# Tschirnhausen Cubic Catacaustic: Difference between revisions

The Tschirnhausen cubic is defined parametrically as

$\displaystyle{ x = a3(t^2-3) }$
$\displaystyle{ y = at(t^2-3) }$

Its catcaustic with radiant point $\displaystyle{ (-8a,p) }$ is the semicubical parabola with parametric equations

$\displaystyle{ x = a6(t^2-1) }$
$\displaystyle{ y = a4t^3 }$

### 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;},
-2, 2
],
{strokeWidth:1, strokeColor:'red'});
brd.unsuspendUpdate();