Polynomial curve of constant width

From JSXGraph Wiki
Revision as of 10:54, 7 June 2011 by A WASSERMANN (talk | contribs)

Jump to: navigation, search

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]. In the visuzalitaion with JSXGraph below [math]k[/math] is determined

[math]k = 2k'+1.[/math]


References

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, suspendUpdate){ 
                              var kk, aa, bb, p, ps, co, si;
                              //if (!suspendUpdate) {
                                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, suspendUpdate){ 
                              var kk, aa, bb, p, ps, co, si;
                              //if (!suspendUpdate) {
                                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();