Difference between revisions of "Tschirnhausen Cubic Catacaustic"
From JSXGraph Wiki
Jump to navigationJump to searchA WASSERMANN (talk | contribs) |
A WASSERMANN (talk | contribs) |
||
Line 11: | Line 11: | ||
:<math> y = a4t^3 </math> | :<math> y = a4t^3 </math> | ||
+ | You can drag the ''point of reflection''. | ||
<jsxgraph width="600" height="600"> | <jsxgraph width="600" height="600"> | ||
(function(){ | (function(){ | ||
Line 25: | Line 26: | ||
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' | + | var reflectionpoint = brd.create('glider',[-7,1,cubic],{name:'point of reflection'}); |
var dir = brd.create('segment',[radpoint,reflectionpoint],{strokeWidth:1}); | var dir = brd.create('segment',[radpoint,reflectionpoint],{strokeWidth:1}); | ||
var infty = brd.create('point', | var infty = brd.create('point', | ||
Line 49: | Line 50: | ||
-4, 4 | -4, 4 | ||
], | ], | ||
− | {strokeWidth: | + | {strokeWidth:3, strokeColor:'red'}); |
brd.unsuspendUpdate(); | brd.unsuspendUpdate(); | ||
})(); | })(); | ||
Line 70: | Line 71: | ||
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'}); | ||
+ | var dir = brd.create('segment',[radpoint,reflectionpoint],{strokeWidth:1}); | ||
+ | var infty = brd.create('point', | ||
+ | [ | ||
+ | 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 [0, diry, -dirx]; | ||
+ | } | ||
+ | ],{name:'', visible:false}); | ||
+ | |||
+ | var reflection = brd.create('line', | ||
+ | [reflectionpoint,infty], | ||
+ | {strokeWidth:1, straightFirst:false, trace:true}); | ||
var cataustic = brd.create('curve', | var cataustic = brd.create('curve', | ||
Line 76: | Line 95: | ||
-4, 4 | -4, 4 | ||
], | ], | ||
− | {strokeWidth: | + | {strokeWidth:3, strokeColor:'red'}); |
brd.unsuspendUpdate(); | brd.unsuspendUpdate(); | ||
</source> | </source> |
Revision as of 15:15, 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]
You can drag the point of reflection.
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 reflectionpoint = brd.create('glider',[-7,1,cubic],{name:'point of reflection'});
var dir = brd.create('segment',[radpoint,reflectionpoint],{strokeWidth:1});
var infty = brd.create('point',
[
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 [0, diry, -dirx];
}
],{name:'', visible:false});
var reflection = brd.create('line',
[reflectionpoint,infty],
{strokeWidth:1, straightFirst:false, trace:true});
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:3, strokeColor:'red'});
brd.unsuspendUpdate();