venn.js

(function (global, factory) {
    typeof exports === 'object' && typeof module !== 'undefined' ? factory(exports, require('d3-selection'), require('d3-transition')) :
    typeof define === 'function' && define.amd ? define(['exports', 'd3-selection', 'd3-transition'], factory) :
    (factory((global.venn = {}),global.d3,global.d3));
}(this, (function (exports,d3Selection,d3Transition) { 'use strict';

    var SMALL = 1e-10;

    /** Returns the intersection area of a bunch of circles (where each circle
     is an object having an x,y and radius property) */
    function intersectionArea(circles, stats) {
        // get all the intersection points of the circles
        var intersectionPoints = getIntersectionPoints(circles);

        // filter out points that aren't included in all the circles
        var innerPoints = intersectionPoints.filter(function (p) {
            return containedInCircles(p, circles);
        });

        var arcArea = 0, polygonArea = 0, arcs = [], i;

        // if we have intersection points that are within all the circles,
        // then figure out the area contained by them
        if (innerPoints.length > 1) {
            // sort the points by angle from the center of the polygon, which lets
            // us just iterate over points to get the edges
            var center = getCenter(innerPoints);
            for (i = 0; i < innerPoints.length; ++i ) {
                var p = innerPoints[i];
                p.angle = Math.atan2(p.x - center.x, p.y - center.y);
            }
            innerPoints.sort(function(a,b) { return b.angle - a.angle;});

            // iterate over all points, get arc between the points
            // and update the areas
            var p2 = innerPoints[innerPoints.length - 1];
            for (i = 0; i < innerPoints.length; ++i) {
                var p1 = innerPoints[i];

                // polygon area updates easily ...
                polygonArea += (p2.x + p1.x) * (p1.y - p2.y);

                // updating the arc area is a little more involved
                var midPoint = {x : (p1.x + p2.x) / 2,
                                y : (p1.y + p2.y) / 2},
                    arc = null;

                for (var j = 0; j < p1.parentIndex.length; ++j) {
                    if (p2.parentIndex.indexOf(p1.parentIndex[j]) > -1) {
                        // figure out the angle halfway between the two points
                        // on the current circle
                        var circle = circles[p1.parentIndex[j]],
                            a1 = Math.atan2(p1.x - circle.x, p1.y - circle.y),
                            a2 = Math.atan2(p2.x - circle.x, p2.y - circle.y);

                        var angleDiff = (a2 - a1);
                        if (angleDiff < 0) {
                            angleDiff += 2*Math.PI;
                        }

                        // and use that angle to figure out the width of the
                        // arc
                        var a = a2 - angleDiff/2,
                            width = distance(midPoint, {
                                x : circle.x + circle.radius * Math.sin(a),
                                y : circle.y + circle.radius * Math.cos(a)
                            });

                        // clamp the width to the largest is can actually be
                        // (sometimes slightly overflows because of FP errors)
                        if (width > circle.radius * 2) {
                            width = circle.radius * 2;
                        }

                        // pick the circle whose arc has the smallest width
                        if ((arc === null) || (arc.width > width)) {
                            arc = { circle : circle,
                                    width : width,
                                    p1 : p1,
                                    p2 : p2};
                        }
                    }
                }

                if (arc !== null) {
                    arcs.push(arc);
                    arcArea += circleArea(arc.circle.radius, arc.width);
                    p2 = p1;
                }
            }
        } else {
            // no intersection points, is either disjoint - or is completely
            // overlapped. figure out which by examining the smallest circle
            var smallest = circles[0];
            for (i = 1; i < circles.length; ++i) {
                if (circles[i].radius < smallest.radius) {
                    smallest = circles[i];
                }
            }

            // make sure the smallest circle is completely contained in all
            // the other circles
            var disjoint = false;
            for (i = 0; i < circles.length; ++i) {
                if (distance(circles[i], smallest) > Math.abs(smallest.radius - circles[i].radius)) {
                    disjoint = true;
                    break;
                }
            }

            if (disjoint) {
                arcArea = polygonArea = 0;

