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# Superellipse

## Introduction

A supercircle is a generalisation of a circle. Where the well-known equation
for a circle is x²+y²=r², for a supercircle it is x^n+y^n=r^n
with n≥2.

For n=2 we have the circle, for n≈10 the supercircle is almost
undistinguishable from a square.

More general the equation for a superellipse with halfaxes a and b can be
written as (x/a)^n+(y/b)^n=1 .

See also
http://en.wikipedia.org/wiki/Super_ellipse
and
http://nl.wikipedia.org/wiki/Superellips
.

Although superellipses as mathematical shapes are 'unnatural',
they are esthetically pleasing as shapes between ellipses and rectangles.
## Computation

The easiest way to compute the detailed shape of a superellipse is with
awk
. Under Linux this is always available. For Microsoft Windows awk.exe
can be downloaded from
http://unxutils.sourceforge.net/
.

Execute

`
awk '{a=123.0;b=456.0;n=7.8;x=0.0+$0;if(x>=0.0&&x<a)print x " " b*exp(log(1.0-exp(n*(log(x)-log(a))))/n);}'
`

where the values for halfwidth a, halfheight b and power n should be replaced by their actual values.
By typing a x-value, the coordinates of a point on the superellipse
are returned. Finish the program by typing ^D (letter d while holding down the Ctrl key).
## A superellipse table

For my personal use I constructed a readingtable of 315 × 57 cm
in the shape of a superellipse (x/1575)^35+(y/285)^35=1 .

First I made a gauge based on a exact drawing on graph-paper of the desired curve.
Limited to the part where the superellipse departs more
than a fraction of a millimeter from the full rectangle,
just 4 × 12 cm.

A corner of the finished table looks like this.

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