Difference between revisions of "Circles on circles"

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This is an example of a parametric curve plot. It shows the orbit of a point on a circle. The circle rotates on a circle which again rotates on the unit circle. The resulting curve is described by the function
 
This is an example of a parametric curve plot. It shows the orbit of a point on a circle. The circle rotates on a circle which again rotates on the unit circle. The resulting curve is described by the function
 
:<math> [0,2\pi]\to{\mathbf R}^2, \quad t\mapsto {\cos(t)\choose \sin(t)}+c_1{\cos(f_1t)\choose \sin(f_1t)}+c_2{\cos(f_2t)\choose \sin(f_2t)}</math>
 
:<math> [0,2\pi]\to{\mathbf R}^2, \quad t\mapsto {\cos(t)\choose \sin(t)}+c_1{\cos(f_1t)\choose \sin(f_1t)}+c_2{\cos(f_2t)\choose \sin(f_2t)}</math>
The sliders to adjust the parameters of this curve are from the jQuery UI package, see [http://jqueryui.com http://jqueryui.com]. This example shows the seamless integration of JSXGraph into the web page.
+
 
 +
This example shows the seamless integration of JSXGraph into the web page.
  
 
<html>
 
<html>
<link rel="Stylesheet" type="text/css" href="http://jsxgraph.uni-bayreuth.de/distrib/css/ui-lightness/jquery-ui-1.7.2.custom.css"/>
+
<div style="margin:5px">
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jquery.min.js"></script>
 
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jquery-ui.min.js"></script>
 
 
 
<style type="text/css">
 
#slider-frame > div.sliders { padding: 10px !important; };
 
</style>
 
<script type="text/javascript">
 
$(function() {
 
$("#sliderc1").slider({
 
orientation: "horizontal",range: "min",min: 0,max: 100,value: 60,
 
slide: function(event, ui) {
 
$("#c1").val(ui.value*0.01);
 
                                c1 = ui.value*0.01;
 
                                board.updateQuality = board.BOARD_QUALITY_HIGH;
 
                                board.update();
 
}
 
});
 
$("#sliderf1").slider({
 
orientation: "horizontal",range: "min",min: 1,max: 100,value: 7,
 
slide: function(event, ui) {
 
$("#f1").val(ui.value);
 
                                f1 = ui.value;
 
                                board.updateQuality = board.BOARD_QUALITY_HIGH;
 
                                board.update();
 
}
 
});
 
$("#c1").val($("#sliderc1").slider("value")*0.01);
 
$("#f1").val($("#sliderf1").slider("value"));
 
 
 
$("#sliderc2").slider({
 
orientation: "horizontal",range: "min",min: 0,max: 100,value: 0,
 
slide: function(event, ui) {
 
$("#c2").val(ui.value*0.01);
 
                                c2 = ui.value*0.01;
 
                                board.updateQuality = board.BOARD_QUALITY_HIGH;
 
                                board.update();
 
}
 
});
 
$("#sliderf2").slider({
 
orientation: "horizontal",range: "min",min: 1,max: 100,value: 17,
 
slide: function(event, ui) {
 
$("#f2").val(ui.value);
 
                                f2 = ui.value;
 
                                board.updateQuality = board.BOARD_QUALITY_HIGH;
 
                                board.update();
 
}
 
});
 
$("#c2").val($("#sliderc2").slider("value")*0.01);
 
$("#f2").val($("#sliderf2").slider("value"));
 
});
 
</script>
 
 
 
<div class="sliders" style="margin:5px">
 
 
   <p>
 
   <p>
 
     <label for="c1">c1:</label>
 
     <label for="c1">c1:</label>
     <input type="text" id="c1" style="border:0; color:#f6931f; font-weight:bold;" />
+
     <input type="range" id="c1" style="border:0; color:#f6931f; font-weight:bold;"  
 +
        min="0" max="100" value="60"
 +
        oninput="c1 = this.value*0.01; board.update();"
 +
    />
 
     <label for="f1">f1:</label>
 
     <label for="f1">f1:</label>
     <input type="text" id="f1" style="border:0; color:#f6931f; font-weight:bold;" />
+
     <input type="range" id="f1" style="border:0; color:#f6931f; font-weight:bold;"  
 +
        min="1" max="100" value="7"
 +
        oninput="f1 = this.value; board.update();"
 +
    />
 
     <label for="c2">c2:</label>
 
     <label for="c2">c2:</label>
     <input type="text" id="c2" style="border:0; color:#f6931f; font-weight:bold;" />
+
     <input type="range" id="c2" style="border:0; color:#f6931f; font-weight:bold;"  
 +
        min="0" max="100" value="0"
 +
        oninput="c2 = this.value*0.01;
 +
                  board.updateQuality = board.BOARD_QUALITY_HIGH;
 +
                  board.update();"
 +
    />
 
     <label for="f2">f2:</label>
 
     <label for="f2">f2:</label>
     <input type="text" id="f2" style="border:0; color:#f6931f; font-weight:bold;" />
+
     <input type="range" id="f2" style="border:0; color:#f6931f; font-weight:bold;"  
 +
        min="1" max="100" value="17"
 +
        oninput="f2 = this.value; board.update();"
 +
    />
 