            } else {
                arcArea = smallest.radius * smallest.radius * Math.PI;
                arcs.push({circle : smallest,
                           p1: { x: smallest.x,        y : smallest.y + smallest.radius},
                           p2: { x: smallest.x - SMALL, y : smallest.y + smallest.radius},
                           width : smallest.radius * 2 });
            }
        }

        polygonArea /= 2;
        if (stats) {
            stats.area = arcArea + polygonArea;
            stats.arcArea = arcArea;
            stats.polygonArea = polygonArea;
            stats.arcs = arcs;
            stats.innerPoints = innerPoints;
            stats.intersectionPoints = intersectionPoints;
        }

        return arcArea + polygonArea;
    }

    /** returns whether a point is contained by all of a list of circles */
    function containedInCircles(point, circles) {
        for (var i = 0; i < circles.length; ++i) {
            if (distance(point, circles[i]) > circles[i].radius + SMALL) {
                return false;
            }
        }
        return true;
    }

    /** Gets all intersection points between a bunch of circles */
    function getIntersectionPoints(circles) {
        var ret = [];
        for (var i = 0; i < circles.length; ++i) {
            for (var j = i + 1; j < circles.length; ++j) {
                var intersect = circleCircleIntersection(circles[i],
                                                              circles[j]);
                for (var k = 0; k < intersect.length; ++k) {
                    var p = intersect[k];
                    p.parentIndex = [i,j];
                    ret.push(p);
                }
            }
        }
        return ret;
    }

    /** Circular segment area calculation. See http://mathworld.wolfram.com/CircularSegment.html */
    function circleArea(r, width) {
        return r * r * Math.acos(1 - width/r) - (r - width) * Math.sqrt(width * (2 * r - width));
    }

    /** euclidean distance between two points */
    function distance(p1, p2) {
        return Math.sqrt((p1.x - p2.x) * (p1.x - p2.x) +
                         (p1.y - p2.y) * (p1.y - p2.y));
    }


    /** Returns the overlap area of two circles of radius r1 and r2 - that
    have their centers separated by distance d. Simpler faster
    circle intersection for only two circles */
    function circleOverlap(r1, r2, d) {
        // no overlap
        if (d >= r1 + r2) {
            return 0;
        }

        // completely overlapped
        if (d <= Math.abs(r1 - r2)) {
            return Math.PI * Math.min(r1, r2) * Math.min(r1, r2);
        }

        var w1 = r1 - (d * d - r2 * r2 + r1 * r1) / (2 * d),
            w2 = r2 - (d * d - r1 * r1 + r2 * r2) / (2 * d);
        return circleArea(r1, w1) + circleArea(r2, w2);
    }

    /** Given two circles (containing a x/y/radius attributes),
    returns the intersecting points if possible.
    note: doesn't handle cases where there are infinitely many
    intersection points (circles are equivalent):, or only one intersection point*/
    function circleCircleIntersection(p1, p2) {
        var d = distance(p1, p2),
            r1 = p1.radius,
            r2 = p2.radius;

        // if to far away, or self contained - can't be done
        if ((d >= (r1 + r2)) || (d <= Math.abs(r1 - r2))) {
            return [];
        }

        var a = (r1 * r1 - r2 * r2 + d * d) / (2 * d),
            h = Math.sqrt(r1 * r1 - a * a),
            x0 = p1.x + a * (p2.x - p1.x) / d,
            y0 = p1.y + a * (p2.y - p1.y) / d,
            rx = -(p2.y - p1.y) * (h / d),
            ry = -(p2.x - p1.x) * (h / d);

        return [{x: x0 + rx, y : y0 - ry },
                {x: x0 - rx, y : y0 + ry }];
    }

    /** Returns the center of a bunch of points */
    function getCenter(points) {
        var center = {x: 0, y: 0};
        for (var i =0; i < points.length; ++i ) {
            center.x += points[i].x;
            center.y += points[i].y;
        }
        center.x /= points.length;
        center.y /= points.length;
        return center;
    }

