Polynomial curve of constant width - Revision history

A WASSERMANN at 20:51, 23 March 2021

2021-03-23T20:51:41Z

← Older revision		Revision as of 20:51, 23 March 2021
Line 1:		Line 1:
	The curve with support defined by		The curve with support defined by

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

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

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

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

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

A WASSERMANN at 10:04, 7 June 2011

2011-06-07T10:04:57Z

← Older revision		Revision as of 10:04, 7 June 2011
Line 22:		Line 22:
	var k = brd.create('slider',[[-1,4],[1,4],[1,1,11]], {name:'k\'', snapWidth:1});		var k = brd.create('slider',[[-1,4],[1,4],[1,1,11]], {name:'k\'', snapWidth:1});

	var c = brd.create('curve',[function(phi~~, suspendUpdate~~){		var c = brd.create('curve',[function(phi){
	var kk, aa, bb, p, ps, co, si;		var kk, aa, bb, p, ps, co, si;
	~~//if (!suspendUpdate) {~~		aa = a.Value();
	aa = a.Value();		bb = b.Value();
	bb = b.Value();		kk = 2*k.Value()+1;
	kk = 2*k.Value()+1;
	~~//}~~
	co = Math.cos(kkphi0.5);		co = Math.cos(kkphi0.5);
	si = Math.sin(kkphi0.5);		si = Math.sin(kkphi0.5);
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	return pMath.cos(phi)-psMath.sin(phi);		return pMath.cos(phi)-psMath.sin(phi);
	},		},
	function(phi~~, suspendUpdate~~){		function(phi){
	var kk, aa, bb, p, ps, co, si;		var kk, aa, bb, p, ps, co, si;
	~~//if (!suspendUpdate) {~~		aa = a.Value();
	aa = a.Value();		bb = b.Value();
	bb = b.Value();		kk = 2*k.Value()+1;
	kk = 2*k.Value()+1;
	~~//}~~
	co = Math.cos(kkphi0.5);		co = Math.cos(kkphi0.5);
	si = Math.sin(kkphi0.5);		si = Math.sin(kkphi0.5);
Line 65:		Line 61:
	var k = brd.create('slider',[[-1,1.4],[1,1.4],[1,1,11]], {name:'k\'', snapWidth:1});		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 c = brd.create('curve',[function(phi){
	var kk, aa, bb, p, ps, co, si;		var kk, aa, bb, p, ps, co, si;
	~~//if (!suspendUpdate) {~~		aa = a.Value();
	aa = a.Value();		bb = b.Value();
	bb = b.Value();		kk = 2*k.Value()+1;
	kk = 2*k.Value()+1;
	~~//}~~
	co = Math.cos(kkphi0.5);		co = Math.cos(kkphi0.5);
	si = Math.sin(kkphi0.5);		si = Math.sin(kkphi0.5);
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	return pMath.cos(phi)-psMath.sin(phi);		return pMath.cos(phi)-psMath.sin(phi);
	},		},
	function(phi~~, suspendUpdate~~){		function(phi){
	var kk, aa, bb, p, ps, co, si;		var kk, aa, bb, p, ps, co, si;
	~~//if (!suspendUpdate) {~~		aa = a.Value();
	aa = a.Value();		bb = b.Value();
	bb = b.Value();		kk = 2*k.Value()+1;
	kk = 2*k.Value()+1;
	~~//}~~
	co = Math.cos(kkphi0.5);		co = Math.cos(kkphi0.5);
	si = Math.sin(kkphi0.5);		si = Math.sin(kkphi0.5);

A WASSERMANN at 09:59, 7 June 2011

2011-06-07T09:59:28Z

← Older revision		Revision as of 09:59, 7 June 2011
Line 1:		Line 1:
	The curve with support defined by		The curve with support defined by

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

	has constant width for odd values of <math>k</math>.		has constant width for odd values of <math>k</math>.

A WASSERMANN at 09:59, 7 June 2011

2011-06-07T09:59:05Z

← Older revision		Revision as of 09:59, 7 June 2011
Line 1:		Line 1:
	The curve with support defined by		The curve with support defined by

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

	has constant width for odd values of <math>k</math>.		has constant width for odd values of <math>k</math>.

