
Born January 10 1938
..... Click the link for more information.

id="CITEREFSokolov2001">Sokolov, D. D. (2001), "LamÃ© curve", Springer Encyclopaedia of Mathematics, <[2]

id="CITEREFGielis2003">Gielis, Johan (2003), Inventing the Circle: The Geometry of Nature, Antwerp: Geniaal Press, ISBN 9789080775619

id="CITEREFGardner1977">Gardner, Martin (1977), "Piet Hein’s Superellipse", Mathematical Carnival. A New RoundUp of Tantalizers and Puzzles from Scientific American, New York: Vintage Press, pp. 240–254, ISBN 9780394723495

id="CITEREFBarr1992">Barr, Alan H. (1992), "Rigid Physically Based Superquadrics", in Kirk, David, Graphics Gems III, Academic Press, pp. 137–159 (code: 472–477), ISBN 9780124096721
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