    /** finds the zeros of a function, given two starting points (which must
     * have opposite signs */
    function bisect(f, a, b, parameters) {
        parameters = parameters || {};
        var maxIterations = parameters.maxIterations || 100,
            tolerance = parameters.tolerance || 1e-10,
            fA = f(a),
            fB = f(b),
            delta = b - a;

        if (fA * fB > 0) {
            throw "Initial bisect points must have opposite signs";
        }

        if (fA === 0) return a;
        if (fB === 0) return b;

        for (var i = 0; i < maxIterations; ++i) {
            delta /= 2;
            var mid = a + delta,
                fMid = f(mid);

            if (fMid * fA >= 0) {
                a = mid;
            }

            if ((Math.abs(delta) < tolerance) || (fMid === 0)) {
                return mid;
            }
        }
        return a + delta;
    }

    // need some basic operations on vectors, rather than adding a dependency,
    // just define here
    function zeros(x) { var r = new Array(x); for (var i = 0; i < x; ++i) { r[i] = 0; } return r; }
    function zerosM(x,y) { return zeros(x).map(function() { return zeros(y); }); }

    function dot(a, b) {
        var ret = 0;
        for (var i = 0; i < a.length; ++i) {
            ret += a[i] * b[i];
        }
        return ret;
    }

    function norm2(a)  {
        return Math.sqrt(dot(a, a));
    }

    function scale(ret, value, c) {
        for (var i = 0; i < value.length; ++i) {
            ret[i] = value[i] * c;
        }
    }

    function weightedSum(ret, w1, v1, w2, v2) {
        for (var j = 0; j < ret.length; ++j) {
            ret[j] = w1 * v1[j] + w2 * v2[j];
        }
    }

    /** minimizes a function using the downhill simplex method */
    function nelderMead(f, x0, parameters) {
        parameters = parameters || {};

        var maxIterations = parameters.maxIterations || x0.length * 200,
            nonZeroDelta = parameters.nonZeroDelta || 1.05,
            zeroDelta = parameters.zeroDelta || 0.001,
            minErrorDelta = parameters.minErrorDelta || 1e-6,
            minTolerance = parameters.minErrorDelta || 1e-5,
            rho = (parameters.rho !== undefined) ? parameters.rho : 1,
            chi = (parameters.chi !== undefined) ? parameters.chi : 2,
            psi = (parameters.psi !== undefined) ? parameters.psi : -0.5,
            sigma = (parameters.sigma !== undefined) ? parameters.sigma : 0.5,
            maxDiff;

        // initialize simplex.
        var N = x0.length,
            simplex = new Array(N + 1);
        simplex[0] = x0;
        simplex[0].fx = f(x0);
        simplex[0].id = 0;
        for (var i = 0; i < N; ++i) {
            var point = x0.slice();
            point[i] = point[i] ? point[i] * nonZeroDelta : zeroDelta;
            simplex[i+1] = point;
            simplex[i+1].fx = f(point);
            simplex[i+1].id = i+1;
        }

        function updateSimplex(value) {
            for (var i = 0; i < value.length; i++) {
                simplex[N][i] = value[i];
            }
            simplex[N].fx = value.fx;
        }

        var sortOrder = function(a, b) { return a.fx - b.fx; };

        var centroid = x0.slice(),
            reflected = x0.slice(),
            contracted = x0.slice(),
            expanded = x0.slice();

        for (var iteration = 0; iteration < maxIterations; ++iteration) {
            simplex.sort(sortOrder);

            if (parameters.history) {
                // copy the simplex (since later iterations will mutate) and
                // sort it to have a consistent order between iterations
                var sortedSimplex = simplex.map(function (x) {
                    var state = x.slice();
                    state.fx = x.fx;
                    state.id = x.id;
                    return state;
                });
                sortedSimplex.sort(function(a,b) { return a.id - b.id; });

                parameters.history.push({x: simplex[0].slice(),
                                         fx: simplex[0].fx,
                                         simplex: sortedSimplex});
            }

            maxDiff = 0;
            for (i = 0; i < N; ++i) {
                maxDiff = Math.max(maxDiff, Math.abs(simplex[0][i] - simplex[1][i]));
            }

            if ((Math.abs(simplex[0].fx - simplex[N].fx) < minErrorDelta) &&
                (maxDiff < minTolerance)) {
                break;
            }

