Polynomial curve of constant width

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

• 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();