   </p>
 
   </p>
  <div id="sliderc1" style="width:300px;margin:10px;"></div>
 
  <div id="sliderf1" style="width:300px;margin:10px;"></div>
 
  <div id="sliderc2" style="width:300px;margin:10px;"></div>
 
  <div id="sliderf2" style="width:300px;margin:10px;"></div>
 
 
</div>
 
</div>
 
</html>
 
</html>
Line 83: Line 41:
 
               function(t) { return Math.cos(t)+ c1*Math.cos(f1*t)+ c2*Math.cos(f2*t);},
 
               function(t) { return Math.cos(t)+ c1*Math.cos(f1*t)+ c2*Math.cos(f2*t);},
 
               function(t) { return Math.sin(t)+ c1*Math.sin(f1*t)+ c2*Math.sin(f2*t);},
 
               function(t) { return Math.sin(t)+ c1*Math.sin(f1*t)+ c2*Math.sin(f2*t);},
               0,2.02*Math.PI],{strokeWidth:2});
+
               0,2.02*Math.PI], {strokeWidth:2});
 
</jsxgraph>
 
</jsxgraph>
  
Line 101: Line 59:
  
 
===The source code of this construction===
 
===The source code of this construction===
This is the first experiment with the jQuery UI package. So, the code may not be optimized, yet. The main difficulty is to read the values of the sliders. This is done via four JavaScript variables <math>c1, c2, f1, f2</math>.
+
The main difficulty is to read the values of the sliders.  
 +
This is done via four JavaScript variables <math>c1, c2, f1, f2</math>.
 
<source lang="html4strict">
 
<source lang="html4strict">
<link rel="Stylesheet" type="text/css" href="http://jsxgraph.uni-bayreuth.de/distrib/css/ui-lightness/jquery-ui-1.7.2.custom.css"/>
+
<div style="margin:5px">
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jquery.min.js"></script>
 
<script type="text/javascript" src="http://jsxgraph.uni-bayreuth.de/distrib/jquery-ui.min.js"></script>
 
 
 
<style type="text/css">
 
#slider-frame > div.sliders { padding: 10px !important; };
 
</style>
 
<script type="text/javascript">
 
$(function() {
 
$("#sliderc1").slider({
 
orientation: "horizontal",range: "min",min: 0,max: 100,value: 60,
 
slide: function(event, ui) {
 
$("#c1").val(ui.value*0.01);
 
                                c1 = ui.value*0.01;
 
                                board.updateQuality = board.BOARD_QUALITY_HIGH;
 
                                board.update();
 
}
 
});
 
$("#sliderf1").slider({
 
orientation: "horizontal",range: "min",min: 1,max: 100,value: 7,
 
slide: function(event, ui) {
 
$("#f1").val(ui.value);
 
                                f1 = ui.value;
 
                                board.updateQuality = board.BOARD_QUALITY_HIGH;
 
                                board.update();
 
}
 
});
 
$("#c1").val($("#sliderc1").slider("value")*0.01);
 
$("#f1").val($("#sliderf1").slider("value"));
 
 
 
$("#sliderc2").slider({
 
orientation: "horizontal",range: "min",min: 0,max: 100,value: 0,
 
slide: function(event, ui) {
 
$("#c2").val(ui.value*0.01);
 
                                c2 = ui.value*0.01;
 
                                board.updateQuality = board.BOARD_QUALITY_HIGH;
 
                                board.update();
 
}
 
});
 
$("#sliderf2").slider({
 
orientation: "horizontal",range: "min",min: 1,max: 100,value: 17,
 
slide: function(event, ui) {
 
$("#f2").val(ui.value);
 
                                f2 = ui.value;
 
                                board.updateQuality = board.BOARD_QUALITY_HIGH;
 
                                board.update();
 
}
 
});
 
$("#c2").val($("#sliderc2").slider("value")*0.01);
 
$("#f2").val($("#sliderf2").slider("value"));
 
});
 
</script>
 
 
 
<div class="sliders" style="margin:5px">
 
 
   <p>
 
   <p>
 
     <label for="c1">c1:</label>
 
     <label for="c1">c1:</label>
     <input type="text" id="c1" style="border:0; color:#f6931f; font-weight:bold;" />
+
     <input type="range" id="c1" style="border:0; color:#f6931f; font-weight:bold;"  
 +
        min="0" max="100" value="60"
 +
        oninput="c1 = this.value*0.01; board.update();"
 +
    />
 
     <label for="f1">f1:</label>
 
     <label for="f1">f1:</label>
     <input type="text" id="f1" style="border:0; color:#f6931f; font-weight:bold;" />
+
     <input type="range" id="f1" style="border:0; color:#f6931f; font-weight:bold;"  
 +
        min="1" max="100" value="7"
 +
        oninput="f1 = this.value; board.update();"
 +
    />
 
