JSXGraph
JSXGraph share

Share

Apollonian circle packing
Show plain example
QR code 
<iframe 
    src="http://jsxgraph.org/share/iframe/apollonian-circle-packing" 
    style="border: 1px solid black; overflow: hidden; width: 550px; aspect-ratio: 55 / 65;" 
    name="JSXGraph example: Apollonian circle packing" 
    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: [-2, 2, 2, -2],
        keepaspectratio: true
    });
    
    var b0, c0, c1, c2, c3,
        a, p1,
        solveQ2, thirdCircleX, thirdCircleY, thirdCircleRadius,
        otherCirc;
    
    solveQ2 = function(x1, x2, x3, off) {
        var a, b, c, d;
        a = 0.5;
        b = -(x1 + x2 + x3);
        c = x1 * x1 + x2 * x2 + x3 * x3 - 0.5 * (x1 + x2 + x3) * (x1 + x2 + x3) - off;
        d = b * b - 4 * a * c;
        if (Math.abs(d) < 0.00000001) d = 0.0;
        return [(-b + Math.sqrt(d)) / (2.0 * a), (-b - Math.sqrt(d)) / (2.0 * a)];
    }
    
    a = board.create('segment', [
        [0, 0],
        [2, 0]
    ], {
        visible: false
    });
    
    p1 = board.create('glider', [1.3, 0, a], {
        name: 'Drag me'
    });
    b0 = -0.5;
    
    c0 = board.create('circle', [
        [0, 0], Math.abs(1.0 / b0)
    ], {
        strokeWidth: 1
    });
    c1 = board.create('circle', [p1, function() {
        return 2 - p1.X();
    }], {
        strokeWidth: 1
    });
    
    c2 = board.create('circle', [
        [function() {
            return p1.X() - 2;
        }, 0],
        function() {
            return p1.X();
        }
    ], {
        strokeWidth: 1
    });
    
    // Constant curvature
    c0.curvature = function() {
        return b0;
    };
    c1.curvature = function() {
        return 1 / (2 - p1.X());
    };
    c2.curvature = function() {
        return 1 / (p1.X());
    };
    
    thirdCircleX = function() {
        var b0, b1, b2, x0, x1, x2, b3, bx3;
        b0 = c0.curvature();
        b1 = c1.curvature();
        b2 = c2.curvature();
        x0 = c0.midpoint.X();
        x1 = c1.midpoint.X();
        x2 = c2.midpoint.X();
    
        b3 = solveQ2(b0, b1, b2, 0);
        bx3 = solveQ2(b0 * x0, b1 * x1, b2 * x2, 2);
        return bx3[0] / b3[0];
    }
    thirdCircleY = function() {
        var b0, b1, b2, y0, y1, y2, b3, by3;
        b0 = c0.curvature();
        b1 = c1.curvature();
        b2 = c2.curvature();
        y0 = c0.midpoint.Y();
        y1 = c1.midpoint.Y();
        y2 = c2.midpoint.Y();
    
        b3 = solveQ2(b0, b1, b2, 0);
        by3 = solveQ2(b0 * y0, b1 * y1, b2 * y2, 2);
        return by3[0] / b3[0];
    }
    thirdCircleRadius = function() {
        var b0, b1, b2, b3, bx3, by3;
        b0 = c0.curvature();
        b1 = c1.curvature();
        b2 = c2.curvature();
        b3 = solveQ2(b0, b1, b2, 0);
        return 1.0 / b3[0];
    }
    
    c3 = board.create('circle', [
        [thirdCircleX, thirdCircleY], thirdCircleRadius
    ], {
        strokeWidth: 1
    });
    c3.curvature = function() {
        return 1.0 / this.radius;
    };
    
    otherCirc = function(circs, level) {
        var c, fx, fy, fr;
        if (level <= 0) return;
    
        fx = function() {
            var b, x, i;
            b = [];
            x = [];
            for (i = 0; i < 4; i++) {
                b[i] = circs[i].curvature();
                x[i] = circs[i].midpoint.X();
            }
    
