Superellipse (original) (raw)

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology

Alphabetical Index New in MathWorld

Superellipses

A superellipse is a curve with Cartesian equation

| $\|x/a|^r+|y/b|^r=1,$ | (1) | | ---------------------------------------------------------------------------------------------------------- | --- |

first discussed in 1818 by Lamé. A superellipse may be described parametrically by

The restriction to r>2 is sometimes made.

The generalization to a three-dimensional surface is known as a superellipsoid.

Superellipses with a=b are also known as Lamé curves or Lamé ovals, and the case a=b with r=4 is sometimes known as the squircle. By analogy, the superellipse with a!=b and r=4 might be termed the rectellipse.

A range of superellipses are shown above, with special cases r=2/3 , 1, and 2 illustrated right above. The following table summarizes a few special cases. Piet Hein used r=5/2 with a number of different a/b ratios for various of his projects. For example, he used a/b=6/5 for Sergels Torg (Sergel's Square) in Stockholm, Sweden (Vestergaard), and a/b=3/2 for his table.

If is a rational, then a superellipse is algebraic. However, for irrational , it is transcendental. For even integers r=n , the curve becomes closer to a rectangle as increases.

The area of a superellipse is given by

SuperellipseRegions

The above plots show generalization of the superellipse given by the function

| $\|x|^p+|y|^q<=1$ | (6) | | ------------------------------------------------------------------------------------------------------ | --- |

for p=1 , ..., 4 and q=1 , ..., 4.

SuperellipsePlants

Gielis (2003) has considered the further generalization of the superellipse given in polar coordinates by

| $r(theta)=[\|(cos(1/4mtheta))/a|^(n_2)+|(sin(1/4mtheta))/b|^(n_3)]^(-1/n_1).$ | (7) | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------ | --- |

Here, introduction of the parameter and use of polar coordinates gives rise to curves with -fold rotational symmetry. A number of curves for different parameters are illustrated above, together with the names of organisms that the curves resemble. While the above equation, dubbed the "superformula" by Gielis (2003), is clearly capable of describing a number of diverse biological shapes having a variety of symmetries, it seems unlikely that this formula has any particularly fundamental biological significance (Peterson 2002, Whitfield 2003) beyond as a possibly convenient parametrization. In fact, while the number of free parameters in the "superformula" is six, Gielis (2003) also applies it as a prefactor by which to multiply other polar curves (e.g., the logarithmic spiral, rose curve curve, etc.), so the number of parameters in the equation is effectively larger. Of course, any formula with a large number of free parameters is capable of describing a very large parameter space. (To emphasize this fact, it is sometimes humorously said that, given eight or so free parameters, it is possible to describe an elephant.)

SuperellipsePlants1

SuperellipsePlants2

Families of curves generated by the "superformula" with a=b=1 and varying from 0 to 2 are illustrated above for values of n=n_1=n_2=n_3 varying from 1 to 8.

Explore with Wolfram|Alpha

References

Gardner, M. "Piet Hein's Superellipse." Ch. 18 in Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage, pp. 240-254, 1977.Gielis, J. "A Generic Geometric Transformation that Unifies a Wide Range of Natural and Abstract Shapes."Amer. J. Botany 90, 333-338, 2003.Gridgeman, N. T. "Lamé Ovals." Math. Gaz. 54, 31-37, 1970.Lin, S.; Zhang, L.; Reddy, G. V.; Hui, C.; Gielis, J.; Ding, Y.; and Shi, P. "A Geometrical Model for Testing Bilateral Symmetry of Bamboo Leaf with a Simplified Gielis Equation." Ecology and Evolution 6, 6798-6806, 2016.Loria, G. Spezielle algebraische und transcendente ebene Kurven. Leipzig: Teubner, 1910.MacTutor History of Mathematics Archive. "Lamé Curves."http://www-groups.dcs.st-and.ac.uk/~history/Curves/Lame.html.Peterson, I. "MathTrek: A Geometric Superformula." May 3, 2002. http://www.sciencenews.org/20030503/mathtrek.asp.Shi, P.-J.; Huang, J.-G.; Hui, C.; Grissino-Mayer, H. D.; Tardif, J. C., Zhai, L.-H.; Wang, F.-S.; and Li, B.-L. "Capturing Spiral Radial Growth of Conifers Using the Superellipse to Model Tree-Ring Geometric Shape." Frontiers in Plant Science 6, Art. 856, 1-13, Oct. 15, 2015.Vestergaard, E. "Piet Heins Superellipse." http://www.matematiksider.dk/piethein.html.Whitfield, J. "Maths Gets into Shape." Nature Science Update. March 31, 2003.http://www.nature.com/nsu/030331/030331-3.html.

Cite this as:

Weisstein, Eric W. "Superellipse." FromMathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Superellipse.html

Superellipse (original) (raw)

See also

Explore with Wolfram|Alpha

References

Cite this as:

Subject classifications