Tschirnhausen Cubic Catacaustic

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 catcaustic (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]

References

• Weisstein, Eric W. "Tschirnhausen Cubic Catacaustic." From MathWorld--A Wolfram Web Resource.

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 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:1, strokeColor:'red'});
brd.unsuspendUpdate();