Tschirnhausen Cubic Catacaustic

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Revision as of 14:19, 13 January 2011 by A WASSERMANN (talk | contribs)

The Tschirnhausen cubic (black curve) is defined parametrically as

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

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

[math]\displaystyle{ x = a6(t^2-1) }[/math]
[math]\displaystyle{ 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.


The underlying JavaScript code

var brd = JXG.JSXGraph.initBoard('jxgbox',{boundingbox:[-10,10,10,-10], keepaspectratio:true, axis:true});
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',
            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',
       {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'});