            b[4] = 2 * (b[0] + b[1] + b[2]) - b[3];
            x[4] = (2 * (b[0] * x[0] + b[1] * x[1] + b[2] * x[2]) - b[3] * x[3]) / b[4];
            return x[4];
        }
        fy = function() {
            var b, y, i;
            b = [];
            y = [];
            for (i = 0; i < 4; i++) {
                b[i] = circs[i].curvature();
                y[i] = circs[i].midpoint.Y();
            }
    
            b[4] = 2 * (b[0] + b[1] + b[2]) - b[3];
            y[4] = (2 * (b[0] * y[0] + b[1] * y[1] + b[2] * y[2]) - b[3] * y[3]) / b[4];
            return y[4];
        }
        fr = function() {
            var b, i;
            b = [];
            for (i = 0; i < 4; i++) {
                b[i] = circs[i].curvature();
            }
            b[4] = 2 * (b[0] + b[1] + b[2]) - b[3];
            if (isNaN(b[4])) {
                return 1000.0;
            } else {
                return 1 / b[4];
            }
        }
        c = board.create('circle', [
            [fx, fy], fr
        ], {
            strokeWidth: 1,
            name: '',
            fillColor: JXG.hsv2rgb(50 * level, 0.8, 0.8),
            highlightFillColor: JXG.hsv2rgb(50 * level, 0.5, 0.8),
            fillOpacity: 0.5,
            highlightFillOpacity: 0.5
        });
        c.curvature = function() {
            return 1 / this.radius;
        };
    
        // Recursion
        otherCirc([circs[0], circs[1], c, circs[2]], level - 1);
        otherCirc([circs[0], circs[2], c, circs[1]], level - 1);
        otherCirc([circs[1], circs[2], c, circs[0]], level - 1);
        return c;
    }
    
    //-------------------------------------------------------
    
    var level = 4;
    otherCirc([c0, c1, c2, c3], level);
    otherCirc([c3, c1, c2, c0], level);
    otherCirc([c0, c2, c3, c1], level);
    otherCirc([c0, c1, c3, c2], level);
 </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: [-2, 2, 2, -2],
    keepaspectratio: true
});

var b0, c0, c1, c2, c3,
    a, p1,
    solveQ2, thirdCircleX, thirdCircleY, thirdCircleRadius,
    otherCirc;

solveQ2 = function(x1, x2, x3, off) {
    var a, b, c, d;
    a = 0.5;
    b = -(x1 + x2 + x3);
    c = x1 * x1 + x2 * x2 + x3 * x3 - 0.5 * (x1 + x2 + x3) * (x1 + x2 + x3) - off;
    d = b * b - 4 * a * c;
    if (Math.abs(d) < 0.00000001) d = 0.0;
    return [(-b + Math.sqrt(d)) / (2.0 * a), (-b - Math.sqrt(d)) / (2.0 * a)];
}

a = board.create('segment', [
    [0, 0],
    [2, 0]
], {
    visible: false
});

p1 = board.create('glider', [1.3, 0, a], {
    name: 'Drag me'
});
b0 = -0.5;

c0 = board.create('circle', [
    [0, 0], Math.abs(1.0 / b0)
], {
    strokeWidth: 1
});
c1 = board.create('circle', [p1, function() {
    return 2 - p1.X();
}], {
    strokeWidth: 1
});

c2 = board.create('circle', [
    [function() {
        return p1.X() - 2;
    }, 0],
    function() {
        return p1.X();
    }
], {
    strokeWidth: 1
});

