superellipse

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Squircle, the superellipse for n = 4, a = b = 1, approximates a chamfered square.


The superellipse (or Lamé curve) is the geometric figure defined in the cartesian coordinate system as the set of all points (x, y) with
where n > 0 and a and b are the radii of the oval shape. The case n = 2 yields an ordinary ellipse; increasing n beyond 2 yields the hyperellipses, which increasingly resemble rectangles; decreasing n below 2 yields hypoellipses which develop pointy corners in the x and y directions and increasingly resemble crosses. The case n = 1 yields a rhombus. For a = b it is also the unit circle in R2 when distance is defined by the n-norm.

Effects of n

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n = 32, a = b = 1 produces a rounder shape that somewhat resembles a chamfered square.
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n = 12, a = b = 1 produces a four-pointed star.
When n is a nonzero rational number pq (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 x4/3 + y4/3=1, then the curve is an algebraic curve of degree twelve and genus three, given by the implicit equation
or by the parametric equations


The area inside the ellipse can be expressed in terms of the gamma function, Γ(x), as

Generalizations

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Example of the generalized superellipse with m ≠ n.
The superellipse is further generalized as:

Superellipsoid

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Brass superegg by Piet Hein.
In three-dimensions, 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 north-south parameter n, an east-west parameter e, and length, width, and depth parameters ax, ay, az. Then an implicit equation for the surface is
Parametric equations in terms of surface parameters u and v (longitude and latitude) are
where the auxiliary functions are
and the signum function sgn(x) is
The volume inside this surface can be expressed in terms of beta functions, β(m,n) = Γ(m)Γ(n)/Γ(m+n), as

History

Though he is often credited with its invention, the Danish poet and scientist Piet Hein (1905–1996) did not discover the super-ellipse. The general Cartesian notation of the form comes from the French mathematician Gabriel Lamé (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 super-egg or superellipsoid by starting with the superellipse
and revolving it about the x-axis. Unlike a regular ellipsoid, the super-ellipsoid can stand upright on a flat surface.

City 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 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. —Piet Hein

See also

  • Astroid, the superellipse with n = 23 and a = b
  • Ellipse
  • Ellipsoid, a higher-dimensional 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 978-0-12-409672-1
      • id="CITEREFGardner1977">Gardner, Martin (1977), "Piet Hein’s Superellipse", Mathematical Carnival. A New Round-Up of Tantalizers and Puzzles from Scientific American, New York: Vintage Press, pp. 240–254, ISBN 978-0-394-72349-5
        • id="CITEREFGielis2003">Gielis, Johan (2003), Inventing the Circle: The Geometry of Nature, Antwerp: Geniaal Press, ISBN 978-90-807756-1-9
          • id="CITEREFSokolov2001">Sokolov, D. D. (2001), "Lamé curve", Springer Encyclopaedia of Mathematics, <[2]

            External links

            Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point.
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            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.
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            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.
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            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.
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            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.
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            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.
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            In algebraic geometry, the geometric genus is a basic birational invariant pg of algebraic varieties, defined for non-singular complex projective varieties (and more generally for complex manifolds) as the Hodge number hn,0
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            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


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            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.

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            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.
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            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").
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            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".
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            Gabriel Lamé (July 22, 1795 - May 1, 1870) was a French mathematician.

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