            // compute the centroid of all but the worst point in the simplex
            for (i = 0; i < N; ++i) {
                centroid[i] = 0;
                for (var j = 0; j < N; ++j) {
                    centroid[i] += simplex[j][i];
                }
                centroid[i] /= N;
            }

            // reflect the worst point past the centroid  and compute loss at reflected
            // point
            var worst = simplex[N];
            weightedSum(reflected, 1+rho, centroid, -rho, worst);
            reflected.fx = f(reflected);

            // if the reflected point is the best seen, then possibly expand
            if (reflected.fx < simplex[0].fx) {
                weightedSum(expanded, 1+chi, centroid, -chi, worst);
                expanded.fx = f(expanded);
                if (expanded.fx < reflected.fx) {
                    updateSimplex(expanded);
                }  else {
                    updateSimplex(reflected);
                }
            }

            // if the reflected point is worse than the second worst, we need to
            // contract
            else if (reflected.fx >= simplex[N-1].fx) {
                var shouldReduce = false;

                if (reflected.fx > worst.fx) {
                    // do an inside contraction
                    weightedSum(contracted, 1+psi, centroid, -psi, worst);
                    contracted.fx = f(contracted);
                    if (contracted.fx < worst.fx) {
                        updateSimplex(contracted);
                    } else {
                        shouldReduce = true;
                    }
                } else {
                    // do an outside contraction
                    weightedSum(contracted, 1-psi * rho, centroid, psi*rho, worst);
                    contracted.fx = f(contracted);
                    if (contracted.fx < reflected.fx) {
                        updateSimplex(contracted);
                    } else {
                        shouldReduce = true;
                    }
                }

                if (shouldReduce) {
                    // if we don't contract here, we're done
                    if (sigma >= 1) break;

                    // do a reduction
                    for (i = 1; i < simplex.length; ++i) {
                        weightedSum(simplex[i], 1 - sigma, simplex[0], sigma, simplex[i]);
                        simplex[i].fx = f(simplex[i]);
                    }
                }
            } else {
                updateSimplex(reflected);
            }
        }

        simplex.sort(sortOrder);
        return {fx : simplex[0].fx,
                x : simplex[0]};
    }

    /// searches along line 'pk' for a point that satifies the wolfe conditions
    /// See 'Numerical Optimization' by Nocedal and Wright p59-60
    /// f : objective function
    /// pk : search direction
    /// current: object containing current gradient/loss
    /// next: output: contains next gradient/loss
    /// returns a: step size taken
    function wolfeLineSearch(f, pk, current, next, a, c1, c2) {
        var phi0 = current.fx, phiPrime0 = dot(current.fxprime, pk),
            phi = phi0, phi_old = phi0,
            phiPrime = phiPrime0,
            a0 = 0;

        a = a || 1;
        c1 = c1 || 1e-6;
        c2 = c2 || 0.1;

        function zoom(a_lo, a_high, phi_lo) {
            for (var iteration = 0; iteration < 16; ++iteration) {
                a = (a_lo + a_high)/2;
                weightedSum(next.x, 1.0, current.x, a, pk);
                phi = next.fx = f(next.x, next.fxprime);
                phiPrime = dot(next.fxprime, pk);

                if ((phi > (phi0 + c1 * a * phiPrime0)) ||
                    (phi >= phi_lo)) {
                    a_high = a;

