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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]]

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