     <label for="c2">c2:</label>
 
     <label for="c2">c2:</label>
     <input type="text" id="c2" style="border:0; color:#f6931f; font-weight:bold;" />
+
     <input type="range" id="c2" style="border:0; color:#f6931f; font-weight:bold;"  
 +
        min="0" max="100" value="0"
 +
        oninput="c2 = this.value*0.01;
 +
                  board.updateQuality = board.BOARD_QUALITY_HIGH;
 +
                  board.update();"
 +
    />
 
     <label for="f2">f2:</label>
 
     <label for="f2">f2:</label>
     <input type="text" id="f2" style="border:0; color:#f6931f; font-weight:bold;" />
+
     <input type="range" id="f2" style="border:0; color:#f6931f; font-weight:bold;"  
 +
        min="1" max="100" value="17"
 +
        oninput="f2 = this.value; board.update();"
 +
    />
 
   </p>
 
   </p>
  <div id="sliderc1" style="width:300px;margin:10px;"></div>
 
  <div id="sliderf1" style="width:300px;margin:10px;"></div>
 
  <div id="sliderc2" style="width:300px;margin:10px;"></div>
 
  <div id="sliderf2" style="width:300px;margin:10px;"></div>
 
 
</div>
 
</div>
</html>
+
 
 
<jsxgraph width="500" height="500" box="jxgbox">
 
<jsxgraph width="500" height="500" box="jxgbox">
 
board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox:[-2.5,2.5,2.5,-2.5], keepaspectratio:true});
 
board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox:[-2.5,2.5,2.5,-2.5], keepaspectratio:true});
Line 182: Line 98:
 
               function(t) { return Math.cos(t)+ c1*Math.cos(f1*t)+ c2*Math.cos(f2*t);},
 
               function(t) { return Math.cos(t)+ c1*Math.cos(f1*t)+ c2*Math.cos(f2*t);},
 
               function(t) { return Math.sin(t)+ c1*Math.sin(f1*t)+ c2*Math.sin(f2*t);},
 
               function(t) { return Math.sin(t)+ c1*Math.sin(f1*t)+ c2*Math.sin(f2*t);},
               0,2.02*Math.PI],{strokeWidth:2});
+
               0,2.02*Math.PI], {strokeWidth:2});
 
</jsxgraph>
 
</jsxgraph>
 
</source>
 
</source>
 
[[Category:Examples]]
 
[[Category:Examples]]
 
[[Category:Curves]]
 
[[Category:Curves]]

Latest revision as of 14:18, 23 June 2020

This is an example of a parametric curve plot. It shows the orbit of a point on a circle. The circle rotates on a circle which again rotates on the unit circle. The resulting curve is described by the function

[math] [0,2\pi]\to{\mathbf R}^2, \quad t\mapsto {\cos(t)\choose \sin(t)}+c_1{\cos(f_1t)\choose \sin(f_1t)}+c_2{\cos(f_2t)\choose \sin(f_2t)}[/math]

This example shows the seamless integration of JSXGraph into the web page.

Variation:

External references

Epicycloidal curves have been used by the ancient greeks to describe the orbits of the planets, see

More on epicycloidal curves:

The source code of this construction

The main difficulty is to read the values of the sliders. This is done via four JavaScript variables [math]c1, c2, f1, f2[/math].

<div style="margin:5px">
  <p>
    <label for="c1">c1:</label>
    <input type="range" id="c1" style="border:0; color:#f6931f; font-weight:bold;" 
        min="0" max="100" value="60" 
        oninput="c1 = this.value*0.01; board.update();" 
    />
    <label for="f1">f1:</label>
    <input type="range" id="f1" style="border:0; color:#f6931f; font-weight:bold;" 
        min="1" max="100" value="7"
        oninput="f1 = this.value; board.update();" 
    />
    <label for="c2">c2:</label>
    <input type="range" id="c2" style="border:0; color:#f6931f; font-weight:bold;" 
        min="0" max="100" value="0"
        oninput="c2 = this.value*0.01; 
                  board.updateQuality = board.BOARD_QUALITY_HIGH;
                  board.update();" 
    />
    <label for="f2">f2:</label>
    <input type="range" id="f2" style="border:0; color:#f6931f; font-weight:bold;" 
        min="1" max="100" value="17"
        oninput="f2 = this.value; board.update();" 
    />
  </p>
</div>

<jsxgraph width="500" height="500" box="jxgbox">
board = JXG.JSXGraph.initBoard('jxgbox', {boundingbox:[-2.5,2.5,2.5,-2.5], keepaspectratio:true});
var c1 = 0.6;
var c2 = 0.0; 
var f1 = 7; 
var f2 = 17;
var c = board.create('curve', [
              function(t) { return Math.cos(t)+ c1*Math.cos(f1*t)+ c2*Math.cos(f2*t);},
              function(t) { return Math.sin(t)+ c1*Math.sin(f1*t)+ c2*Math.sin(f2*t);},
              0,2.02*Math.PI], {strokeWidth:2});
</jsxgraph>