View source for Polynomial curve of constant width

The curve with support defined by

:<math> p(\phi)= a\cdot \cos^2(k\cdot\phi/2)+b </math>

has constant width for odd values of <math>k</math>.
It defines the parametric curve

:<math> x(\phi) = p(\phi)\cos(\phi) - p'\sin(\phi)</math>

:<math> y(\phi) = p(\phi)\sin(\phi) + p'\cos(\phi)</math>

In the visualization with JSXGraph below <math>k</math> is determined 

:<math>k = 2k'+1.</math>


<jsxgraph width="600" height="600">
var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-8,8,8,-8], keepaspectratio:true});
brd.suspendUpdate();
var a = brd.create('slider',[[-1,6],[1,6],[-5,0.20,8]], {name:'a'});
var b = brd.create('slider',[[-1,5],[1,5],[-5,1.15,20]], {name:'b'});
var k = brd.create('slider',[[-1,4],[1,4],[1,1,11]], {name:'k\'', snapWidth:1});

var c = brd.create('curve',[function(phi){ 
                              var kk, aa, bb, p, ps, co, si;
                              aa = a.Value();
                              bb = b.Value();
                              kk = 2*k.Value()+1;
                              co = Math.cos(kk*phi*0.5);
                              si = Math.sin(kk*phi*0.5);
                              p  =  aa*co*co+bb;
                              ps = -aa*kk*co*si;
                              return p*Math.cos(phi)-ps*Math.sin(phi);
                             },
                             function(phi){ 
                              var kk, aa, bb, p, ps, co, si;
                              aa = a.Value();
                              bb = b.Value();
                              kk = 2*k.Value()+1;
                              co = Math.cos(kk*phi*0.5);
                              si = Math.sin(kk*phi*0.5);
                              p  =  aa*co*co+bb;
                              ps = -aa*kk*co*si;
                              return p*Math.sin(phi)+ps*Math.cos(phi);
                             },
                             0, Math.PI*2], {strokeWidth:10, strokeColor:'#ad5544'});


brd.unsuspendUpdate();
</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===
<source lang="javascript">
var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-8,8,8,-8], 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 c = brd.create('curve',[function(phi){ 
                              var kk, aa, bb, p, ps, co, si;
                              aa = a.Value();
                              bb = b.Value();
                              kk = 2*k.Value()+1;
                              co = Math.cos(kk*phi*0.5);
                              si = Math.sin(kk*phi*0.5);
                              p  =  aa*co*co+bb;
                              ps = -aa*kk*co*si;
                              return p*Math.cos(phi)-ps*Math.sin(phi);
                             },
                             function(phi){ 
                              var kk, aa, bb, p, ps, co, si;
                              aa = a.Value();
                              bb = b.Value();
                              kk = 2*k.Value()+1;
                              co = Math.cos(kk*phi*0.5);
                              si = Math.sin(kk*phi*0.5);
                              p  =  aa*co*co+bb;
                              ps = -aa*kk*co*si;
                              return p*Math.sin(phi)+ps*Math.cos(phi);
                             },
                             0, Math.PI*2], {strokeWidth:10, strokeColor:'#ad5544'});

brd.unsuspendUpdate();
</source>

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