A superellipse is a curve with Cartesian equation
(1)
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first discussed in 1818 by Lamé. A superellipse may be described parametrically by
(2)
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(3)
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The restriction to is sometimes made.
The generalization to a three-dimensional surface is known as a superellipsoid.
Superellipses with are also known as Lamé curves or Lamé ovals, and the case with is sometimes known as the squircle. By analogy, the superellipse with and might be termed the rectellipse.
A range of superellipses are shown above, with special cases , 1, and 2 illustrated right above. The following table summarizes a few special cases. Piet Hein used with a number of different ratios for various of his projects. For example, he used for Sergels Torg (Sergel's Square) in Stockholm, Sweden (Vestergaard), and for his table.
curve | |
(squashed) astroid | |
1 | (squashed) diamond |
2 | ellipse |
Piet Hein's "superellipse" | |
4 | rectellipse |
If is a rational, then a superellipse is algebraic. However, for irrational , it is transcendental. For even integers , the curve becomes closer to a rectangle as increases.
The area of a superellipse is given by
(4)
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(5)
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The above plots show generalization of the superellipse given by the function
(6)
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for , ..., 4 and , ..., 4.
Gielis (2003) has considered the further generalization of the superellipse given in polar coordinates by
(7)
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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.)
Families of curves generated by the "superformula" with and varying from 0 to 2 are illustrated above for values of varying from 1 to 8.