Tschirnhausen Cubic Catacaustic - Revision history

A WASSERMANN at 14:26, 13 January 2011

2011-01-13T14:26:54Z

← Older revision		Revision as of 14:26, 13 January 2011
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	The catacaustic is the envelope of the rays reflected by the Tschirnhausen cubic.		The catacaustic is the envelope of the rays reflected by the Tschirnhausen cubic.
	The ray's source is the ''radiant point''.		The ray's source is the ''radiant point''.
	You can drag the ''point of reflection''.
			You can get a feeling why the red curve is called ''envelope'' of the blue line if you drag the ''point of reflection''.
	<jsxgraph width="600" height="600">		<jsxgraph width="600" height="600">
	(function(){		(function(){

A WASSERMANN at 14:19, 13 January 2011

2011-01-13T14:19:46Z

← Older revision		Revision as of 14:19, 13 January 2011
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	:<math> y = at(t^2-3) </math>		:<math> y = at(t^2-3) </math>
	Its ~~catcaustic~~ (red curve) with radiant point <math>(-8a,p)</math>		Its catacaustic (red curve) with radiant point <math>(-8a,p)</math>
	is the semicubical parabola with parametric equations		is the semicubical parabola with parametric equations

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	:<math> y = a4t^3 </math>		:<math> y = a4t^3 </math>

			The catacaustic is the envelope of the rays reflected by the Tschirnhausen cubic.
			The ray's source is the ''radiant point''.
	You can drag the ''point of reflection''.		You can drag the ''point of reflection''.
	<jsxgraph width="600" height="600">		<jsxgraph width="600" height="600">
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	===References===		===References===
	* [http://mathworld.wolfram.com/TschirnhausenCubicCatacaustic.html Weisstein, Eric W. "Tschirnhausen Cubic Catacaustic." From MathWorld--A Wolfram Web Resource.]		* [http://mathworld.wolfram.com/TschirnhausenCubicCatacaustic.html Weisstein, Eric W. "Tschirnhausen Cubic Catacaustic." From MathWorld--A Wolfram Web Resource.]
			* [http://en.wikipedia.org/wiki/Caustic_%28mathematics%29 Wikipedia on Caustics]
	===The underlying JavaScript code===		===The underlying JavaScript code===
	<source lang="javascript">		<source lang="javascript">

A WASSERMANN at 14:15, 13 January 2011

2011-01-13T14:15:52Z

← Older revision		Revision as of 14:15, 13 January 2011
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	:<math> y = a4t^3 </math>		:<math> y = a4t^3 </math>

			You can drag the ''point of reflection''.
	<jsxgraph width="600" height="600">		<jsxgraph width="600" height="600">
	(function(){		(function(){
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	var radpoint = brd.create('point',[function(){ return -a.Value()*8;},0],{name:'radiant point'});		var radpoint = brd.create('point',[function(){ return -a.Value()*8;},0],{name:'radiant point'});
	var reflectionpoint = brd.create('glider',[-7,1,cubic],{name:'point of reflection'~~, size:1~~});		var reflectionpoint = brd.create('glider',[-7,1,cubic],{name:'point of reflection'});
	var dir = brd.create('segment',[radpoint,reflectionpoint],{strokeWidth:1});		var dir = brd.create('segment',[radpoint,reflectionpoint],{strokeWidth:1});
	var infty = brd.create('point',		var infty = brd.create('point',
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	-4, 4		-4, 4
	],		],
	{strokeWidth:1, strokeColor:'red'});		{strokeWidth:3, strokeColor:'red'});
	brd.unsuspendUpdate();		brd.unsuspendUpdate();
	})();		})();
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	var radpoint = brd.create('point',[function(){ return -a.Value()*8;},0],{name:'radiant point'});		var radpoint = brd.create('point',[function(){ return -a.Value()*8;},0],{name:'radiant point'});
			var reflectionpoint = brd.create('glider',[-7,1,cubic],{name:'point of reflection'});
			var dir = brd.create('segment',[radpoint,reflectionpoint],{strokeWidth:1});
			var infty = brd.create('point',
			[
			function(){
			var a = dir.stdform[1], b = dir.stdform[2],
			t = reflectionpoint.position,
			u = JXG.Math.Numerics.D(cubic.X)(t),
			v = JXG.Math.Numerics.D(cubic.Y)(t),
			dirx = avv-2buv-au*u,
			diry = buu-2auv-bv*v;
			return [0, diry, -dirx];
			}
			],{name:'', visible:false});

			var reflection = brd.create('line',
			[reflectionpoint,infty],
			{strokeWidth:1, straightFirst:false, trace:true});

	var cataustic = brd.create('curve',		var cataustic = brd.create('curve',
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	-4, 4		-4, 4
	],		],
	{strokeWidth:1, strokeColor:'red'});		{strokeWidth:3, strokeColor:'red'});
	brd.unsuspendUpdate();		brd.unsuspendUpdate();
	</source>		</source>

