Difference between revisions of "Tschirnhausen Cubic Catacaustic"
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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 dir = brd.create('segment',[radpoint,reflectionpoint],{strokeWidth:1}); | ||
var infty = brd.create('point', | var infty = brd.create('point', | ||
[ | [ | ||
function(){ | function(){ | ||
| − | + | var a = dir.stdform[1], b = dir.stdform[2], | |
| − | |||
| − | |||
t = reflectionpoint.position, | t = reflectionpoint.position, | ||
u = JXG.Math.Numerics.D(cubic.X)(t), | u = JXG.Math.Numerics.D(cubic.X)(t), | ||
| Line 40: | Line 39: | ||
} | } | ||
],{name:'', visible:false}); | ],{name:'', visible:false}); | ||
| − | |||
var reflection = brd.create('line', | var reflection = brd.create('line', | ||
Revision as of 16:13, 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();
