NationMaster - Encyclopedia: Superellipse

The superellipse (or Lamé curve) is the geometric figure defined in the Cartesian coordinate system as the set of all points (x, y) with Image File history File links Superellipse_chamfered_square. ... Image File history File links Superellipse_chamfered_square. ... A squircle A squircle is a mathematical shape with properties between those of a square and those of a circle. ... Example of a chamfer A Chamfer is a beveled edge connecting two surfaces. ... For other uses, see Square. ... Fig. ...

$left|frac{x}{a}right|^n! + left|frac{y}{b}right|^n! = 1$

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. For other uses, see Ellipse (disambiguation). ... In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles. ... Two rhombi. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

1 Effects of n
2 Generalizations
- 2.1 Superellipsoid
3 History
4 See also
5 References
6 External links

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. Image File history File links Superellipse_rounded_diamond. ... Image File history File links Superellipse_star. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... 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 Xn + Yn = Zn. ... 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. ...

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 equation In mathematics, the geometric genus in algebraic geometry 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 (equal to h0,n by Serre duality). ...

$(x^4+y^4)^3-3(x^4-3x^2y^2+y^4)(x^4+3x^2y^2+y^4)+3(x^4+y^4)-1=0 , ,!$

or by the parametric equations

$left. begin{align} xleft(thetaright) &= plusmn acos^{frac{2}{n}} theta yleft(thetaright) &= plusmn bsin^{frac{2}{n}} theta end{align} right} qquad 0 le theta < frac{pi}{2} .$

The area inside the ellipse can be expressed in terms of the gamma function, Γ(x), as The Gamma function along part of the real axis In mathematics, the Gamma function (represented by the capitalized Greek letter Î“) is an extension of the factorial function to real and complex numbers. ...

$mathrm{Area} = 4 a b frac{left(Gamma left(1+tfrac{1}{n}right)right)^2}{Gamma left(1+tfrac{2}{n}right)} .$

Generalizations

Example of the generalized superellipse with m ≠ n.

The superellipse is further generalized as: Image File history File links Superellipse_generalization. ... Image File history File links Superellipse_generalization. ...

$left|frac{x}{a}right|^m + left|frac{y}{b}right|^n = 1; qquad m, n > 0.$

Superellipsoid

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 Barr (1992), it is convenient to distinguish a north-south parameter n, an east-west parameter e, and length, width, and depth parameters a_x, a_y, a_z. Then an implicit equation for the surface is Image File history File links Superaeg. ... Image File history File links Superaeg. ... Piet Hein (December 16, 1905 - April 18, 1996) was a scientist, mathematician, inventor, author, and poet, often writing under the Old Norse pseudonym Kumbel meaning tombstone. His short poems, gruks (or grooks), first started to appear in the daily newspaper Politiken shortly after the Nazi Occupation in April 1940 under... The parabola y=x2 rotated about the z-axis A 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. ... 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. ...

$left( |x/a_x|^{2/e} + |y/a_y|^{2/e} right)^{e/n} + |z/a_z|^{2/n} = 1 . ,!$

Parametric equations in terms of surface parameters u and v (longitude and latitude) are

$begin{align} x(u,v) &{}= a_x c(v,n) c(u,e) y(u,v) &{}= a_y c(v,n) s(u,e) z(u,v) &{}= a_z s(v,n) & -pi/2 le v le pi/2, quad -pi le u < pi , end{align}$

where the auxiliary functions are

$begin{align} c(omega,m) &{}= sgn(cos omega) |cos omega|^m s(omega,m) &{}= sgn(sin omega) |sin omega|^m end{align}$

and the signum function sgn(x) is Signum function In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. ...

$sgn(x) = begin{cases} -1, & x < 0 0, & x = 0 +1, & x > 0 . end{cases}$

The volume inside this surface can be expressed in terms of beta functions, β(m,n) = Γ(m)Γ(n)/Γ(m+n), as In theoretical physics, specifically quantum field theory, a beta-function Î²(g) encodes the dependence of a coupling parameter, g, on the energy scale, of a given physical process. ...

$V = frac23 a_x a_y a_z e n beta left( frac{e}2,frac{e}2 right) beta left(n,frac{n}2 right) .$

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 Piet Hein (December 16, 1905 - April 18, 1996) was a scientist, mathematician, inventor, author, and poet, often writing under the Old Norse pseudonym Kumbel meaning tombstone. His short poems, gruks (or grooks), first started to appear in the daily newspaper Politiken shortly after the Nazi Occupation in April 1940 under... Gabriel LamÃ© (July 22, 1795, Tours, France - May 1, 1870, Paris, France) was a French mathematician. ...