// Constant curvature
c0.curvature = function() {
    return b0;
};
c1.curvature = function() {
    return 1 / (2 - p1.X());
};
c2.curvature = function() {
    return 1 / (p1.X());
};

thirdCircleX = function() {
    var b0, b1, b2, x0, x1, x2, b3, bx3;
    b0 = c0.curvature();
    b1 = c1.curvature();
    b2 = c2.curvature();
    x0 = c0.midpoint.X();
    x1 = c1.midpoint.X();
    x2 = c2.midpoint.X();

    b3 = solveQ2(b0, b1, b2, 0);
    bx3 = solveQ2(b0 * x0, b1 * x1, b2 * x2, 2);
    return bx3[0] / b3[0];
}
thirdCircleY = function() {
    var b0, b1, b2, y0, y1, y2, b3, by3;
    b0 = c0.curvature();
    b1 = c1.curvature();
    b2 = c2.curvature();
    y0 = c0.midpoint.Y();
    y1 = c1.midpoint.Y();
    y2 = c2.midpoint.Y();

    b3 = solveQ2(b0, b1, b2, 0);
    by3 = solveQ2(b0 * y0, b1 * y1, b2 * y2, 2);
    return by3[0] / b3[0];
}
thirdCircleRadius = function() {
    var b0, b1, b2, b3, bx3, by3;
    b0 = c0.curvature();
    b1 = c1.curvature();
    b2 = c2.curvature();
    b3 = solveQ2(b0, b1, b2, 0);
    return 1.0 / b3[0];
}

c3 = board.create('circle', [
    [thirdCircleX, thirdCircleY], thirdCircleRadius
], {
    strokeWidth: 1
});
c3.curvature = function() {
    return 1.0 / this.radius;
};

otherCirc = function(circs, level) {
    var c, fx, fy, fr;
    if (level <= 0) return;

    fx = function() {
        var b, x, i;
        b = [];
        x = [];
        for (i = 0; i < 4; i++) {
            b[i] = circs[i].curvature();
            x[i] = circs[i].midpoint.X();
        }

        b[4] = 2 * (b[0] + b[1] + b[2]) - b[3];
        x[4] = (2 * (b[0] * x[0] + b[1] * x[1] + b[2] * x[2]) - b[3] * x[3]) / b[4];
        return x[4];
    }
    fy = function() {
        var b, y, i;
        b = [];
        y = [];
        for (i = 0; i < 4; i++) {
            b[i] = circs[i].curvature();
            y[i] = circs[i].midpoint.Y();
        }

        b[4] = 2 * (b[0] + b[1] + b[2]) - b[3];
        y[4] = (2 * (b[0] * y[0] + b[1] * y[1] + b[2] * y[2]) - b[3] * y[3]) / b[4];
        return y[4];
    }
    fr = function() {
        var b, i;
        b = [];
        for (i = 0; i < 4; i++) {
            b[i] = circs[i].curvature();
        }
        b[4] = 2 * (b[0] + b[1] + b[2]) - b[3];
        if (isNaN(b[4])) {
            return 1000.0;
        } else {
            return 1 / b[4];
        }
    }
    c = board.create('circle', [
        [fx, fy], fr
    ], {
        strokeWidth: 1,
        name: '',
        fillColor: JXG.hsv2rgb(50 * level, 0.8, 0.8),
        highlightFillColor: JXG.hsv2rgb(50 * level, 0.5, 0.8),
        fillOpacity: 0.5,
        highlightFillOpacity: 0.5
    });
    c.curvature = function() {
        return 1 / this.radius;
    };

    // Recursion
    otherCirc([circs[0], circs[1], c, circs[2]], level - 1);
    otherCirc([circs[0], circs[2], c, circs[1]], level - 1);
    otherCirc([circs[1], circs[2], c, circs[0]], level - 1);
    return c;
}

