The math behind the Superellipse
In his search for a harmonic shape Piet Hein was inspired by the usual ellipse made up of the the points (x,y) satisfying the following equation:
However he wanted the shape to be more rectangular. His idea was then to try other exponents than 2. The absolute values must be used here:
(*)
where the absolute value, is defined by:
It is not difficult to see, that the following values for x and y satisfies (*):
because if you insert it in (*) you will get the well-known trigonometric formula
which is universally true. In the expression for x and y, we will let t run in the interval from -pi to pi, and we could draw four different curves corresponding to the four possible combinations of + (plus) and - (minus) in the expression above. However we can do it in one turn: if we use the signum function sgn defined by
then
Or, if you prefer the notation with abs:
For simplicity we could define a new function hein, which takes one variable x and two parameters a and b:
This is convenient way to handle it, because both functions, sgn and abs, are predefined functions in the function plotter. Then the solutions to (*) would obtain the following harmonic look:
Now using n = 2 would yield an ellipse with half-axes a and b. Piet choose n = 2.5 for his superellipse. For example a superellipse with half-axes 3 and 2 can be produced by the following functions:
The Astroid
The formula (**) not only produces the superellipse but also the well-known curve called the Astroid. The expressions
produces the Astroid with half-axes 1 and 1.
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