Difference between revisions of "Tschirnhausen Cubic Catacaustic"

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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(){

Latest revision as of 16:26, 13 January 2011

The Tschirnhausen cubic (black curve) is defined parametrically as

[math] x = a3(t^2-3) [/math]
[math] y = at(t^2-3) [/math]

Its catacaustic (red curve) with radiant point [math](-8a,p)[/math] is the semicubical parabola with parametric equations

[math] x = a6(t^2-1) [/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 get a feeling why the red curve is called envelope of the blue line if you drag the point of reflection.

References

The underlying JavaScript code

var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-10,10,10,-10], keepaspectratio:true, axis:true});
brd.suspendUpdate();
var a = brd.create('slider',[[-5,6],[5,6],[-5,1,5]], {name:'a'});

var cubic = brd.create('curve',
             [function(t){ return a.Value()*3*(t*t-3);},
              function(t){ return a.Value()*t*(t*t-3);},
              -5, 5
             ],
             {strokeWidth:1, strokeColor:'black'});

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 = a*v*v-2*b*u*v-a*u*u,
                diry = b*u*u-2*a*u*v-b*v*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',
                 [function(t){ return a.Value()*6*(t*t-1);},
                  function(t){ return a.Value()*4*t*t*t;},
                 -4, 4
                 ],
                 {strokeWidth:3, strokeColor:'red'});
brd.unsuspendUpdate();