Lesson 2 - Drawing a Superellipse
Brief History of the Superellipse
In 1959 a city planning team in Stockholm, Sweden, was considering a problem with a rectangular square in the center of the city. The architects wanted the traffic to flow around this square, therefore a rectangular shape was not suitable. However an ellipse did not work either, because the pointed ends of the ellipse would prevent smooth traffic. Not knowing what to do, the team consulted Piet Hein, a known Danish writer, scientist, inventor and philosopher. They believed it was the kind of a problem, that appealed to his combined mathematical and artistic imagination, his sense of humor and his ability to think creatively in unexpected ways.
Piet Heins was searching for a curve, that was something in between an ellipse and a rectangle. He was especially interested in a curve that was defined by mathematically means, because he wanted the curve to be smooth and harmonic. The usual ellipse is defined by the equation
Piet Hein considered changing the exponents to a more generic value n:
He realized that increasing the exponent would make the shape like more and more like a rectangle. He then defined the Superellipse as any curve satisfying the last equation with n greater than 2. Especially he choose n = 2.5 for the actual problem, because he found that with this value the curve was looking especially pleasing for the eye. The parameters a and b happens to be half the width and half the height of the curve. For this particular case he chose a/b = 6/5.
Piet Heins solution was accepted and today you can find the Superellipse Square in Stockholm, called Sergels Torg. Since then more people have been inspired by the curve. Just to mention a few, the Superellipse was used in the design of the Olympic Stadium in Mexico City. At the top of his popularity, Piet Hein was on the front page of Life Magazine with his Superellipse (1966).
It has to be mentioned that the curves considered by Hein were in fact looked at earlier by the French mathematician Gabriel Lamé (1795 - 1870). Therefore these curves go under the name of Lamé curves.
The Superellipse created with the Function Plotter
If you are interested in a detailed explanation of math used in the following example, please click here: The math behind the Superellipse.
The advantage of using custom functions
Above we did not have to define our new custom function hein(x,a,n). We could as well have written the following coordinate functions from the very beginning:
X(t) = 3*sign(cos(t)*(abs(cos(t)))^(2/2.5)
Y(t) = 2*sign(sin(t)*(abs(sin(t)))^(2/2.5)
However if we want to make changes it is not at all as "streamlined" compared to using custom defined functions. Often you have to make more substitutions and you don't get the same overview. And in fact the functions used could have been much more complicated than in this example. As an added bonus to using custom functions, you can save them in an external file for future use.
Let's experiment more with our hein function. Let's use a = b = 1 in all of following examples and see what effect the parameter n has on the shape of the superellipse.
In fact if we use n = 2/3 we get the well-known mathematical curve called The Astroid. Using n = 1 will yield a square. n = 2 generally results in an ellipse, but because a = b here, we get a precise circle. When using n = 2.5 we get the Superellipse of Piet Hein and because a = b we can call it a Supercircle. With n = 10 to show that the shape will approximate a square more and more when you increase the value of n.
All these curves you can get by just changing a few parameters. For example you get the Astroid by using the following coordinate functions for x(t) and y(t) respectively: hein(cos(t),1,2/3) and hein(sin(t),1,2/3). So very easily you get a whole bunch of curves just by changing a few parameters. Good luck with your investigations!
Copyright © 2002 by Alex Vakulenko. All rights reserved.