Difference between revisions of "Euler's spiral (Clothoid)"

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Line 9: Line 9:
  
 
<jsxgraph width="500" height="500" box="box2">
 
<jsxgraph width="500" height="500" box="box2">
  b2 = JXG.JSXGraph.initBoard('box2', {originX: 250, originY: 250, unitX: 25, unitY: 25});
+
  var b2 = JXG.JSXGraph.initBoard('box2', {originX: 250, originY: 250, unitX: 25, unitY: 25});
  var k1 = b2.createElement('slider', [[1,8],[5,8],[0,2,8]],{name:'k'});
+
  var k1 = b2.create('slider', [[1,8],[5,8],[0,2,8]],{name:'k'});
  var len = b2.createElement('slider', [[1,7],[5,7],[1,2,3.2]],{name:'len'});
+
  var len = b2.create('slider', [[1,7],[5,7],[1,2,3.2]],{name:'len'});
  var c = b2.createElement('curve',  
+
  var c = b2.create('curve',  
 
             [function(t){ return k1.Value()*b2.I([0,t],function(u){return Math.sin(u*u*0.5);}); },  
 
             [function(t){ return k1.Value()*b2.I([0,t],function(u){return Math.sin(u*u*0.5);}); },  
 
               function(t){ return k1.Value()*b2.I([0,t],function(u){return Math.cos(u*u*0.5);}); },
 
               function(t){ return k1.Value()*b2.I([0,t],function(u){return Math.cos(u*u*0.5);}); },
 
               function(){return -len.Value()*Math.PI;}, function(){return len.Value()*Math.PI;}],
 
               function(){return -len.Value()*Math.PI;}, function(){return len.Value()*Math.PI;}],
 
             {strokewidth:1});     
 
             {strokewidth:1});     
  var p = b2.createElement('glider',[c],{name:''});
+
  var p = b2.create('glider',[c],{name:''});
  var t = b2.createElement('tangent',[p],{dash:3,strokeWidth:1,strokeColor:'red'});
+
  var t = b2.create('tangent',[p],{dash:3,strokeWidth:1,strokeColor:'red'});
 
</jsxgraph>
 
</jsxgraph>
  
Line 25: Line 25:
 
<source lang="xml">
 
<source lang="xml">
 
<jsxgraph width="500" height="500" box="box2">
 
<jsxgraph width="500" height="500" box="box2">
  b2 = JXG.JSXGraph.initBoard('box2', {originX: 250, originY: 250, unitX: 25, unitY: 25});
+
  var b2 = JXG.JSXGraph.initBoard('box2', {originX: 250, originY: 250, unitX: 25, unitY: 25});
  var k1 = b2.createElement('slider', [[1,8],[5,8],[0,2,8]],{name:'k'});
+
  var k1 = b2.create('slider', [[1,8],[5,8],[0,2,8]],{name:'k'});
  var len = b2.createElement('slider', [[1,7],[5,7],[1,2,3.2]],{name:'len'});
+
  var len = b2.create('slider', [[1,7],[5,7],[1,2,3.2]],{name:'len'});
  var c = b2.createElement('curve',  
+
  var c = b2.create('curve',  
 
             [function(t){ return k1.Value()*b2.I([0,t],function(u){return Math.sin(u*u*0.5);}); },  
 
             [function(t){ return k1.Value()*b2.I([0,t],function(u){return Math.sin(u*u*0.5);}); },  
 
               function(t){ return k1.Value()*b2.I([0,t],function(u){return Math.cos(u*u*0.5);}); },
 
               function(t){ return k1.Value()*b2.I([0,t],function(u){return Math.cos(u*u*0.5);}); },
 
               function(){return -len.Value()*Math.PI;}, function(){return len.Value()*Math.PI;}],
 
               function(){return -len.Value()*Math.PI;}, function(){return len.Value()*Math.PI;}],
 
             {strokewidth:1});       
 
             {strokewidth:1});       
  var p = b2.createElement('glider',[c],{name:''});
+
  var p = b2.create('glider',[c],{name:''});
  var t = b2.createElement('tangent',[p],{dash:3,strokeWidth:1,strokeColor:'red'});
+
  var t = b2.create('tangent',[p],{dash:3,strokeWidth:1,strokeColor:'red'});
 
</jsxgraph>
 
</jsxgraph>
 
</source>
 
</source>

Revision as of 21:54, 29 September 2010

Euler's spiral is defined as a curve whose curvature changes linearly with its curve length. Other names for the spiral are clothoid and spiral of Cornu or Cornu spiral.

Euler spirals are widely used as transition curve in rail track / highway engineering for connecting and transiting the geometry between a tangent and a circular curve.

The curve can be described in parametric form by

[math] x = k\cdot\int_0^t \sin(\frac{u^2}{2}) du\; [/math]
[math] y = k\cdot\int_0^t \cos(\frac{u^2}{2}) du\; [/math]

The JavaScript code to produce this picture

<jsxgraph width="500" height="500" box="box2">
 var b2 = JXG.JSXGraph.initBoard('box2', {originX: 250, originY: 250, unitX: 25, unitY: 25});
 var k1 = b2.create('slider', [[1,8],[5,8],[0,2,8]],{name:'k'});
 var len = b2.create('slider', [[1,7],[5,7],[1,2,3.2]],{name:'len'});
 var c = b2.create('curve', 
             [function(t){ return k1.Value()*b2.I([0,t],function(u){return Math.sin(u*u*0.5);}); }, 
              function(t){ return k1.Value()*b2.I([0,t],function(u){return Math.cos(u*u*0.5);}); },
              function(){return -len.Value()*Math.PI;}, function(){return len.Value()*Math.PI;}],
             {strokewidth:1});      
 var p = b2.create('glider',[c],{name:''});
 var t = b2.create('tangent',[p],{dash:3,strokeWidth:1,strokeColor:'red'});
</jsxgraph>

External links