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Euler's spiral (Clothoid)
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<iframe 
    src="https://jsxgraph.uni-bayreuth.de/share/iframe/eulers-spiral-clothoid" 
    style="border: 1px solid black; overflow: hidden; width: 550px; aspect-ratio: 55 / 65;" 
    name="JSXGraph example: Euler's spiral (Clothoid)" 
    allowfullscreen
></iframe>
This code has to 
<div id="board-0-wrapper" class="jxgbox-wrapper " style="width: 100%; ">
   <div id="board-0" class="jxgbox" style="aspect-ratio: 1 / 1; width: 100%;" data-ar="1 / 1"></div>
</div>

<script type = "text/javascript"> 
    /*
    This example is licensed under a 
    Creative Commons Attribution 4.0 International License.
    https://creativecommons.org/licenses/by/4.0/
    
    Please note you have to mention 
    The Center of Mobile Learning with Digital Technology
    in the credits.
    */
    
    const BOARDID = 'board-0';

    const board = JXG.JSXGraph.initBoard(BOARDID, {
        boundingbox: [-10, 10, 10, -10],
        axis: true
    });
    
    var k = board.create('slider', [[1, 8], [5, 8], [0, 2, 8]], {name: 'k'});
    var len = board.create('slider', [[1, 7], [5, 7], [1, 2, 3.2]], {name: 'len'});
    
    var c = board.create('curve', [
            (t) => k.Value() * JXG.Math.Numerics.I([0, t], (u) => Math.sin(u * u * 0.5)), // Curve x(t)
            (t) => k.Value() * JXG.Math.Numerics.I([0, t], (u) => Math.cos(u * u * 0.5)), // Curve y(t)
            () => -len.Value() * Math.PI, // Start
            () => len.Value() * Math.PI], // End
        {strokewidth: 1});
    
    var p = board.create('glider', [1, 3, c], {name: ''});
    var t = board.create('tangent', [p], {dash: 3, strokeWidth: 1, strokeColor: 'red'});
    
 </script> 
/*
This example is licensed under a 
Creative Commons Attribution 4.0 International License.
https://creativecommons.org/licenses/by/4.0/

Please note you have to mention 
The Center of Mobile Learning with Digital Technology
in the credits.
*/

const BOARDID = 'your_div_id'; // Insert your id here!

const board = JXG.JSXGraph.initBoard(BOARDID, {
    boundingbox: [-10, 10, 10, -10],
    axis: true
});

var k = board.create('slider', [[1, 8], [5, 8], [0, 2, 8]], {name: 'k'});
var len = board.create('slider', [[1, 7], [5, 7], [1, 2, 3.2]], {name: 'len'});

var c = board.create('curve', [
        (t) => k.Value() * JXG.Math.Numerics.I([0, t], (u) => Math.sin(u * u * 0.5)), // Curve x(t)
        (t) => k.Value() * JXG.Math.Numerics.I([0, t], (u) => Math.cos(u * u * 0.5)), // Curve y(t)
        () => -len.Value() * Math.PI, // Start
        () => len.Value() * Math.PI], // End
    {strokewidth: 1});

var p = board.create('glider', [1, 3, c], {name: ''});
var t = board.create('tangent', [p], {dash: 3, strokeWidth: 1, strokeColor: 'red'});

<jsxgraph width="100%" aspect-ratio="1 / 1" title="Euler's spiral (Clothoid)" description="This construction was copied from JSXGraph examples database: BTW HERE SHOULD BE A GENERATED LINKuseGlobalJS="false">
   /*
   This example is licensed under a 
   Creative Commons Attribution 4.0 International License.
   https://creativecommons.org/licenses/by/4.0/
   
   Please note you have to mention 
   The Center of Mobile Learning with Digital Technology
   in the credits.
   */
   
   const board = JXG.JSXGraph.initBoard(BOARDID, {
       boundingbox: [-10, 10, 10, -10],
       axis: true
   });
   
   var k = board.create('slider', [[1, 8], [5, 8], [0, 2, 8]], {name: 'k'});
   var len = board.create('slider', [[1, 7], [5, 7], [1, 2, 3.2]], {name: 'len'});
   
   var c = board.create('curve', [
           (t) => k.Value() * JXG.Math.Numerics.I([0, t], (u) => Math.sin(u * u * 0.5)), // Curve x(t)
           (t) => k.Value() * JXG.Math.Numerics.I([0, t], (u) => Math.cos(u * u * 0.5)), // Curve y(t)
           () => -len.Value() * Math.PI, // Start
           () => len.Value() * Math.PI], // End
       {strokewidth: 1});
   
   var p = board.create('glider', [1, 3, c], {name: ''});
   var t = board.create('tangent', [p], {dash: 3, strokeWidth: 1, strokeColor: 'red'});
   
</jsxgraph>
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 $$ \begin{aligned} x(t) &= k\cdot\int_0^t \sin(\frac{u^2}{2}) du \\ y(t) &= k\cdot\int_0^t \cos(\frac{u^2}{2}) du \end{aligned} $$ 
// Define the id of your board in BOARDID

const board = JXG.JSXGraph.initBoard(BOARDID, {
    boundingbox: [-10, 10, 10, -10],
    axis: true
});

var k = board.create('slider', [[1, 8], [5, 8], [0, 2, 8]], {name: 'k'});
var len = board.create('slider', [[1, 7], [5, 7], [1, 2, 3.2]], {name: 'len'});

var c = board.create('curve', [
        (t) => k.Value() * JXG.Math.Numerics.I([0, t], (u) => Math.sin(u * u * 0.5)), // Curve x(t)
        (t) => k.Value() * JXG.Math.Numerics.I([0, t], (u) => Math.cos(u * u * 0.5)), // Curve y(t)
        () => -len.Value() * Math.PI, // Start
        () => len.Value() * Math.PI], // End
    {strokewidth: 1});

var p = board.create('glider', [1, 3, c], {name: ''});
var t = board.create('tangent', [p], {dash: 3, strokeWidth: 1, strokeColor: 'red'});

license

This example is licensed under a Creative Commons Attribution 4.0 International License.
Please note you have to mention The Center of Mobile Learning with Digital Technology in the credits.