                } else  {
                    if (Math.abs(phiPrime) <= -c2 * phiPrime0) {
                        return a;
                    }

                    if (phiPrime * (a_high - a_lo) >=0) {
                        a_high = a_lo;
                    }

                    a_lo = a;
                    phi_lo = phi;
                }
            }

            return 0;
        }

        for (var iteration = 0; iteration < 10; ++iteration) {
            weightedSum(next.x, 1.0, current.x, a, pk);
            phi = next.fx = f(next.x, next.fxprime);
            phiPrime = dot(next.fxprime, pk);
            if ((phi > (phi0 + c1 * a * phiPrime0)) ||
                (iteration && (phi >= phi_old))) {
                return zoom(a0, a, phi_old);
            }

            if (Math.abs(phiPrime) <= -c2 * phiPrime0) {
                return a;
            }

            if (phiPrime >= 0 ) {
                return zoom(a, a0, phi);
            }

            phi_old = phi;
            a0 = a;
            a *= 2;
        }

        return a;
    }

    function conjugateGradient(f, initial, params) {
        // allocate all memory up front here, keep out of the loop for perfomance
        // reasons
        var current = {x: initial.slice(), fx: 0, fxprime: initial.slice()},
            next = {x: initial.slice(), fx: 0, fxprime: initial.slice()},
            yk = initial.slice(),
            pk, temp,
            a = 1,
            maxIterations;

        params = params || {};
        maxIterations = params.maxIterations || initial.length * 20;

        current.fx = f(current.x, current.fxprime);
        pk = current.fxprime.slice();
        scale(pk, current.fxprime,-1);

        for (var i = 0; i < maxIterations; ++i) {
            a = wolfeLineSearch(f, pk, current, next, a);

            // todo: history in wrong spot?
            if (params.history) {
                params.history.push({x: current.x.slice(),
                                     fx: current.fx,
                                     fxprime: current.fxprime.slice(),
                                     alpha: a});
            }

            if (!a) {
                // faiiled to find point that satifies wolfe conditions.
                // reset direction for next iteration
                scale(pk, current.fxprime, -1);

            } else {
                // update direction using Polak–Ribiere CG method
                weightedSum(yk, 1, next.fxprime, -1, current.fxprime);

                var delta_k = dot(current.fxprime, current.fxprime),
                    beta_k = Math.max(0, dot(yk, next.fxprime) / delta_k);

                weightedSum(pk, beta_k, pk, -1, next.fxprime);

                temp = current;
                current = next;
                next = temp;
            }

            if (norm2(current.fxprime) <= 1e-5) {
                break;
            }
        }

        if (params.history) {
            params.history.push({x: current.x.slice(),
                                 fx: current.fx,
                                 fxprime: current.fxprime.slice(),
                                 alpha: a});
        }

        return current;
    }

    /** given a list of set objects, and their corresponding overlaps.
    updates the (x, y, radius) attribute on each set such that their positions
    roughly correspond to the desired overlaps */
    function venn(areas, parameters) {
        parameters = parameters || {};
        parameters.maxIterations = parameters.maxIterations || 500;
        var initialLayout = parameters.initialLayout || bestInitialLayout;
        var loss = parameters.lossFunction || lossFunction;

        // add in missing pairwise areas as having 0 size
        areas = addMissingAreas(areas);

        // initial layout is done greedily
        var circles = initialLayout(areas, parameters);

        // transform x/y coordinates to a vector to optimize
        var initial = [], setids = [], setid;
        for (setid in circles) {
            if (circles.hasOwnProperty(setid)) {
                initial.push(circles[setid].x);
                initial.push(circles[setid].y);
                setids.push(setid);
            }
        }
        var solution = nelderMead(
            function(values) {
                var current = {};
                for (var i = 0; i < setids.length; ++i) {
                    var setid = setids[i];
                    current[setid] = {x: values[2 * i],
                                      y: values[2 * i + 1],
                                      radius : circles[setid].radius,
                                     // size : circles[setid].size
                                     };
                }
                return loss(current, areas);
            },
            initial,
            parameters);

        // transform solution vector back to x/y points
        var positions = solution.x;
        for (var i = 0; i < setids.length; ++i) {
            setid = setids[i];
            circles[setid].x = positions[2 * i];
            circles[setid].y = positions[2 * i + 1];
        }

        return circles;
    }

    var SMALL$1 = 1e-10;

