A superellipse, or Lamé curve, is drawn from the function
$$\big(\frac{x}{a}\big)^n + \big(\frac{y}{b}\big)^n = 1\label{lame}$$
where $n$ is any real number. This is of course the equation of an ellipse for $n=2$, and a circle with radius $a$
for $n=2$ with $a=b$.
There are a number of things interesting about this curve:
- For instance, if $n<1$, it forms a 4-pointed star-like
object with inward curving sides (parabolas for $n=\half$).
- For $1\lt n\lt 2$, the sides are outward curving, like a rhombus.
- Perhaps most famously, for $n>2$ and gets large, the ellipse is "squared". (Try setting the value
below next to "n =" to 100!) You can
imagine this by first setting $n=2$ and $a=b$ (so it's a circle), then let $n$ get large. For any value
of $x \ne a$, then $y$ will have be very close to $y=a$. And vice versa for
$y\ne a$, all values of $x$ will be close to $x=a$.
- The area as a function of $n$ can be expressed in terms of the Gamma function, which has
to do with how you define factorial operations on complex numbers:
$A = 4ab\frac{(\Gamma(1+\frac{1}{n}))^2}{\Gamma(1+\frac{2}{n})}$
Drew Baden October, 2024
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