Apollonian circle packing

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Revision as of 10:34, 7 June 2011 by Michael (talk | contribs)

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The underlying JavaScript code

var brd = JXG.JSXGraph.initBoard('box', {boundingbox: [-2, 2, 2, -2]});
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 = brd.create('segment',[[0,0],[2,0]],{visible:false});
p1 = brd.create('glider',[1.3,0,a],{name:'Drag me'});
b0 = -0.5;

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

c2 = brd.create('circle',[[function(){return p1.X()-2;},0],function(){return p1.X();}],{strokeWidth:1});
c0.curvature = function(){ return b0;}; // constant
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 = brd.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 = brd.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;
}

//-------------------------------------------------------
brd.suspendUpdate();
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);
brd.unsuspendUpdate();

References