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superformula

The superformula is a generalisation of the circle, the ellipse, and the superellipse, published by Johan Gielis in 2003.

The equation of a circle of radius +++c+++ is:

$$ x^2 + y^2 = c^2 \mbox{ ; or } \left(\frac{x}{c}\right)^2 + \left(\frac{y}{c}\right)^2 = 1 \mbox{ ; or } r = c $$

where +++x = r \sin \phi+++ and +++y = r \cos \phi+++.

The equation of an ellipse with semi-axes +++a+++ and +++b+++ is:

$$ \left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1 \mbox{ ; or } r = \left[\left(\frac{\cos\phi}{a}\right)^2 + \left(\frac{\sin\phi}{b}\right)^2\right]^{-1/2} $$

The superellipse is a generalisation; the exponents can be any +++n+++, not just +++2+++:

$$ \left|\frac{x}{a}\right|^n + \left|\frac{y}{b}\right|^n = 1 \mbox{ ; or } r = \left(\left|\frac{\cos\phi}{a}\right|^n + \left|\frac{\sin\phi}{b}\right|^n\right)^{-1/n} $$

Gielis' superformula is a further generalisation, allowing the three occurences of the exponent +++n+++ to vary independently, and also allowing the frequency of the trigonometric functions to vary:

$$ r = \left(\left|\frac{\cos (m\phi/4)}{a}\right|^{n_a} + \left|\frac{\sin (m\phi/4)}{b}\right|^{n_b}\right)^{-1/n} $$

With its many parameters, the superformula can describe a multitude of curves. For example (from the wikipedia superformula page, individual curves labelled by +++(m,n,n_a,n_b)+++, all with +++a=b=1+++)

Below is a utility (written in Processing, and using processing.js for execution is a browser). The scrollbars allow variation of +++m,n,n_a,n_b,a,b+++, plus a further parameter +++cy+++ described in examples 6,7,8 below. The buttons labelled +++0..24+++ provide +++25+++ predefined value sets, displaying a range of possible superformula images. The values of +++m+++ are restricted to be multiples of +++\frac{1}{4}+++; the values of +++n,n_a,n_b+++ are restricted to be multiples of +++\frac{1}{2}+++.

Predefined images (selected using the numbered buttons):

  1. +++(4, 2, 2, 2, 1, 1, 1)+++ : A circle; +++m=4+++ removes the frequency change; +++n = n_a = n_b = 2+++ is the squared exponent; +++a = b+++ makes it a circle.
  2. +++(4, 2, 2, 2, 1.8, 0.5, 1)+++ : An ellipse; +++m=4+++ removes the frequency change; +++n = n_a = n_b = 2+++ is the squared exponent; +++a \neq b+++ makes it an ellipse rather than a circle.
  3. +++(4, 3, 3, 3, 1.8, 0.5, 1)+++ : A superellipse; +++m=4+++ removes the frequency change; +++n = n_a = n_b \neq 2+++ is the non-squared exponent. If +++n = n_a = n_b = 1+++, the superellipse is a rhombus (see example 16); if +++n = n_a = n_b \lt 1+++, the sides of superellipse are concave.
  4. +++(8, 2, 10, 10, 1, 1, 1)+++ : An eight-pointed star.
  5. +++(3, 5, 18, 18, 1, 1, 1)+++ : A three-pointed star.
  6. +++(5, 1, 2, 15, 1, 1, 1)+++ : An asymmetric five-pointed star.
  7. +++(7, 4, 4, 17, 1, 1, 1)+++ : An asymmetric seven-pointed star.
  8. +++(7, 4, 4, 17, 1.3, 1, 1)+++ : An asymmetric seven-pointed star with a gap; +++a \neq b+++. Because +++m+++ is not an even integer, the frequency term means the value of +++r+++ is different when +++\phi = 0+++ and when +++\phi = 2\pi+++. When +++m+++ an odd integer, as here, it is easy to show that +++r(0) = a^{n_a/n}; r(2\pi)=b^{n_b/n}+++. For the curve to join up, we need +++a^{n_a/n}=b^{n_b/n}+++. One way to achieve this is +++a=b=1+++ (as in the previous example); another is +++a=b\neq 1, n_a=n_b+++.
  9. +++(7, 4, 4, 17, 1.3, 1, 2)+++ : An symmetric seven-pointed star with two cycles; +++cy = 2+++. A further way to make the curve join up for odd +++m+++ is to plot +++\phi+++ from +++0+++ to +++4\pi+++, or +++2+++ cycles, +++cy = 2+++. When +++m+++ is half-integral, we need +++cy=4+++; when it is quarter-integral, we need +++cy = 8+++. Other fractional values (not supported here) would require different values of +++cy+++; irrational values of +++m+++ would never join up, no matter how large +++cy+++.
  10. +++(12, 15, 20, 3, 1, 1, 1)+++ : A fancy dinner plate.
  11. +++(19, 9, 14, 11, 1, 1, 1)+++ : Nineteen spike.
  12. +++(18, 9, 18, 17, 1, 1.5, 1)+++ : Crinkly spikes.
  13. +++(17.75, 9, 18, 17, 1, 1.5, 8)+++ : Crinkly spikey flowerhead; +++m=17.75+++, so +++cy=8+++ for a closed curve.
  14. +++(16, 1, 1, 19, 1, 1, 1)+++ : Crinkly, less spikey.
  15. +++(2, 1, 4, 8, 1, 1, 1)+++ : A bean.
  16. +++(3, 5, 20, 20, 0.7, 1.2, 2)+++ : Overlapping trefoils.
  17. +++(4, 1, 1, 1, 1.5, 1.5, 1)+++ : The diamond (rhombus) superellipse; +++m=4;+++ +++n = n_a = n_b = 1+++.
  18. +++(4, 1, 8, 8, 0.9, 0.9, 1)+++ : A four-pointed star.
  19. +++(6, 1, 1, 1, 1.8, 1.1, 1)+++ : A lumpy triangle: six-pointed, but very differing sizes.
  20. +++(10, 1, 1, 1, 1.8, 1.1, 1)+++ : A lumpy starfish.
  21. +++(5, 2, 17, 4, 0.8, 1.5, 2)+++ : A five-petalled flower.
  22. +++(9, 2, 17, 4, 0.8, 1.5, 2)+++ : A nine-petalled flower.
  23. +++(2.5, 0.5, 0.5, 0.5, 2, 2, 2)+++ : Rose sepals.
  24. +++(8, 0.5, 0.5, 8, 1.9, 1.05, 1)+++ : A spikey four-leafed clover.
  25. +++(18, 0.5, 0.5, 4, 1.2, 1.1, 1)+++ : A virus.