A WASSERMANN at 14:14, 13 January 2011

2011-01-13T14:14:00Z

← Older revision		Revision as of 14:14, 13 January 2011
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	var reflection = brd.create('line',		var reflection = brd.create('line',
	[reflectionpoint,infty],		[reflectionpoint,infty],
	{strokeWidth:1, straightFirst:false});		{strokeWidth:1, straightFirst:false, trace:true});

	var cataustic = brd.create('curve',		var cataustic = brd.create('curve',

A WASSERMANN at 14:13, 13 January 2011

2011-01-13T14:13:31Z

← Older revision		Revision as of 14:13, 13 January 2011
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	var radpoint = brd.create('point',[function(){ return -a.Value()*8;},0],{name:'radiant point'});		var radpoint = brd.create('point',[function(){ return -a.Value()*8;},0],{name:'radiant point'});
	var reflectionpoint = brd.create('glider',[-7,1,cubic],{name:'point of reflection', size:1});		var reflectionpoint = brd.create('glider',[-7,1,cubic],{name:'point of reflection', size:1});
			var dir = brd.create('segment',[radpoint,reflectionpoint],{strokeWidth:1});
	var infty = brd.create('point',		var infty = brd.create('point',
	[		[
	function(){		function(){
	//var a = dir.stdform[1], b = dir.stdform[2],		var a = dir.stdform[1], b = dir.stdform[2],
	~~var a = -reflectionpoint.X()+radpoint.X(),~~
	~~b = -reflectionpoint.Y()+radpoint.Y()~~,
	t = reflectionpoint.position,		t = reflectionpoint.position,
	u = JXG.Math.Numerics.D(cubic.X)(t),		u = JXG.Math.Numerics.D(cubic.X)(t),
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	}		}
	],{name:'', visible:false});		],{name:'', visible:false});
	~~var dir = brd.create('segment',[radpoint,reflectionpoint],{strokeWidth:1});~~

	var reflection = brd.create('line',		var reflection = brd.create('line',

A WASSERMANN at 14:11, 13 January 2011

2011-01-13T14:11:12Z

← Older revision		Revision as of 14:11, 13 January 2011
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	function(){		function(){
	//var a = dir.stdform[1], b = dir.stdform[2],		//var a = dir.stdform[1], b = dir.stdform[2],
	var a = reflectionpoint.X()-radpoint.X(),		var a = -reflectionpoint.X()+radpoint.X(),
	b = reflectionpoint.Y()-radpoint.Y(),		b = -reflectionpoint.Y()+radpoint.Y(),
	t = reflectionpoint.position,		t = reflectionpoint.position,
	u = JXG.Math.Numerics.D(cubic.X)(t),		u = JXG.Math.Numerics.D(cubic.X)(t),

A WASSERMANN at 14:10, 13 January 2011

2011-01-13T14:10:25Z

@@ Line 29: / Line 29: @@
+      [
         function(){
-             var a = dir.stdform[1], b = dir.stdform[2],
+             //var a = dir.stdform[1], b = dir.stdform[2],
                  t = reflectionpoint.position,
                  u = JXG.Math.Numerics.D(cubic.X)(t),
@@ Line 44: / Line 46: @@
         {strokeWidth:1, straightFirst:false});
 var cataustic = brd.create('curve',
                   [function(t){ return a.Value()*6*(t*t-1);},

A WASSERMANN at 14:07, 13 January 2011

2011-01-13T14:07:23Z

← Older revision		Revision as of 14:07, 13 January 2011
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	dirx = avv-2buv-au*u,		dirx = avv-2buv-au*u,
	diry = buu-2auv-bv*v;		diry = buu-2auv-bv*v;
	return [0,-diry,dirx];		return [0, diry, -dirx];
	}		}
	],{name:'', visible:false});		],{name:'', visible:false});

A WASSERMANN at 14:06, 13 January 2011

2011-01-13T14:06:59Z

← Older revision		Revision as of 14:06, 13 January 2011
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	var infty = brd.create('point',		var infty = brd.create('point',
	[		[
	function(){ return [1,1]; }		function(){
			var a = dir.stdform[1], b = dir.stdform[2],
			t = reflectionpoint.position,
			u = JXG.Math.Numerics.D(cubic.X)(t),
			v = JXG.Math.Numerics.D(cubic.Y)(t),
			dirx = avv-2buv-au*u,
			diry = buu-2auv-bv*v;
			return [0,-diry,dirx];
			}
	],{name:'', visible:false});		],{name:'', visible:false});
	var dir = brd.create('segment',[radpoint,reflectionpoint],{strokeWidth:1});		var dir = brd.create('segment',[radpoint,reflectionpoint],{strokeWidth:1});

A WASSERMANN at 14:05, 13 January 2011

2011-01-13T14:05:19Z

← Older revision		Revision as of 14:05, 13 January 2011
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	var radpoint = brd.create('point',[function(){ return -a.Value()*8;},0],{name:'radiant point'});		var radpoint = brd.create('point',[function(){ return -a.Value()*8;},0],{name:'radiant point'});
	var reflectionpoint = brd.create('glider',[-7,1,cubic],{name:'point of reflection', size:1});		var reflectionpoint = brd.create('glider',[-7,1,cubic],{name:'point of reflection', size:1});
			var infty = brd.create('point',
			[
			function(){ return [1,1]; }
			],{name:'', visible:false});
	var dir = brd.create('segment',[radpoint,reflectionpoint],{strokeWidth:1});		var dir = brd.create('segment',[radpoint,reflectionpoint],{strokeWidth:1});

	var reflection = brd.create('line',		var reflection = brd.create('line',
	[reflectionpoint,		[reflectionpoint,infty],
	~~function(){ return [0,0]; }~~
	],
	{strokeWidth:1, straightFirst:false});		{strokeWidth:1, straightFirst:false});