Difference between revisions of "Tschirnhausen Cubic Catacaustic"
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− | + | 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(); | ||
})(); | })(); | ||
Line 32: | Line 37: | ||
===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 11:06, 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();