Difference between revisions of "Tschirnhausen Cubic Catacaustic"

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A semicubical parabola is a curve defined parametrically as
+
The Tschirnhausen cubic is defined parametrically as
  
:<math> x = t^2 </math>
+
:<math> x = a3(t^2-3) </math>
  
:<math> y = at^3 </math>
+
:<math> y = at(t^2-3) </math>
 +
Its catcaustic 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>
  
 
<jsxgraph width="600" height="600">
 
<jsxgraph width="600" height="600">
Line 26: Line 32:
 
                 ],
 
                 ],
 
                 {strokeWidth:1, strokeColor:'red'});
 
                 {strokeWidth:1, strokeColor:'red'});
 
 
brd.unsuspendUpdate();
 
brd.unsuspendUpdate();
 
})();
 
})();
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===References===
 
===References===
 
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* [http://mathworld.wolfram.com/TschirnhausenCubicCatacaustic.html Weisstein, Eric W. "Tschirnhausen Cubic Catacaustic." From MathWorld--A Wolfram Web Resource.]
 
===The underlying JavaScript code===
 
===The underlying JavaScript code===
 
<source lang="javascript">
 
<source lang="javascript">
 +
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();
 
</source>
 
</source>
  
 
[[Category:Examples]]
 
[[Category:Examples]]
 
[[Category:Curves]]
 
[[Category:Curves]]

Revision as of 12:06, 13 January 2011

The Tschirnhausen cubic is defined parametrically as

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

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