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Tschirnhausen Cubic Catacaustic
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<iframe 
    src="https://www.jsxgraph.uni-bayreuth.de/share/iframe/tschirnhausen-cubic-catacaustic" 
    style="border: 1px solid black; overflow: hidden; width: 550px; aspect-ratio: 55 / 65;" 
    name="JSXGraph example: Tschirnhausen Cubic Catacaustic" 
    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 ShareAlike 4.0 International License.
    https://creativecommons.org/licenses/by-sa/4.0/
    
    Please note you have to mention 
    The Center of Mobile Learning with Digital Technology
    in the credits.
    */
    
    const BOARDID = 'board-0';

    var board = JXG.JSXGraph.initBoard(BOARDID, {
        boundingbox: [-10, 10, 10, -10],
        showClearTraces: true,
        axis: true
    });
    
    board.suspendUpdate();
    
    var a = board.create('slider', [[-5, 6], [5, 6], [-5, 1, 5]], { name: 'a' });
    
    var cubic = board.create('curve',
        [function(t) { return a.Value() * 3 * (t * t - 3); },
         function(t) { return a.Value() * t * (t * t - 3); },
         -5, 5
        ], { strokeWidth: 1, strokeColor: 'black' }
    );
    
    var radpoint = board.create('point', [function() { return -a.Value() * 8; }, 0], { name: 'radiant point' });
    var reflectionpoint = board.create('glider', [-7, 1, cubic], { name: 'point of reflection' });
    var dir = board.create('segment', [radpoint, reflectionpoint], { strokeWidth: 1 });
    
    var infty = board.create('point',
        [function() {
            var A = dir.stdform[1],
                B = dir.stdform[2],
                t = reflectionpoint.position,
                u = JXG.Math.Numerics.D(cubic.X)(t),
                v = JXG.Math.Numerics.D(cubic.Y)(t),
                dirx = A * v * v - 2 * B * u * v - A * u * u,
                diry = B * u * u - 2 * A * u * v - B * v * v;
            return [0, diry, -dirx];
        }], { name: '', visible: false }
    );
    
    var reflection = board.create('line',
        [reflectionpoint, infty], { strokeWidth: 1, straightFirst: false, trace: true }
    );
    
    var cataustic = board.create('curve',
        [function(t) { return a.Value() * 6 * (t * t - 1); },
         function(t) { return a.Value() * 4 * t * t * t; },
         -4, 4
        ], { strokeWidth: 3, strokeColor: 'red' }
    );
    
    board.unsuspendUpdate();
 </script> 
/*
This example is licensed under a 
Creative Commons Attribution ShareAlike 4.0 International License.
https://creativecommons.org/licenses/by-sa/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!

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

board.suspendUpdate();

var a = board.create('slider', [[-5, 6], [5, 6], [-5, 1, 5]], { name: 'a' });

var cubic = board.create('curve',
    [function(t) { return a.Value() * 3 * (t * t - 3); },
     function(t) { return a.Value() * t * (t * t - 3); },
     -5, 5
    ], { strokeWidth: 1, strokeColor: 'black' }
);

var radpoint = board.create('point', [function() { return -a.Value() * 8; }, 0], { name: 'radiant point' });
var reflectionpoint = board.create('glider', [-7, 1, cubic], { name: 'point of reflection' });
var dir = board.create('segment', [radpoint, reflectionpoint], { strokeWidth: 1 });

var infty = board.create('point',
    [function() {
        var A = dir.stdform[1],
            B = dir.stdform[2],
            t = reflectionpoint.position,
            u = JXG.Math.Numerics.D(cubic.X)(t),
            v = JXG.Math.Numerics.D(cubic.Y)(t),
            dirx = A * v * v - 2 * B * u * v - A * u * u,
            diry = B * u * u - 2 * A * u * v - B * v * v;
        return [0, diry, -dirx];
    }], { name: '', visible: false }
);

var reflection = board.create('line',
    [reflectionpoint, infty], { strokeWidth: 1, straightFirst: false, trace: true }
);

var cataustic = board.create('curve',
    [function(t) { return a.Value() * 6 * (t * t - 1); },
     function(t) { return a.Value() * 4 * t * t * t; },
     -4, 4
    ], { strokeWidth: 3, strokeColor: 'red' }
);

board.unsuspendUpdate();
<jsxgraph width="100%" aspect-ratio="1 / 1" title="Tschirnhausen Cubic Catacaustic" 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 ShareAlike 4.0 International License.
   https://creativecommons.org/licenses/by-sa/4.0/
   