//-------------------------------------------------------

var level = 4;
otherCirc([c0, c1, c2, c3], level);
otherCirc([c3, c1, c2, c0], level);
otherCirc([c0, c2, c3, c1], level);
otherCirc([c0, c1, c3, c2], level);
<jsxgraph width="100%" aspect-ratio="1 / 1" title="Apollonian circle packing" 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: [-2, 2, 2, -2],
       keepaspectratio: true
   });
   
   var b0, c0, c1, c2, c3,
       a, p1,
       solveQ2, thirdCircleX, thirdCircleY, thirdCircleRadius,
       otherCirc;
   
   solveQ2 = function(x1, x2, x3, off) {
       var a, b, c, d;
       a = 0.5;
       b = -(x1 + x2 + x3);
       c = x1 * x1 + x2 * x2 + x3 * x3 - 0.5 * (x1 + x2 + x3) * (x1 + x2 + x3) - off;
       d = b * b - 4 * a * c;
       if (Math.abs(d) < 0.00000001) d = 0.0;
       return [(-b + Math.sqrt(d)) / (2.0 * a), (-b - Math.sqrt(d)) / (2.0 * a)];
   }
   
   a = board.create('segment', [
       [0, 0],
       [2, 0]
   ], {
       visible: false
   });
   
   p1 = board.create('glider', [1.3, 0, a], {
       name: 'Drag me'
   });
   b0 = -0.5;
   
   c0 = board.create('circle', [
       [0, 0], Math.abs(1.0 / b0)
   ], {
       strokeWidth: 1
   });
   c1 = board.create('circle', [p1, function() {
       return 2 - p1.X();
   }], {
       strokeWidth: 1
   });
   
   c2 = board.create('circle', [
       [function() {
           return p1.X() - 2;
       }, 0],
       function() {
           return p1.X();
       }
   ], {
       strokeWidth: 1
   });
   
   // Constant curvature
   c0.curvature = function() {
       return b0;
   };
   c1.curvature = function() {
       return 1 / (2 - p1.X());
   };
   c2.curvature = function() {
       return 1 / (p1.X());
   };
   
   thirdCircleX = function() {
       var b0, b1, b2, x0, x1, x2, b3, bx3;
       b0 = c0.curvature();
       b1 = c1.curvature();
       b2 = c2.curvature();
       x0 = c0.midpoint.X();
       x1 = c1.midpoint.X();
       x2 = c2.midpoint.X();
   
       b3 = solveQ2(b0, b1, b2, 0);
       bx3 = solveQ2(b0 * x0, b1 * x1, b2 * x2, 2);
       return bx3[0] / b3[0];
   }
   thirdCircleY = function() {
       var b0, b1, b2, y0, y1, y2, b3, by3;
       b0 = c0.curvature();
       b1 = c1.curvature();
       b2 = c2.curvature();
       y0 = c0.midpoint.Y();
       y1 = c1.midpoint.Y();
       y2 = c2.midpoint.Y();
   
       b3 = solveQ2(b0, b1, b2, 0);
       by3 = solveQ2(b0 * y0, b1 * y1, b2 * y2, 2);
       return by3[0] / b3[0];
   }
   thirdCircleRadius = function() {
       var b0, b1, b2, b3, bx3, by3;
       b0 = c0.curvature();
       b1 = c1.curvature();
       b2 = c2.curvature();
       b3 = solveQ2(b0, b1, b2, 0);
       return 1.0 / b3[0];
   }
   
   c3 = board.create('circle', [
       [thirdCircleX, thirdCircleY], thirdCircleRadius
   ], {
       strokeWidth: 1
   });
   c3.curvature = function() {
       return 1.0 / this.radius;
   };
   
   otherCirc = function(circs, level) {
       var c, fx, fy, fr;
       if (level <= 0) return;
   
       fx = function() {
           var b, x, i;
           b = [];
           x = [];
           for (i = 0; i < 4; i++) {
               b[i] = circs[i].curvature();
               x[i] = circs[i].midpoint.X();
           }
   
           b[4] = 2 * (b[0] + b[1] + b[2]) - b[3];
           x[4] = (2 * (b[0] * x[0] + b[1] * x[1] + b[2] * x[2]) - b[3] * x[3]) / b[4];
           return x[4];
       }
       fy = function() {
           var b, y, i;
           b = [];
           y = [];
           for (i = 0; i < 4; i++) {
               b[i] = circs[i].curvature();
               y[i] = circs[i].midpoint.Y();
           }
   
