Polynomial curve of constant width

The curve with support defined by

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

has constant width for odd values of [math]\displaystyle{ k }[/math]. It defines the parametric curve

[math]\displaystyle{ x(\phi) = p(\phi)\cos(\phi) - p'\sin(\phi) }[/math]

[math]\displaystyle{ y(\phi) = p(\phi)\sin(\phi) + p'\cos(\phi) }[/math]

In the visualization with JSXGraph below [math]\displaystyle{ k }[/math] is determined

[math]\displaystyle{ k = 2k'+1. }[/math]

References

• 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:[-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();