Superformula: Difference between revisions

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Line 9: Line 9:
<jsxgraph width="500" height="500" box="box2">
<jsxgraph width="500" height="500" box="box2">
  var b2 = JXG.JSXGraph.initBoard('box2', {axis:true,originX: 250, originY: 250, unitX: 25, unitY: 25});
  var b2 = JXG.JSXGraph.initBoard('box2', {axis:true,originX: 250, originY: 250, unitX: 25, unitY: 25});
  var a = b2.createElement('slider', [[1,8],[5,8],[0,1,4]],{name:'a'});
  var a = b2.createElement('slider', [[1,8],[7,8],[0,1,4]],{name:'a'});
  var b = b2.createElement('slider', [[1,7],[5,7],[0,1,4]],{name:'b'});
  var b = b2.createElement('slider', [[1,7],[7,7],[0,1,4]],{name:'b'});
  var m = b2.createElement('slider', [[1,6],[5,6],[0,4,100]],{name:'m'});
  var m = b2.createElement('slider', [[1,6],[7,6],[0,4,40]],{name:'m'});
  var n1 = b2.createElement('slider', [[1,5],[5,5],[0,4,40]],{name:'n_1'});
  var n1 = b2.createElement('slider', [[1,5],[7,5],[0,4,20]],{name:'n_1'});
  var n2 = b2.createElement('slider', [[1,4],[5,4],[0,4,40]],{name:'n_2'});
  var n2 = b2.createElement('slider', [[1,4],[7,4],[0,4,20]],{name:'n_2'});
  var n3 = b2.createElement('slider', [[1,3],[5,3],[0,4,40]],{name:'n_3'});
  var n3 = b2.createElement('slider', [[1,3],[7,3],[0,4,20]],{name:'n_3'});
  var len = b2.createElement('slider', [[1,2],[5,2],[0,2,2]],{name:'len'});  
  var len = b2.createElement('slider', [[1,2],[7,2],[0,2,2]],{name:'len'});  
  var c = b2.createElement('curve', [
  var c = b2.createElement('curve', [
         function(phi){return b2.pow(
         function(phi){return b2.pow(
Line 22: Line 22:
                             -1/n1.Value()); },  
                             -1/n1.Value()); },  
         [0, 0],0, function(){return len.Value()*Math.PI;}],
         [0, 0],0, function(){return len.Value()*Math.PI;}],
         {curveType:'polar', strokewidth:1,fillColor:'#765412',fillOpacity:0.6});       
         {curveType:'polar', strokewidth:1,fillColor:'#765412',fillOpacity:0.5});       
</jsxgraph>
</jsxgraph>


Line 28: Line 28:
<source lang="xml">
<source lang="xml">
  var b2 = JXG.JSXGraph.initBoard('box2', {axis:true,originX: 250, originY: 250, unitX: 25, unitY: 25});
  var b2 = JXG.JSXGraph.initBoard('box2', {axis:true,originX: 250, originY: 250, unitX: 25, unitY: 25});
  var a = b2.createElement('slider', [[1,8],[5,8],[0,1,4]],{name:'a'});
  var a = b2.createElement('slider', [[1,8],[7,8],[0,1,4]],{name:'a'});
  var b = b2.createElement('slider', [[1,7],[5,7],[0,1,4]],{name:'b'});
  var b = b2.createElement('slider', [[1,7],[7,7],[0,1,4]],{name:'b'});
  var m = b2.createElement('slider', [[1,6],[5,6],[0,4,20]],{name:'m'});
  var m = b2.createElement('slider', [[1,6],[7,6],[0,4,40]],{name:'m'});
  var n1 = b2.createElement('slider', [[1,5],[5,5],[0,4,20]],{name:'n_1'});
  var n1 = b2.createElement('slider', [[1,5],[7,5],[0,4,20]],{name:'n_1'});
  var n2 = b2.createElement('slider', [[1,4],[5,4],[0,4,20]],{name:'n_2'});
  var n2 = b2.createElement('slider', [[1,4],[7,4],[0,4,20]],{name:'n_2'});
  var n3 = b2.createElement('slider', [[1,3],[5,3],[0,4,20]],{name:'n_3'});
  var n3 = b2.createElement('slider', [[1,3],[7,3],[0,4,20]],{name:'n_3'});
  var len = b2.createElement('slider', [[1,2],[5,2],[0,2,2]],{name:'len'});  
  var len = b2.createElement('slider', [[1,2],[7,2],[0,2,2]],{name:'len'});  
  var c = b2.createElement('curve', [
  var c = b2.createElement('curve', [
         function(phi){return b2.pow(
         function(phi){return b2.pow(
Line 41: Line 41:
                             -1/n1.Value()); },  
                             -1/n1.Value()); },  
         [0, 0],0, function(){return len.Value()*Math.PI;}],
         [0, 0],0, function(){return len.Value()*Math.PI;}],
         {curveType:'polar', strokewidth:1,fillColor:'#765412',fillOpacity:0.6});       
         {curveType:'polar', strokewidth:1,fillColor:'#765412',fillOpacity:0.5});       
</source>
</source>



Revision as of 15:55, 18 March 2009

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:

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}}}

The JavaScript code to produce this picture

 var b2 = JXG.JSXGraph.initBoard('box2', {axis:true,originX: 250, originY: 250, unitX: 25, unitY: 25});
 var a = b2.createElement('slider', [[1,8],[7,8],[0,1,4]],{name:'a'});
 var b = b2.createElement('slider', [[1,7],[7,7],[0,1,4]],{name:'b'});
 var m = b2.createElement('slider', [[1,6],[7,6],[0,4,40]],{name:'m'});
 var n1 = b2.createElement('slider', [[1,5],[7,5],[0,4,20]],{name:'n_1'});
 var n2 = b2.createElement('slider', [[1,4],[7,4],[0,4,20]],{name:'n_2'});
 var n3 = b2.createElement('slider', [[1,3],[7,3],[0,4,20]],{name:'n_3'});
 var len = b2.createElement('slider', [[1,2],[7,2],[0,2,2]],{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:0.5});

External links