           b[4] = 2 * (b[0] + b[1] + b[2]) - b[3];
           y[4] = (2 * (b[0] * y[0] + b[1] * y[1] + b[2] * y[2]) - b[3] * y[3]) / b[4];
           return y[4];
       }
       fr = function() {
           var b, i;
           b = [];
           for (i = 0; i < 4; i++) {
               b[i] = circs[i].curvature();
           }
           b[4] = 2 * (b[0] + b[1] + b[2]) - b[3];
           if (isNaN(b[4])) {
               return 1000.0;
           } else {
               return 1 / b[4];
           }
       }
       c = board.create('circle', [
           [fx, fy], fr
       ], {
           strokeWidth: 1,
           name: '',
           fillColor: JXG.hsv2rgb(50 * level, 0.8, 0.8),
           highlightFillColor: JXG.hsv2rgb(50 * level, 0.5, 0.8),
           fillOpacity: 0.5,
           highlightFillOpacity: 0.5
       });
       c.curvature = function() {
           return 1 / this.radius;
       };
   
       // Recursion
       otherCirc([circs[0], circs[1], c, circs[2]], level - 1);
       otherCirc([circs[0], circs[2], c, circs[1]], level - 1);
       otherCirc([circs[1], circs[2], c, circs[0]], level - 1);
       return c;
   }
   
   //-------------------------------------------------------
   
   var level = 4;
   otherCirc([c0, c1, c2, c3], level);
   otherCirc([c3, c1, c2, c0], level);
   otherCirc([c0, c2, c3, c1], level);
   otherCirc([c0, c1, c3, c2], level);
</jsxgraph>

Apollonian circle packing

Euclidean
Geometry
__References__ - [https://www.ams.org/featurecolumn/archive/kissing.html](https://www.ams.org/featurecolumn/archive/kissing.html) - Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks: Beyond the Descartes circle theorem - [https://en.wikipedia.org/wiki/Apollonian_gasket](https://en.wikipedia.org/wiki/Apollonian_gasket) - Weisstein, Eric W. ["Apollonian Gasket."](https://mathworld.wolfram.com/ApollonianGasket.html) From MathWorld - A Wolfram Web Resource - Weisstein, Eric W. ["Soddy Circles."](https://mathworld.wolfram.com/SoddyCircles.html) From MathWorld - A Wolfram Web Resource 
// Define the id of your board in BOARDID

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

var b0, c0, c1, c2, c3,
    a, p1,
    solveQ2, thirdCircleX, thirdCircleY, thirdCircleRadius,
    otherCirc;

solveQ2 = function(x1, x2, x3, off) {
    var a, b, c, d;
    a = 0.5;
    b = -(x1 + x2 + x3);
    c = x1 * x1 + x2 * x2 + x3 * x3 - 0.5 * (x1 + x2 + x3) * (x1 + x2 + x3) - off;
    d = b * b - 4 * a * c;
    if (Math.abs(d) < 0.00000001) d = 0.0;
    return [(-b + Math.sqrt(d)) / (2.0 * a), (-b - Math.sqrt(d)) / (2.0 * a)];
}

a = board.create('segment', [
    [0, 0],
    [2, 0]
], {
    visible: false
});

p1 = board.create('glider', [1.3, 0, a], {
    name: 'Drag me'
});
b0 = -0.5;

c0 = board.create('circle', [
    [0, 0], Math.abs(1.0 / b0)
], {
    strokeWidth: 1
});
c1 = board.create('circle', [p1, function() {
    return 2 - p1.X();
}], {
    strokeWidth: 1
});

c2 = board.create('circle', [
    [function() {
        return p1.X() - 2;
    }, 0],
    function() {
        return p1.X();
    }
], {
    strokeWidth: 1
});

