Polynomial curve of constant width: Difference between revisions

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The curve defined by
:<math> p(\phi) = a\cdot cos(k\cdot\phi/2)+b </math>
in polar form is smooth and of constant width for odd values of <math>k</math>.
<jsxgraph width="600" height="600">
<jsxgraph width="600" height="600">
var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-2,2,2,-2], keepaspectratio:true});
var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-2,2,2,-2], keepaspectratio:true});
Line 15: Line 21:
                               var co = Math.cos(kk*phi*0.5);
                               var co = Math.cos(kk*phi*0.5);
                               return aa*co*co+bb;
                               return aa*co*co+bb;
                             },[0,0], 0,Math.PI*2], {curveType:'polar', strokeWidth:6, strokeColor:'#3d1c24'});
                             },[0,0], 0,Math.PI*2], {curveType:'polar', strokeWidth:6, strokeColor:'#ad5544'});
brd.unsuspendUpdate();
brd.unsuspendUpdate();
</jsxgraph>
</jsxgraph>
===References===
* [http://www.mathpropress.com/stan/bibliography/ Stanley Rabinowitz, A Polynomial Curve of Constant Width. Missouri Journal of Mathematical Sciences, 9(1997)23-27.]
===The underlying JavaScript code===
var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-2,2,2,-2], keepaspectratio:true});
brd.suspendUpdate();
var a = brd.create('slider',[[-1,1.8],[1,1.8],[-5,0.20,5]], {name:'a'});
var b = brd.create('slider',[[-1,1.6],[1,1.6],[-5,1.15,10]], {name:'b'});
var k = brd.create('slider',[[-1,1.4],[1,1.4],[1,1,11]], {name:'k\'', snapWidth:1});
var p = brd.create('curve',[function(phi, suspendUpdate){
                              var kk, aa, bb;
                              if (!suspendUpdate) {
                                aa = a.Value();
                                bb = b.Value();
                                kk = 2*k.Value()+1;
                              }
                              var co = Math.cos(kk*phi*0.5);
                              return aa*co*co+bb;
                            },[0,0], 0,Math.PI*2], {curveType:'polar', strokeWidth:6, strokeColor:'#ad5544'});
brd.unsuspendUpdate();


[[Category:Examples]]
[[Category:Examples]]
[[Category:Curves]]
[[Category:Curves]]

Revision as of 14:39, 6 October 2010

The curve defined by

[math]\displaystyle{ p(\phi) = a\cdot cos(k\cdot\phi/2)+b }[/math]

in polar form is smooth and of constant width for odd values of [math]\displaystyle{ k }[/math].

References

The underlying JavaScript code

var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-2,2,2,-2], keepaspectratio:true}); brd.suspendUpdate(); var a = brd.create('slider',[[-1,1.8],[1,1.8],[-5,0.20,5]], {name:'a'}); var b = brd.create('slider',[[-1,1.6],[1,1.6],[-5,1.15,10]], {name:'b'}); var k = brd.create('slider',[[-1,1.4],[1,1.4],[1,1,11]], {name:'k\, snapWidth:1});

var p = brd.create('curve',[function(phi, suspendUpdate){

                             var kk, aa, bb;
                             if (!suspendUpdate) {
                               aa = a.Value();
                               bb = b.Value();
                               kk = 2*k.Value()+1;
                             }
                             var co = Math.cos(kk*phi*0.5);
                             return aa*co*co+bb;
                            },[0,0], 0,Math.PI*2], {curveType:'polar', strokeWidth:6, strokeColor:'#ad5544'});

brd.unsuspendUpdate();