Difference between revisions of "Elliptic curves"

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An elliptic curve can be written as a plane algebraic curve defined by an equation of the form
 
An elliptic curve can be written as a plane algebraic curve defined by an equation of the form
 
:<math>    y^2=x^3+ax+b\,. </math>
 
:<math>    y^2=x^3+ax+b\,. </math>
<html>
+
<jsxgraph width="600" height="400" box="jxgbox">
<link rel="stylesheet" type="text/css" href="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraph.css" />
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JXG.Options.axis.strokeColor = '#cccccc';
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/prototype.js"></script>
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brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraphcore.js"></script>
 
<div id="jxgbox" class="jxgbox" style="width:600px; height:400px;"></div>
 
<script language="JavaScript">
 
brd = JXG.JSXGraph.initBoard('jxgbox', {originX: 300, originY: 200, unitX: 20, unitY: 20});
 
 
brd.suspendUpdate();
 
brd.suspendUpdate();
brd.createElement('axis', [[0,0], [1,0]], {});
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var b = brd.create('slider',[[1,-3],[10,-3],[-10,2.10,10]],{name:'a'});
brd.createElement('axis', [[0,0], [0,1]], {});
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var a = brd.create('slider',[[1,-4],[10,-4],[-10,-9.52,10]],{name:'b'});
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var b = brd.createElement('slider',[[1,-3],[10,-3],[-10,1,10]],{name:'a'});
 
brd.createElement('text',[12,-3,'(a)']);
 
var a = brd.createElement('slider',[[1,-4],[10,-4],[-10,-8,10]],{name:'b'});
 
brd.createElement('text',[12,-4,'(b)']);
 
 
 
var c1 = brd.createElement('curve', ['x', function(x){ return Math.sqrt(x*x*x+a.Value()*x+b.Value()); }, 'x'], {strokeWidth:3,strokeColor:'black'});
 
var c2 = brd.createElement('curve', ['x', function(x){ return -Math.sqrt(x*x*x+a.Value()*x+b.Value()); }, 'x'], {strokeWidth:3,strokeColor:'black'});
 
  
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var c2 = brd.create('functiongraph', [function(x){ return -Math.sqrt(x*x*x+a.Value()*x+b.Value()); }],
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                      {strokeWidth:3,strokeColor:'black'});
 
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brd.unsuspendUpdate();
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</jsxgraph>  
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<source lang="html4strict">
 
<link rel="stylesheet" type="text/css" href="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraph.css" />
 
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/prototype.js"></script>
 
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jsxgraphcore.js"></script>
 
<div id="jxgbox" class="jxgbox" style="width:600px; height:400px;"></div>
 
</source>
 
 
<source lang="javascript">
 
<source lang="javascript">
brd = JXG.JSXGraph.initBoard('jxgbox', {originX: 300, originY: 200, unitX: 20, unitY: 20});
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JXG.Options.axis.strokeColor = '#cccccc';
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brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});
 
brd.suspendUpdate();
 
brd.suspendUpdate();
brd.createElement('axis', [[0,0], [1,0]], {});
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var b = brd.create('slider',[[1,-3],[10,-3],[-10,0,10]],{name:'a'});
brd.createElement('axis', [[0,0], [0,1]], {});
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var a = brd.create('slider',[[1,-4],[10,-4],[-10,0,10]],{name:'b'});
//
 
var b = brd.createElement('slider',[[1,-3],[10,-3],[-10,1,10]],{name:'a'});
 
brd.createElement('text',[12,-3,'(a)']);
 
var a = brd.createElement('slider',[[1,-4],[10,-4],[-10,-8,10]],{name:'b'});
 
brd.createElement('text',[12,-4,'(b)']);
 
 
 
var c1 = brd.createElement('curve', ['x', function(x){ return Math.sqrt(x*x*x+a.Value()*x+b.Value()); }, 'x'], {strokeWidth:3,strokeColor:'black'});
 
var c2 = brd.createElement('curve', ['x', function(x){ return -Math.sqrt(x*x*x+a.Value()*x+b.Value()); }, 'x'], {strokeWidth:3,strokeColor:'black'});
 
  
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var c1 = brd.create('functiongraph', [function(x){ return Math.sqrt(x*x*x+a.Value()*x+b.Value()); }],
 +
                      {strokeWidth:3,strokeColor:'black'});
 +
var c2 = brd.create('functiongraph', [function(x){ return -Math.sqrt(x*x*x+a.Value()*x+b.Value()); }],
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                      {strokeWidth:3,strokeColor:'black'});
 
brd.unsuspendUpdate();
 
brd.unsuspendUpdate();
 
</source>
 
</source>
 +
 +
===References===
 +
* [http://en.wikipedia.org/wiki/Elliptic_curves http://en.wikipedia.org/wiki/Elliptic_curves]
 +
  
 
[[Category:Examples]]
 
[[Category:Examples]]
 +
[[Category:Curves]]

Latest revision as of 13:42, 28 February 2014

An elliptic curve can be written as a plane algebraic curve defined by an equation of the form

[math] y^2=x^3+ax+b\,. [/math]

JXG.Options.axis.strokeColor = '#cccccc';
brd = JXG.JSXGraph.initBoard('jxgbox', {boundingbox: [-15, 10, 15, -10], axis:true});
brd.suspendUpdate();
var b = brd.create('slider',[[1,-3],[10,-3],[-10,0,10]],{name:'a'});
var a = brd.create('slider',[[1,-4],[10,-4],[-10,0,10]],{name:'b'});

var c1 = brd.create('functiongraph', [function(x){ return Math.sqrt(x*x*x+a.Value()*x+b.Value()); }],
                       {strokeWidth:3,strokeColor:'black'});
var c2 = brd.create('functiongraph', [function(x){ return -Math.sqrt(x*x*x+a.Value()*x+b.Value()); }], 
                       {strokeWidth:3,strokeColor:'black'});
brd.unsuspendUpdate();

References