    /** Returns the distance necessary for two circles of radius r1 + r2 to
    have the overlap area 'overlap' */
    function distanceFromIntersectArea(r1, r2, overlap) {
        // handle complete overlapped circles
        if (Math.min(r1, r2) * Math.min(r1,r2) * Math.PI <= overlap + SMALL$1) {
            return Math.abs(r1 - r2);
        }

        return bisect(function(distance$$1) {
            return circleOverlap(r1, r2, distance$$1) - overlap;
        }, 0, r1 + r2);
    }

    /** Missing pair-wise intersection area data can cause problems:
     treating as an unknown means that sets will be laid out overlapping,
     which isn't what people expect. To reflect that we want disjoint sets
     here, set the overlap to 0 for all missing pairwise set intersections */
    function addMissingAreas(areas) {
        areas = areas.slice();

        // two circle intersections that aren't defined
        var ids = [], pairs = {}, i, j, a, b;
        for (i = 0; i < areas.length; ++i) {
            var area = areas[i];
            if (area.sets.length == 1) {
                ids.push(area.sets[0]);
            } else if (area.sets.length == 2) {
                a = area.sets[0];
                b = area.sets[1];
                pairs[[a, b]] = true;
                pairs[[b, a]] = true;
            }
        }
        ids.sort(function(a, b) { return a > b; });

        for (i = 0; i < ids.length; ++i) {
            a = ids[i];
            for (j = i + 1; j < ids.length; ++j) {
                b = ids[j];
                if (!([a, b] in pairs)) {
                    areas.push({'sets': [a, b],
                                'size': 0});
                }
            }
        }
        return areas;
    }

    /// Returns two matrices, one of the euclidean distances between the sets
    /// and the other indicating if there are subset or disjoint set relationships
    function getDistanceMatrices(areas, sets, setids) {
        // initialize an empty distance matrix between all the points
        var distances = zerosM(sets.length, sets.length),
            constraints = zerosM(sets.length, sets.length);

        // compute required distances between all the sets such that
        // the areas match
        areas.filter(function(x) { return x.sets.length == 2; })
            .map(function(current) {
            var left = setids[current.sets[0]],
                right = setids[current.sets[1]],
                r1 = Math.sqrt(sets[left].size / Math.PI),
                r2 = Math.sqrt(sets[right].size / Math.PI),
                distance$$1 = distanceFromIntersectArea(r1, r2, current.size);

            distances[left][right] = distances[right][left] = distance$$1;

            // also update constraints to indicate if its a subset or disjoint
            // relationship
            var c = 0;
            if (current.size + 1e-10 >= Math.min(sets[left].size,
                                                 sets[right].size)) {
                c = 1;
            } else if (current.size <= 1e-10) {
                c = -1;
            }
            constraints[left][right] = constraints[right][left] = c;
        });

        return {distances: distances, constraints: constraints};
    }

    /// computes the gradient and loss simulatenously for our constrained MDS optimizer
    function constrainedMDSGradient(x, fxprime, distances, constraints) {
        var loss = 0, i;
        for (i = 0; i < fxprime.length; ++i) {
            fxprime[i] = 0;
        }

        for (i = 0; i < distances.length; ++i) {
            var xi = x[2 * i], yi = x[2 * i + 1];
            for (var j = i + 1; j < distances.length; ++j) {
                var xj = x[2 * j], yj = x[2 * j + 1],
                    dij = distances[i][j],
                    constraint = constraints[i][j];

                var squaredDistance = (xj - xi) * (xj - xi) + (yj - yi) * (yj - yi),
                    distance$$1 = Math.sqrt(squaredDistance),
                    delta = squaredDistance - dij * dij;

                if (((constraint > 0) && (distance$$1 <= dij)) ||
                    ((constraint < 0) && (distance$$1 >= dij))) {
                    continue;
                }

                loss += 2 * delta * delta;

                fxprime[2*i]     += 4 * delta * (xi - xj);
                fxprime[2*i + 1] += 4 * delta * (yi - yj);

                fxprime[2*j]     += 4 * delta * (xj - xi);
                fxprime[2*j + 1] += 4 * delta * (yj - yi);
            }
        }
        return loss;
    }

