The supershape equation is an extension of the both the equation of the sphere and ellipse
(x / a)2 + (y / b)2 = r2
and even the superellipse given here. The general formula for the supershape is given below.
Where r and phi are polar coordinates (radius,angle). n1, n2, n3, and m are real numbers. a and b are real numbers excluding zero.
m = 0
This results in circles, namely r = 1
n1 = n2 = n3 = 1
Increasing m adds rotational symmetry to the shape. This is generally the case for other values of the n parameters. The curves are repeated in sections of the circle of angle 2pi/m, this is apparent in most of the following examples for integer values of m. m=1 m=2
m=3 m=4
m=5 m=6
n1 = n2 = n3 = 0.3
As the n's are kept equal but reduced the form becomes increasingly pinched. m=1 m=2
m=3 m=4
m=5 m=6
If n1 is slightly larger than n2 and n3 then bloated forms result. The examples on the right have n1 = 40 and n2 = n3 = 10. m=1 m=2
m=3 m=4
m=5 m=6
Polygonal shapes are achieved with very large values of n1 and large but equal values for n2 and n3. m=3 m=4
m=5 m=6
Asymmetric forms can be created by using different values for the n's. The following example have n1 = 60, n2 = 55 and n3 = 30. m=3 m=4
m=5 m=6
For non integral values of m the form is still closed for rational values. The following are example with n1 = n2 = n3 = 0.3. The angle phi needs to extend from 0 to 12 pi. m=1/6 m=7/6
m=13/6 m=19/6
Smooth starfish shapes result from smaller values of n1 than the n2 and n3. The following examples have m=5 and n2 = n3 = 1.7. n1=0.50 n1=0.20
n1=0.10 n1=0.02
Other examples
Source code
Given a value of phi the following function evaluates the supershape function and calculates (x,y).
