Tschirnhausen Cubic Catacaustic: Difference between revisions

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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">
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                 ],
                 ],
                 {strokeWidth:1, strokeColor:'red'});
                 {strokeWidth:1, strokeColor:'red'});
brd.unsuspendUpdate();
brd.unsuspendUpdate();
})();
})();
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===References===
===References===
 
* [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 10:06, 13 January 2011

The Tschirnhausen cubic is defined parametrically as

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

Its catcaustic 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

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();