Difference between revisions of "Tschirnhausen Cubic Catacaustic"

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Line 26: Line 26:
 
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});
  

Revision as of 16:05, 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 catcaustic (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]

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 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();