Difference between revisions of "Trochoid"

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<html>
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The Trochoid curve (blue) and its dual curve (red).
<link rel="stylesheet" type="text/css" href="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraph.css" />
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The equation of the trochoid is
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jquery.min.js"></script>
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:<math>x = a\phi-b\sin(\phi)</math>
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraphcore.js"></script>
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:<math>y = a-b\cos(\phi)</math>
<div id="jsxgbox" class="jxgbox" style="width:500px; height:500px;"></div>
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<script language="JavaScript">
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<jsxgraph box="jsxgbox" width="550" height="500">
board = JXG.JSXGraph.initBoard('jsxgbox', {originX: 250, originY: 250, unitX: 25, unitY: 25});
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board = JXG.JSXGraph.initBoard('jsxgbox', {boundingbox:[-10,10,10,-10], axis:true});
board.createElement('axis', [[0,0],[1,0]]);
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board.suspendUpdate();
board.createElement('axis', [[0,0],[0,1]]);
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var D = JXG.Math.Numerics.D;
var a = board.createElement('slider', [[1,-1],[8,-1],[-5,1,5]], {style:6});
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var a = board.create('slider', [[1,-1],[8,-1],[-5,1,5]], {style:6,name:'a'});
var b = board.createElement('slider', [[1,-2],[8,-2],[-5,1,5]], {style:6});
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var b = board.create('slider', [[1,-2],[8,-2],[-5,1,5]], {style:6,name:'b'});
var x = function(phi) { return a.Value()*phi-b.Value()*Math.sin(phi); }
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var x = function(phi) { return a.Value()*phi-b.Value()*Math.sin(phi); }
var y = function(phi) { return a.Value()-b.Value()*Math.cos(phi); }
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var y = function(phi) { return a.Value()-b.Value()*Math.cos(phi); }
var c1 = board.createElement('curve', [x,y,'phi',-Math.PI*4,Math.PI*4],{strokeWidth:3});
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var c1 = board.create('curve', [x,y,-Math.PI*4,Math.PI*4],{strokeWidth:3});
 
   
 
   
var X = function(phi) { return board.D(y)(phi)/(y(phi)*board.D(x)(phi)-x(phi)*board.D(y)(phi)); }
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var dualCurve = function(x,y,board) {
var Y = function(phi) { return board.D(x)(phi)/(x(phi)*board.D(y)(phi)-y(phi)*board.D(x)(phi)); }
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    var X = function(phi) { return D(y)(phi)/(y(phi)*D(x)(phi)-x(phi)*D(y)(phi)); }
var c2 = board.createElement('curve', [X,Y,'phi',-Math.PI*4,Math.PI*4],{strokeWidth:3});
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    var Y = function(phi) { return D(x)(phi)/(x(phi)*D(y)(phi)-y(phi)*D(x)(phi)); }
</script>
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    return [X,Y];
</html>
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}
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var dual = dualCurve(x,y,board);
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var c2 = board.create('curve', [dual[0],dual[1],-Math.PI*1,Math.PI*1],{strokeWidth:3, strokeColor:'red'});
 +
board.unsuspendUpdate();
 +
</jsxgraph>
 +
 
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===References===
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* [http://en.wikipedia.org/wiki/Trochoid http://en.wikipedia.org/wiki/Trochoid]
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* [http://en.wikipedia.org/wiki/Dual_curve http://en.wikipedia.org/wiki/Dual_curve]
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===The underlying JavaScript code===
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<source lang="javascript">
 +
board = JXG.JSXGraph.initBoard('jsxgbox', {boundingbox:[-10,10,10,-10], axis:true});
 +
board.suspendUpdate();
 +
var D = JXG.Math.Numerics.D;
 +
var a = board.create('slider', [[1,-1],[8,-1],[-5,1,5]], {style:6,name:'a'});
 +
var b = board.create('slider', [[1,-2],[8,-2],[-5,1,5]], {style:6,name:'b'});
 +
var x = function(phi) { return a.Value()*phi-b.Value()*Math.sin(phi); }
 +
var y = function(phi) { return a.Value()-b.Value()*Math.cos(phi); }
 +
var c1 = board.create('curve', [x,y,-Math.PI*4,Math.PI*4],{strokeWidth:3});
 +
 +
var dualCurve = function(x,y,board) {
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    var X = function(phi) { return D(y)(phi)/(y(phi)*D(x)(phi)-x(phi)*D(y)(phi)); }
 +
    var Y = function(phi) { return D(x)(phi)/(x(phi)*D(y)(phi)-y(phi)*D(x)(phi)); }
 +
    return [X,Y];
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}
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var dual = dualCurve(x,y,board);
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var c2 = board.create('curve', [dual[0],dual[1],-Math.PI*1,Math.PI*1],{strokeWidth:3, strokeColor:'red'});
 +
board.unsuspendUpdate();
 +
</source>
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[[Category:Examples]]
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[[Category:Curves]]

Latest revision as of 17:12, 20 February 2013

The Trochoid curve (blue) and its dual curve (red). The equation of the trochoid is

[math]x = a\phi-b\sin(\phi)[/math]
[math]y = a-b\cos(\phi)[/math]

References

The underlying JavaScript code

board = JXG.JSXGraph.initBoard('jsxgbox', {boundingbox:[-10,10,10,-10], axis:true});
board.suspendUpdate();
var D = JXG.Math.Numerics.D;
var a = board.create('slider', [[1,-1],[8,-1],[-5,1,5]], {style:6,name:'a'});
var b = board.create('slider', [[1,-2],[8,-2],[-5,1,5]], {style:6,name:'b'});
var x = function(phi) { return a.Value()*phi-b.Value()*Math.sin(phi); }
var y = function(phi) { return a.Value()-b.Value()*Math.cos(phi); }
var c1 = board.create('curve', [x,y,-Math.PI*4,Math.PI*4],{strokeWidth:3});
 
var dualCurve = function(x,y,board) {
    var X = function(phi) { return D(y)(phi)/(y(phi)*D(x)(phi)-x(phi)*D(y)(phi)); }
    var Y = function(phi) { return D(x)(phi)/(x(phi)*D(y)(phi)-y(phi)*D(x)(phi)); }
    return [X,Y];
}
var dual = dualCurve(x,y,board);
var c2 = board.create('curve', [dual[0],dual[1],-Math.PI*1,Math.PI*1],{strokeWidth:3, strokeColor:'red'});
board.unsuspendUpdate();