// Constant curvature
c0.curvature = function() {
    return b0;
};
c1.curvature = function() {
    return 1 / (2 - p1.X());
};
c2.curvature = function() {
    return 1 / (p1.X());
};

thirdCircleX = function() {
    var b0, b1, b2, x0, x1, x2, b3, bx3;
    b0 = c0.curvature();
    b1 = c1.curvature();
    b2 = c2.curvature();
    x0 = c0.midpoint.X();
    x1 = c1.midpoint.X();
    x2 = c2.midpoint.X();

    b3 = solveQ2(b0, b1, b2, 0);
    bx3 = solveQ2(b0 * x0, b1 * x1, b2 * x2, 2);
    return bx3[0] / b3[0];
}
thirdCircleY = function() {
    var b0, b1, b2, y0, y1, y2, b3, by3;
    b0 = c0.curvature();
    b1 = c1.curvature();
    b2 = c2.curvature();
    y0 = c0.midpoint.Y();
    y1 = c1.midpoint.Y();
    y2 = c2.midpoint.Y();

    b3 = solveQ2(b0, b1, b2, 0);
    by3 = solveQ2(b0 * y0, b1 * y1, b2 * y2, 2);
    return by3[0] / b3[0];
}
thirdCircleRadius = function() {
    var b0, b1, b2, b3, bx3, by3;
    b0 = c0.curvature();
    b1 = c1.curvature();
    b2 = c2.curvature();
    b3 = solveQ2(b0, b1, b2, 0);
    return 1.0 / b3[0];
}

c3 = board.create('circle', [
    [thirdCircleX, thirdCircleY], thirdCircleRadius
], {
    strokeWidth: 1
});
c3.curvature = function() {
    return 1.0 / this.radius;
};

otherCirc = function(circs, level) {
    var c, fx, fy, fr;
    if (level <= 0) return;

    fx = function() {
        var b, x, i;
        b = [];
        x = [];
        for (i = 0; i < 4; i++) {
            b[i] = circs[i].curvature();
            x[i] = circs[i].midpoint.X();
        }

        b[4] = 2 * (b[0] + b[1] + b[2]) - b[3];
        x[4] = (2 * (b[0] * x[0] + b[1] * x[1] + b[2] * x[2]) - b[3] * x[3]) / b[4];
        return x[4];
    }
    fy = function() {
        var b, y, i;
        b = [];
        y = [];
        for (i = 0; i < 4; i++) {
            b[i] = circs[i].curvature();
            y[i] = circs[i].midpoint.Y();
        }

        b[4] = 2 * (b[0] + b[1] + b[2]) - b[3];
        y[4] = (2 * (b[0] * y[0] + b[1] * y[1] + b[2] * y[2]) - b[3] * y[3]) / b[4];
        return y[4];
    }
    fr = function() {
        var b, i;
        b = [];
        for (i = 0; i < 4; i++) {
            b[i] = circs[i].curvature();
        }
        b[4] = 2 * (b[0] + b[1] + b[2]) - b[3];
        if (isNaN(b[4])) {
            return 1000.0;
        } else {
            return 1 / b[4];
        }
    }
    c = board.create('circle', [
        [fx, fy], fr
    ], {
        strokeWidth: 1,
        name: '',
        fillColor: JXG.hsv2rgb(50 * level, 0.8, 0.8),
        highlightFillColor: JXG.hsv2rgb(50 * level, 0.5, 0.8),
        fillOpacity: 0.5,
        highlightFillOpacity: 0.5
    });
    c.curvature = function() {
        return 1 / this.radius;
    };

    // Recursion
    otherCirc([circs[0], circs[1], c, circs[2]], level - 1);
    otherCirc([circs[0], circs[2], c, circs[1]], level - 1);
    otherCirc([circs[1], circs[2], c, circs[0]], level - 1);
    return c;
}

//-------------------------------------------------------

var level = 4;
otherCirc([c0, c1, c2, c3], level);
otherCirc([c3, c1, c2, c0], level);
otherCirc([c0, c2, c3, c1], level);
otherCirc([c0, c1, c3, c2], level);

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.