   Please note you have to mention 
   The Center of Mobile Learning with Digital Technology
   in the credits.
   */
   
   var board = JXG.JSXGraph.initBoard(BOARDID, {
       boundingbox: [-10, 10, 10, -10],
       showClearTraces: true,
       axis: true
   });
   
   board.suspendUpdate();
   
   var a = board.create('slider', [[-5, 6], [5, 6], [-5, 1, 5]], { name: 'a' });
   
   var cubic = board.create('curve',
       [function(t) { return a.Value() * 3 * (t * t - 3); },
        function(t) { return a.Value() * t * (t * t - 3); },
        -5, 5
       ], { strokeWidth: 1, strokeColor: 'black' }
   );
   
   var radpoint = board.create('point', [function() { return -a.Value() * 8; }, 0], { name: 'radiant point' });
   var reflectionpoint = board.create('glider', [-7, 1, cubic], { name: 'point of reflection' });
   var dir = board.create('segment', [radpoint, reflectionpoint], { strokeWidth: 1 });
   
   var infty = board.create('point',
       [function() {
           var A = dir.stdform[1],
               B = dir.stdform[2],
               t = reflectionpoint.position,
               u = JXG.Math.Numerics.D(cubic.X)(t),
               v = JXG.Math.Numerics.D(cubic.Y)(t),
               dirx = A * v * v - 2 * B * u * v - A * u * u,
               diry = B * u * u - 2 * A * u * v - B * v * v;
           return [0, diry, -dirx];
       }], { name: '', visible: false }
   );
   
   var reflection = board.create('line',
       [reflectionpoint, infty], { strokeWidth: 1, straightFirst: false, trace: true }
   );
   
   var cataustic = board.create('curve',
       [function(t) { return a.Value() * 6 * (t * t - 1); },
        function(t) { return a.Value() * 4 * t * t * t; },
        -4, 4
       ], { strokeWidth: 3, strokeColor: 'red' }
   );
   
   board.unsuspendUpdate();
</jsxgraph>

Tschirnhausen Cubic Catacaustic

Calculus
Curves
The Tschirnhausen cubic (black curve) is defined parametrically as $$x = a3(t^2-3)$$ $$y = at(t^2-3)$$ Its __catacaustic__ (red curve) with radiant point (-8a,p) is the semicubical parabola with parametric equations $$x = a6(t^2-1)$$ $$y = a4t^3$$ The __catacaustic__ is the envelope of the rays reflected by the Tschirnhausen cubic. The ray's source is the *radiant point*. You can get a feeling why the red curve is called *envelope* of the blue line if you __drag__ the __point of reflection__.
// Define the id of your board in BOARDID

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

board.suspendUpdate();

var a = board.create('slider', [[-5, 6], [5, 6], [-5, 1, 5]], { name: 'a' });

var cubic = board.create('curve',
    [function(t) { return a.Value() * 3 * (t * t - 3); },
     function(t) { return a.Value() * t * (t * t - 3); },
     -5, 5
    ], { strokeWidth: 1, strokeColor: 'black' }
);

var radpoint = board.create('point', [function() { return -a.Value() * 8; }, 0], { name: 'radiant point' });
var reflectionpoint = board.create('glider', [-7, 1, cubic], { name: 'point of reflection' });
var dir = board.create('segment', [radpoint, reflectionpoint], { strokeWidth: 1 });

var infty = board.create('point',
    [function() {
        var A = dir.stdform[1],
            B = dir.stdform[2],
            t = reflectionpoint.position,
            u = JXG.Math.Numerics.D(cubic.X)(t),
            v = JXG.Math.Numerics.D(cubic.Y)(t),
            dirx = A * v * v - 2 * B * u * v - A * u * u,
            diry = B * u * u - 2 * A * u * v - B * v * v;
        return [0, diry, -dirx];
    }], { name: '', visible: false }
);

var reflection = board.create('line',
    [reflectionpoint, infty], { strokeWidth: 1, straightFirst: false, trace: true }
);

var cataustic = board.create('curve',
    [function(t) { return a.Value() * 6 * (t * t - 1); },
     function(t) { return a.Value() * 4 * t * t * t; },
     -4, 4
    ], { strokeWidth: 3, strokeColor: 'red' }
);

board.unsuspendUpdate();

license

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