A WASSERMANN at 09:58, 7 June 2011

2011-06-07T09:58:37Z

← Older revision		Revision as of 09:58, 7 June 2011
Line 1:		Line 1:
	The curve with support defined by		The curve with support defined by

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

	has constant width for odd values of <math>k</math>.		has constant width for odd values of <math>k</math>.

A WASSERMANN at 09:57, 7 June 2011

2011-06-07T09:57:43Z

← Older revision		Revision as of 09:57, 7 June 2011
Line 1:		Line 1:
	The curve defined by		The curve with support defined by

	:<math> p(\phi) = a\cdot cos(k\cdot\phi/2)+b </math>		:<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>.		has constant width for odd values of <math>k</math>.
	In the ~~visuzalitaion~~ with JSXGraph below <math>k</math> is determined		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>		:<math>k = 2k'+1.</math>
Line 13:		Line 19:
	brd.suspendUpdate();		brd.suspendUpdate();
	var a = brd.create('slider',[[-1,6],[1,6],[-5,0.20,8]], {name:'a'});		var a = brd.create('slider',[[-1,6],[1,6],[-5,0.20,8]], {name:'a'});
	var b = brd.create('slider',[[-1,4],[1,5],[-5,1.15,20]], {name:'b'});		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 k = brd.create('slider',[[-1,4],[1,4],[1,1,11]], {name:'k\'', snapWidth:1});

A WASSERMANN at 09:54, 7 June 2011

2011-06-07T09:54:10Z

← Older revision		Revision as of 09:54, 7 June 2011
Line 12:		Line 12:
	var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-8,8,8,-8], keepaspectratio:true});		var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-8,8,8,-8], keepaspectratio:true});
	brd.suspendUpdate();		brd.suspendUpdate();
	var a = brd.create('slider',[[-1,~~1.8~~],[1,~~1.8~~],[-5,0.20,5]], {name:'a'});		var a = brd.create('slider',[[-1,6],[1,6],[-5,0.20,8]], {name:'a'});
	var b = brd.create('slider',[[-1,~~1.6~~],[1,~~1.6~~],[-5,1.15,10]], {name:'b'});		var b = brd.create('slider',[[-1,4],[1,5],[-5,1.15,20]], {name:'b'});
	var k = brd.create('slider',[[-1,1.4],[1,1.4],[1,1,11]], {name:'k\'', snapWidth:1});		var k = brd.create('slider',[[-1,4],[1,4],[1,1,11]], {name:'k\'', snapWidth:1});

	var c = brd.create('curve',[function(phi, suspendUpdate){		var c = brd.create('curve',[function(phi, suspendUpdate){

A WASSERMANN at 09:52, 7 June 2011

2011-06-07T09:52:58Z

← Older revision		Revision as of 09:52, 7 June 2011
Line 10:		Line 10:

	<jsxgraph width="600" height="600">		<jsxgraph width="600" height="600">
	var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-2,2,2,-2], keepaspectratio:true});		var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-8,8,8,-8], keepaspectratio:true});
	brd.suspendUpdate();		brd.suspendUpdate();
	var a = brd.create('slider',[[-1,1.8],[1,1.8],[-5,0.20,5]], {name:'a'});		var a = brd.create('slider',[[-1,1.8],[1,1.8],[-5,0.20,5]], {name:'a'});

A WASSERMANN at 09:52, 7 June 2011

2011-06-07T09:52:44Z

← Older revision		Revision as of 09:52, 7 June 2011
Line 16:		Line 16:
	var k = brd.create('slider',[[-1,1.4],[1,1.4],[1,1,11]], {name:'k\'', snapWidth:1});		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 ~~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(kkphi0.5);~~
	~~return aacoco+bb;~~
	~~},[0,0], 0,Math.PI*2], {curveType:'polar', strokeWidth:10, strokeColor:'#ad5544'});~~
	*/

