Packing different sized circles into rectangle - d3.js

Question

I was trying to pack circles of different sizes into a rectangular container, not packing in circular container that d3.js bundled with, under

Answer 1

Here is a go at the implementation of your algorithm.

I tweaked it quite a bit, but I think it does basically the same thing.

Bounding circles

I used a trick to make the computation more regular.

Instead of segments defining the bounding box, I used circles with "infinite" radii, that can be considered a good approximation of lines:

The picture shows what the 4 bounding circles look like when the radius is decreased. They are computed to pass through the corners of the bounding box and converge toward the actual sides when the radius grows.

The "corner" circles (as the algorithm calls them) are all computed as tangents to a pair of circles, thus eliminating the special circle+segment or segment+segment cases.

This also simplifies the start condition greatly.
The algorithm simply starts with the four bounding circles and adds one circle at a time, using the greedy heuristic lambda parameter to pick the "best" location.

Best fit strategy

The original algorithm does not produce the smallest rectangle to hold all the circles
(it simply tries to fit a bunch of circles into a given rectangle).

I have added a simple dichotomic search on top of it to guess the minimal surface (which yields the smallest bounding rectangle for a given aspect ratio).

The code

Here is a fiddle

var Packer = function (circles, ratio)
{
    this.circles = circles;
    this.ratio   = ratio || 1;
    this.list = this.solve();
}

Packer.prototype = {
    // try to fit all circles into a rectangle of a given surface
    compute: function (surface)
    {
        // check if a circle is inside our rectangle
        function in_rect (radius, center)
        {
            if (center.x - radius < - w/2) return false;
            if (center.x + radius >   w/2) return false;
            if (center.y - radius < - h/2) return false;
            if (center.y + radius >   h/2) return false;
            return true;
        }

        // approximate a segment with an "infinite" radius circle
        function bounding_circle (x0, y0, x1, y1)
        {
            var xm = Math.abs ((x1-x0)*w);
            var ym = Math.abs ((y1-y0)*h);
            var m = xm > ym ? xm : ym;
            var theta = Math.asin(m/4/bounding_r);
            var r = bounding_r * Math.cos (theta);
            return new Circle (bounding_r, 
                new Point (r*(y0-y1)/2+(x0+x1)*w/4, 
                           r*(x1-x0)/2+(y0+y1)*h/4));
        }

        // return the corner placements for two circles
        function corner (radius, c1, c2)
        {
            var u = c1.c.vect(c2.c); // c1 to c2 vector
            var A = u.norm();
            if (A == 0) return [] // same centers
            u = u.mult(1/A); // c1 to c2 unary vector
            // compute c1 and c2 intersection coordinates in (u,v) base
            var B = c1.r+radius;
            var C = c2.r+radius;
            if (A > (B + C)) return []; // too far apart
            var x = (A + (B*B-C*C)/A)/2;
            var y = Math.sqrt (B*B - x*x);
            var base = c1.c.add (u.mult(x));

            var res = [];
            var p1 = new Point (base.x -u.y * y, base.y + u.x * y);
            var p2 = new Point (base.x +u.y * y, base.y - u.x * y);
            if (in_rect(radius, p1)) res.push(new Circle (radius, p1));
            if (in_rect(radius, p2)) res.push(new Circle (radius, p2));
            return res;
        }

        /////////////////////////////////////////////////////////////////

        // deduce starting dimensions from surface
        var bounding_r = Math.sqrt(surface) * 100; // "infinite" radius
        var w = this.w = Math.sqrt (surface * this.ratio);
        var h = this.h = this.w/this.ratio;

        // place our bounding circles
        var placed=[
            bounding_circle ( 1,  1,  1, -1),
            bounding_circle ( 1, -1, -1, -1),
            bounding_circle (-1, -1, -1,  1),
            bounding_circle (-1,  1,  1,  1)];

        // Initialize our rectangles list
        var unplaced = this.circles.slice(0); // clones the array
        while (unplaced.length > 0)
        {
            // compute all possible placements of the unplaced circles
            var lambda = {};
            var circle = {};
            for (var i = 0 ; i != unplaced.length ; i++)
            {
                var lambda_min = 1e10;
                lambda[i] = -1e10;
                // match current circle against all possible pairs of placed circles
                for (var j = 0   ; j < placed.length ; j++)
                for (var k = j+1 ; k < placed.length ; k++)
                {
                    // find corner placement
                    var corners = corner (unplaced[i], placed[j], placed[k]);

                    // check each placement
                    for (var c = 0 ; c != corners.length ; c++)
                    {
                        // check for overlap and compute min distance
                        var d_min = 1e10;
                        for (var l = 0 ; l != placed.length ; l++)
                        {
                            // skip the two circles used for the placement
                            if (l==j || l==k) continue;

                            // compute distance from current circle
                            var d = placed[l].distance (corners[c]);
                            if (d < 0) break; // circles overlap

                            if (d < d_min) d_min = d;
                        }
                        if (l == placed.length) // no overlap
                        {
                            if (d_min < lambda_min)
                            {
                                lambda_min = d_min;
                                lambda[i] = 1- d_min/unplaced[i];
                                circle[i] = corners[c];
                            }
                        }
                    }
                }
            }

            // select the circle with maximal gain
            var lambda_max = -1e10;
            var i_max = -1;
            for (var i = 0 ; i != unplaced.length ; i++)
            {
                if (lambda[i] > lambda_max)
                {
                    lambda_max = lambda[i];
                    i_max = i;
                }
            }

            // failure if no circle fits
            if (i_max == -1) break;

            // place the selected circle
            unplaced.splice(i_max,1);
            placed.push (circle[i_max]);
        }

        // return all placed circles except the four bounding circles
        this.tmp_bounds = placed.splice (0, 4);
        return placed;
    },

    // find the smallest rectangle to fit all circles
    solve: function ()
    {
        // compute total surface of the circles
        var surface = 0;
        for (var i = 0 ; i != this.circles.length ; i++)
        {
            surface += Math.PI * Math.pow(this.circles[i],2);
        }

        // set a suitable precision
        var limit = surface/1000;

        var step = surface/2;
        var res = [];
        while (step > limit)
        {
            var placement = this.compute.call (this, surface);
console.log ("placed",placement.length,"out of",this.circles.length,"for surface", surface);
            if (placement.length != this.circles.length)
            {
                surface += step;
            }
            else
            {
                res = placement;
                this.bounds = this.tmp_bounds;
                surface -= step;
            }
            step /= 2;
        }
        return res; 
    }
};

Performance

The code is not optimized, to favor readability (or so I hope :)).

The computation time rises pretty steeply.
You can safely place about 20 circles, but anything above 100 will make your browser crawl.

For some reason, it is way faster on FireFox than on IE11.

Packing efficiency

The algorithm works quite poorly on identically-sized circles (it cannot find the famous honeycomb pattern for 20 circles in a square), but pretty well on a wide distribution of random radii.

Aesthetics

The result is pretty ungainly for identical-sized circles.
There is no attempt to bunch the circles together, so if two possibilities are deemed equivalent by the algorithm, one is just picked at random.

I suspect the lambda parameter could be refined a bit to allow for a more aesthetic choice in case of equal values.

Possible evolutions

With the "infinite radii" trick, it becomes possible to define an arbitrary bounding polygon.

If you provide a function to check if a circle fits into the said polygon, there is no reason the algorithm should not produce a result.

Whether this result would be efficient is another question :).