JSXGraph
JSXGraph share

Share

Approximate circular arc by a Bézier curve
Show plain example
QR code 
<iframe 
    src="http://jsxgraph.org/share/iframe/approximate-circular-arc-by-a-b-zier-curve" 
    style="border: 1px solid black; overflow: hidden; width: 550px; aspect-ratio: 55 / 65;" 
    name="JSXGraph example: Approximate circular arc by a Bézier curve" 
    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, {
        axis: false,
        boundingbox: [-2, 2, 2, -2],
        keepaspectratio: true
    });
    
    // Create circle through D with center M and to gliders A and B
    var M = board.create('point', [0, 0], {
        name: 'M'
    });
    var C = board.create('point', [0, -1], {
        name: 'D'
    });
    var c = board.create('circle', [M, C], {
        strokeWidth: 1
    });
    var A = board.create('glider', [1, 0, c], {
        name: 'A'
    });
    var B = board.create('glider', [0, 1, c], {
        name: 'B'
    });
    
    // Determine two optimal control points
    var k = function(M, A, B) {
        var ax = A.X() - M.X(),
            ay = A.Y() - M.Y(),
            bx = B.X() - M.X(),
            by = B.Y() - M.Y(),
            d, r;
        r = M.Dist(A);
        d = Math.sqrt((ax + bx) * (ax + bx) + (ay + by) * (ay + by));
        if (JXG.Math.Geometry.rad(A, M, B) > Math.PI) {
            d *= -1;
        }
    
        if (Math.abs(by - ay) > JXG.Math.eps) {
            return (ax + bx) * (r / d - 0.5) * 8.0 / 3.0 / (by - ay);
        } else {
            return (ay + by) * (r / d - 0.5) * 8.0 / 3.0 / (ax - bx);
        }
    };
    
    var P1 = board.create('point', [
        () => A.X() - k(M, A, B) * (A.Y() - M.Y()),
        () => A.Y() + k(M, A, B) * (A.X() - M.X())
    ], {
        color: 'blue'
    });
    var P2 = board.create('point', [
        () => B.X() + k(M, A, B) * (B.Y() - M.Y()),
        () => B.Y() - k(M, A, B) * (B.X() - M.X())
    ], {
        color: 'blue'
    });
    
    // Create the Bezier segment
    var b = board.create('curve', JXG.Math.Numerics.bezier([A, P1, P2, B]), {
        strokecolor: 'black',
        strokeOpacity: 1,
        strokeWidth: 3
    });
    
    var l1 = board.create('segment', [A, P1], {
        dash: 2
    });
    var l2 = board.create('segment', [B, P2], {
        dash: 2
    });
 </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, {
    axis: false,
    boundingbox: [-2, 2, 2, -2],
    keepaspectratio: true
});

// Create circle through D with center M and to gliders A and B
var M = board.create('point', [0, 0], {
    name: 'M'
});
var C = board.create('point', [0, -1], {
    name: 'D'
});
var c = board.create('circle', [M, C], {
    strokeWidth: 1
});
var A = board.create('glider', [1, 0, c], {
    name: 'A'
});
var B = board.create('glider', [0, 1, c], {
    name: 'B'
});

// Determine two optimal control points
var k = function(M, A, B) {
    var ax = A.X() - M.X(),
        ay = A.Y() - M.Y(),
        bx = B.X() - M.X(),
        by = B.Y() - M.Y(),
        d, r;
    r = M.Dist(A);
    d = Math.sqrt((ax + bx) * (ax + bx) + (ay + by) * (ay + by));
    if (JXG.Math.Geometry.rad(A, M, B) > Math.PI) {
        d *= -1;
    }

    if (Math.abs(by - ay) > JXG.Math.eps) {
        return (ax + bx) * (r / d - 0.5) * 8.0 / 3.0 / (by - ay);
    } else {
        return (ay + by) * (r / d - 0.5) * 8.0 / 3.0 / (ax - bx);
    }
};

var P1 = board.create('point', [
    () => A.X() - k(M, A, B) * (A.Y() - M.Y()),
    () => A.Y() + k(M, A, B) * (A.X() - M.X())
], {
    color: 'blue'
});
var P2 = board.create('point', [
    () => B.X() + k(M, A, B) * (B.Y() - M.Y()),
    () => B.Y() - k(M, A, B) * (B.X() - M.X())
], {
    color: 'blue'
});

// Create the Bezier segment
var b = board.create('curve', JXG.Math.Numerics.bezier([A, P1, P2, B]), {
    strokecolor: 'black',
    strokeOpacity: 1,
    strokeWidth: 3
});

var l1 = board.create('segment', [A, P1], {
    dash: 2
});
var l2 = board.create('segment', [B, P2], {
    dash: 2
});
<jsxgraph width="100%" aspect-ratio="1 / 1" title="Approximate circular arc by a Bézier curve" 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, {
       axis: false,
       boundingbox: [-2, 2, 2, -2],
       keepaspectratio: true
   });
   
