Difference between revisions of "Tschirnhausen Cubic Catacaustic"

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Line 29: Line 29:
 
     [
 
     [
 
       function(){  
 
       function(){  
             var a = dir.stdform[1], b = dir.stdform[2],
+
             //var a = dir.stdform[1], b = dir.stdform[2],
 +
            var a = reflectionpoint.X()-radpoint.X(),
 +
                b = reflectionpoint.Y()-radpoint.Y(),
 
                 t = reflectionpoint.position,
 
                 t = reflectionpoint.position,
 
                 u = JXG.Math.Numerics.D(cubic.X)(t),  
 
                 u = JXG.Math.Numerics.D(cubic.X)(t),  
Line 44: Line 46:
 
       {strokeWidth:1, straightFirst:false});
 
       {strokeWidth:1, straightFirst:false});
  
/*
 
      [
 
        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 -(reflectionpoint.X()*dirx+reflectionpoint.Y()*diry)/reflectionpoint.Z();
 
        },
 
        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);
 
            return a*v*v-2*b*u*v-a*u*u ;
 
        },
 
        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);
 
            return b*u*u-2*a*u*v-b*v*v;
 
        }
 
      ],
 
*/
 
 
var cataustic = brd.create('curve',
 
var cataustic = brd.create('curve',
 
                 [function(t){ return a.Value()*6*(t*t-1);},
 
                 [function(t){ return a.Value()*6*(t*t-1);},

Revision as of 15:10, 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();