	~~var p2~~ = brd.create('curve',[function(phi, suspendUpdate){
	var kk, aa, bb, p, ps, co, si;		var kk, aa, bb, p, ps, co, si;
	if (!suspendUpdate) {		//if (!suspendUpdate) {
	aa = a.Value();		aa = a.Value();
	bb = b.Value();		bb = b.Value();
	kk = 2*k.Value()+1;		kk = 2*k.Value()+1;
	}		//}
	co = Math.cos(kkphi0.5);		co = Math.cos(kkphi0.5);
	si = Math.sin(kkphi0.5);		si = Math.sin(kkphi0.5);
Line 44:		Line 31:
	function(phi, suspendUpdate){		function(phi, suspendUpdate){
	var kk, aa, bb, p, ps, co, si;		var kk, aa, bb, p, ps, co, si;
	if (!suspendUpdate) {		//if (!suspendUpdate) {
	aa = a.Value();		aa = a.Value();
	bb = b.Value();		bb = b.Value();
	kk = 2*k.Value()+1;		kk = 2*k.Value()+1;
	}		//}
	co = Math.cos(kkphi0.5);		co = Math.cos(kkphi0.5);
	si = Math.sin(kkphi0.5);		si = Math.sin(kkphi0.5);
Line 66:		Line 53:
	===The underlying JavaScript code===		===The underlying JavaScript code===
	<source lang="javascript">		<source lang="javascript">
	var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-2,2,2,-2], keepaspectratio:true});		var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-8,8,8,-8], keepaspectratio:true});
	brd.suspendUpdate();		brd.suspendUpdate();
	var a = brd.create('slider',[[-1,1.8],[1,1.8],[-5,0.20,5]], {name:'a'});		var a = brd.create('slider',[[-1,1.8],[1,1.8],[-5,0.20,5]], {name:'a'});
Line 72:		Line 59:
	var k = brd.create('slider',[[-1,1.4],[1,1.4],[1,1,11]], {name:'k\'', snapWidth:1});		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 c = brd.create('curve',[function(phi, suspendUpdate){
	var kk, aa, bb;		var kk, aa, bb, p, ps, co, si;
	if (!suspendUpdate) {		//if (!suspendUpdate) {
			aa = a.Value();
			bb = b.Value();
			kk = 2*k.Value()+1;
			//}
			co = Math.cos(kkphi0.5);
			si = Math.sin(kkphi0.5);
			p = aacoco+bb;
			ps = -aakkco*si;
			return pMath.cos(phi)-psMath.sin(phi);
			},
			function(phi, suspendUpdate){
			var kk, aa, bb, p, ps, co, si;
			//if (!suspendUpdate) {
	aa = a.Value();		aa = a.Value();
	bb = b.Value();		bb = b.Value();
	kk = 2*k.Value()+1;		kk = 2*k.Value()+1;
	}		//}
	~~var~~ co = Math.cos(kkphi0.5);		co = Math.cos(kkphi0.5);
	~~return~~ aacoco+bb;		si = Math.sin(kkphi0.5);
	},~~[0,0],~~ 0,Math.PI*2], {~~curveType:'polar',~~ strokeWidth:10, strokeColor:'#ad5544'});		p = aacoco+bb;
			ps = -aakkco*si;
			return pMath.sin(phi)+psMath.cos(phi);
			},
			0, Math.PI*2], {strokeWidth:10, strokeColor:'#ad5544'});

	brd.unsuspendUpdate();		brd.unsuspendUpdate();
	</source>		</source>

A WASSERMANN at 09:51, 7 June 2011

2011-06-07T09:51:16Z

← Older revision		Revision as of 09:51, 7 June 2011
Line 31:		Line 31:
	var p2 = brd.create('curve',[function(phi, suspendUpdate){		var p2 = brd.create('curve',[function(phi, suspendUpdate){
	var kk, aa, bb, p, ps, co, si;		var kk, aa, bb, p, ps, co, si;
	if (~~true \|\|~~ !suspendUpdate) {		if (!suspendUpdate) {
	aa = a.Value();		aa = a.Value();
	bb = b.Value();		bb = b.Value();
Line 44:		Line 44:
	function(phi, suspendUpdate){		function(phi, suspendUpdate){
	var kk, aa, bb, p, ps, co, si;		var kk, aa, bb, p, ps, co, si;
	if (~~true \|\|~~ !suspendUpdate) {		if (!suspendUpdate) {
	aa = a.Value();		aa = a.Value();
	bb = b.Value();		bb = b.Value();