   // Create circle through D with center M and to gliders A and B
   var M = board.create('point', [0, 0], {
       name: 'M'
   });
   var C = board.create('point', [0, -1], {
       name: 'D'
   });
   var c = board.create('circle', [M, C], {
       strokeWidth: 1
   });
   var A = board.create('glider', [1, 0, c], {
       name: 'A'
   });
   var B = board.create('glider', [0, 1, c], {
       name: 'B'
   });
   
   // Determine two optimal control points
   var k = function(M, A, B) {
       var ax = A.X() - M.X(),
           ay = A.Y() - M.Y(),
           bx = B.X() - M.X(),
           by = B.Y() - M.Y(),
           d, r;
       r = M.Dist(A);
       d = Math.sqrt((ax + bx) * (ax + bx) + (ay + by) * (ay + by));
       if (JXG.Math.Geometry.rad(A, M, B) > Math.PI) {
           d *= -1;
       }
   
       if (Math.abs(by - ay) > JXG.Math.eps) {
           return (ax + bx) * (r / d - 0.5) * 8.0 / 3.0 / (by - ay);
       } else {
           return (ay + by) * (r / d - 0.5) * 8.0 / 3.0 / (ax - bx);
       }
   };
   
   var P1 = board.create('point', [
       () => A.X() - k(M, A, B) * (A.Y() - M.Y()),
       () => A.Y() + k(M, A, B) * (A.X() - M.X())
   ], {
       color: 'blue'
   });
   var P2 = board.create('point', [
       () => B.X() + k(M, A, B) * (B.Y() - M.Y()),
       () => B.Y() - k(M, A, B) * (B.X() - M.X())
   ], {
       color: 'blue'
   });
   
   // Create the Bezier segment
   var b = board.create('curve', JXG.Math.Numerics.bezier([A, P1, P2, B]), {
       strokecolor: 'black',
       strokeOpacity: 1,
       strokeWidth: 3
   });
   
   var l1 = board.create('segment', [A, P1], {
       dash: 2
   });
   var l2 = board.create('segment', [B, P2], {
       dash: 2
   });
</jsxgraph>
Approximating a circular by a single Bézier curve only is sufficiently exact if the arc is less or equal than a quarter circle.
Have also a look at the examples 
// Define the id of your board in BOARDID

const board = JXG.JSXGraph.initBoard(BOARDID, {
    axis: false,
    boundingbox: [-2, 2, 2, -2],
    keepaspectratio: true
});

// Create circle through D with center M and to gliders A and B
var M = board.create('point', [0, 0], {
    name: 'M'
});
var C = board.create('point', [0, -1], {
    name: 'D'
});
var c = board.create('circle', [M, C], {
    strokeWidth: 1
});
var A = board.create('glider', [1, 0, c], {
    name: 'A'
});
var B = board.create('glider', [0, 1, c], {
    name: 'B'
});

// Determine two optimal control points
var k = function(M, A, B) {
    var ax = A.X() - M.X(),
        ay = A.Y() - M.Y(),
        bx = B.X() - M.X(),
        by = B.Y() - M.Y(),
        d, r;
    r = M.Dist(A);
    d = Math.sqrt((ax + bx) * (ax + bx) + (ay + by) * (ay + by));
    if (JXG.Math.Geometry.rad(A, M, B) > Math.PI) {
        d *= -1;
    }

    if (Math.abs(by - ay) > JXG.Math.eps) {
        return (ax + bx) * (r / d - 0.5) * 8.0 / 3.0 / (by - ay);
    } else {
        return (ay + by) * (r / d - 0.5) * 8.0 / 3.0 / (ax - bx);
    }
};

var P1 = board.create('point', [
    () => A.X() - k(M, A, B) * (A.Y() - M.Y()),
    () => A.Y() + k(M, A, B) * (A.X() - M.X())
], {
    color: 'blue'
});
var P2 = board.create('point', [
    () => B.X() + k(M, A, B) * (B.Y() - M.Y()),
    () => B.Y() - k(M, A, B) * (B.X() - M.X())
], {
    color: 'blue'
});

// Create the Bezier segment
var b = board.create('curve', JXG.Math.Numerics.bezier([A, P1, P2, B]), {
    strokecolor: 'black',
    strokeOpacity: 1,
    strokeWidth: 3
});

var l1 = board.create('segment', [A, P1], {
    dash: 2
});
var l2 = board.create('segment', [B, P2], {
    dash: 2
});

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.