$left|frac{x}{4}right|^{2.5} + left|frac{y}{3}right|^{2.5} = 1$

and revolving it about the x-axis. Unlike a regular ellipsoid, the super-ellipsoid can stand upright on a flat surface. 3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ...

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, in Mexico City. For other uses, see Stockholm (disambiguation). ... A roundabout is a type of road junction at which traffic enters a one-way stream around a central island. ... Sergels Torg with Kulturhuset (to the left) at night Sergels torg (translated Sergels Square) is a well-known public square in the centre of Stockholm, Sweden. ... Year 1968 (MCMLXVIII) was a leap year starting on Monday (link will display full calendar) of the Gregorian calendar. ... This article is about the capital of France. ... Combatants Republic of Vietnam United States Republic of Korea Thailand Australia New Zealand The Philippines National Front for the Liberation of South Vietnam Democratic Republic of Vietnam Peopleâ€™s Republic of China Democratic Peoples Republic of Korea Strength US 1,000,000 South Korea 300,000 Australia 48,000... The New York Times is an internationally known daily newspaper published in New York City and distributed in the United States and many other nations worldwide. ... Year 1968 (MCMLXVIII) was a leap year starting on Monday (link will display full calendar) of the Gregorian calendar. ... The Estadio Azteca is the home stadium of the Mexico national team and the Mexican club America, and the one-time home of Atlante, Cruz Azul, and Necaxa. ... Nickname: Motto: Capital en movimiento Location of Mexico City in south central Mexico Coordinates: , Country Federal entity Boroughs The 16 delegaciones Founded c. ...

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). Hermann Zapf (born in Nuremberg, Germany on November 8, 1918) is a prolific German typeface designer. ... â€œFontâ€ redirects here. ... Donald Ervin Knuth ( or Ka-NOOTH[1], Chinese: [2]) (b. ... Sample text in Computer Modern Computer 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. ... In the mathematical subfield of numerical analysis a spline is a special curve defined piecewise by polynomials. ...

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

References

Barr, Alan H. (1983), Geometric Modeling and Fluid Dynamic Analysis of Swimming Spermatozoa, Rensselaer Polytechnic Institute (Ph.D. dissertation using superellipsoids)
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
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
Gielis, Johan (2003), Inventing the Circle: The Geometry of Nature, Antwerp: Geniaal Press, ISBN 978-90-807756-1-9
Sokolov, D. D. (2001), "Lamé curve", Springer Encyclopaedia of Mathematics, <http://eom.springer.de/L/l057390.htm>

Rensselaer Polytechnic Institute, or RPI, is a nonsectarian, coeducational private research university in Troy, New York, a city lying just outside the state capital of Albany. ... Academic Press (London, New York and San Diego) was an academic book publisher that is now part of Elsevier. ... Martin Gardner (b. ... The Vintage Press is an American publisher. ... The Encyclopaedia of Mathematics is a large reference work in mathematics. ...

External links

Superellipse (MathWorld)
Lamé's Super Ellipse (Java-Applet)
Super Ellipsoid (Java-Applet)
Johan Gielis' and Bert Beirinckx' "Superformula".

Categories: Articles with sections needing expansion | Curves | Algebraic curves

Results from FactBites:

superellipse (187 words)
	Superellipses have a form partway between an ellipse and a rounded rectangle (or, if a = b, partway between a circle and a rounded square).
	The Danish poet and architect Piet Hein decided that the superellipse with n = 5/2 and a/b = 6/5 is the most pleasing the eye.
	The surface of revolution of a superellipse is a superellipsoid, one special form of which has been nicknamed the superegg.

SuperEllipse and SuperEllipsoid (537 words)
	This is the form most easily employed to draw a superellipse, the angle ø is varied from 0 to 2pi in suitably small steps, a line is drawn to each subsequent value of ø.
	Note that the superellipse always passes through the same points at ø = 0, pi/2, pi, 3pi/2 and so it is possible to have different values of n in each quadrant.
	The superellipse is name given to a family of shapes formed from the spherical product of two superquadratric curves.

More results at FactBites »

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