Superformula

The superformula is a generalization of the superellipse and was first proposed by Johan Gielis.

Gielis suggested that the formula can be used to describe many complex shapes and curves that are found in nature. Others point out that the same can be said about many formulas with a sufficient number of parameters.

In polar coordinates, with r the radius and φ the angle, the superformula is:

[math]\displaystyle{ r\left(\phi\right) = \left[ \left| \frac{\cos\left(\frac{m\phi}{4}\right)}{a} \right| ^{n_{2}} + \left| \frac{\sin\left(\frac{m\phi}{4}\right)}{b} \right| ^{n_{3}} \right] ^{-\frac{1}{n_{1}}} }[/math]

The JavaScript code to produce this picture

<jsxgraph width="550" height="500" box="box2">
 var b2 = JXG.JSXGraph.initBoard('box2', {axis:true,originX: 250, originY: 250, unitX: 25, unitY: 25});
 b2.suspendUpdate();
 var a = b2.createElement('slider', [[-7,8],[7,8],[0,1,4]],{name:'a'});
 var b = b2.createElement('slider', [[-7,7],[7,7],[0,1,4]],{name:'b'});
 var m = b2.createElement('slider', [[-7,6],[7,6],[0,4,40]],{name:'m'});
 var n1 = b2.createElement('slider', [[-7,5],[7,5],[0,4,20]],{name:'n_1'});
 var n2 = b2.createElement('slider', [[-7,4],[7,4],[0,4,20]],{name:'n_2'});
 var n3 = b2.createElement('slider', [[-7,3],[7,3],[0,4,20]],{name:'n_3'});
 var len = b2.createElement('slider', [[1,2],[7,2],[0,2,20]],{name:'len'}); 
 var c = b2.createElement('curve', [
        function(phi){return b2.pow(
                               b2.pow(Math.abs(Math.cos( m.Value()*phi*0.25/a.Value() )), n2.Value())+
                               b2.pow(Math.abs(Math.sin( m.Value()*phi*0.25/b.Value() )), n3.Value()),
                             -1/n1.Value()); }, 
        [0, 0],0, function(){return len.Value()*Math.PI;}],
        {curveType:'polar', strokewidth:1,fillColor:'#765412',fillOpacity:0.3});      
 b2.unsuspendUpdate();
</jsxgraph>

External links

• http://en.wikipedia.org/wiki/Superformula