    /// takes the best working variant of either constrained MDS or greedy
    function bestInitialLayout(areas, params) {
        var initial = greedyLayout(areas, params);
        var loss = params.lossFunction || lossFunction;

        // greedylayout is sufficient for all 2/3 circle cases. try out
        // constrained MDS for higher order problems, take its output
        // if it outperforms. (greedy is aesthetically better on 2/3 circles
        // since it axis aligns)
        if (areas.length >= 8) {
            var constrained  = constrainedMDSLayout(areas, params),
                constrainedLoss = loss(constrained, areas),
                greedyLoss = loss(initial, areas);

            if (constrainedLoss + 1e-8 < greedyLoss) {
                initial = constrained;
            }
        }
        return initial;
    }

    /// use the constrained MDS variant to generate an initial layout
    function constrainedMDSLayout(areas, params) {
        params = params || {};
        var restarts = params.restarts || 10;

        // bidirectionally map sets to a rowid  (so we can create a matrix)
        var sets = [], setids = {}, i;
        for (i = 0; i < areas.length; ++i ) {
            var area = areas[i];
            if (area.sets.length == 1) {
                setids[area.sets[0]] = sets.length;
                sets.push(area);
            }
        }

        var matrices = getDistanceMatrices(areas, sets, setids),
            distances = matrices.distances,
            constraints = matrices.constraints;

        // keep distances bounded, things get messed up otherwise.
        // TODO: proper preconditioner?
        var norm = norm2(distances.map(norm2))/(distances.length);
        distances = distances.map(function (row) {
            return row.map(function (value) { return value / norm; });});

        var obj = function(x, fxprime) {
            return constrainedMDSGradient(x, fxprime, distances, constraints);
        };

        var best, current;
        for (i = 0; i < restarts; ++i) {
            var initial = zeros(distances.length*2).map(Math.random);

            current = conjugateGradient(obj, initial, params);
            if (!best || (current.fx < best.fx)) {
                best = current;
            }
        }
        var positions = best.x;

        // translate rows back to (x,y,radius) coordinates
        var circles = {};
        for (i = 0; i < sets.length; ++i) {
            var set = sets[i];
            circles[set.sets[0]] = {
                x: positions[2*i] * norm,
                y: positions[2*i + 1] * norm,
                radius:  Math.sqrt(set.size / Math.PI)
            };
        }

        if (params.history) {
            for (i = 0; i < params.history.length; ++i) {
                scale(params.history[i].x, norm);
            }
        }
        return circles;
    }

    /** Lays out a Venn diagram greedily, going from most overlapped sets to
    least overlapped, attempting to position each new set such that the
    overlapping areas to already positioned sets are basically right */
    function greedyLayout(areas, params) {
        var loss = params && params.lossFunction ? params.lossFunction : lossFunction;
        // define a circle for each set
        var circles = {}, setOverlaps = {}, set;
        for (var i = 0; i < areas.length; ++i) {
            var area = areas[i];
            if (area.sets.length == 1) {
                set = area.sets[0];
                circles[set] = {x: 1e10, y: 1e10,
                                rowid: circles.length,
                                size: area.size,
                                radius: Math.sqrt(area.size / Math.PI)};
                setOverlaps[set] = [];
            }
        }
        areas = areas.filter(function(a) { return a.sets.length == 2; });

        // map each set to a list of all the other sets that overlap it
        for (i = 0; i < areas.length; ++i) {
            var current = areas[i];
            var weight = current.hasOwnProperty('weight') ? current.weight : 1.0;
            var left = current.sets[0], right = current.sets[1];

            // completely overlapped circles shouldn't be positioned early here
            if (current.size + SMALL$1 >= Math.min(circles[left].size,
                                                 circles[right].size)) {
                weight = 0;
            }

            setOverlaps[left].push ({set:right, size:current.size, weight:weight});
            setOverlaps[right].push({set:left,  size:current.size, weight:weight});
        }

        // get list of most overlapped sets
        var mostOverlapped = [];
        for (set in setOverlaps) {
            if (setOverlaps.hasOwnProperty(set)) {
                var size = 0;
                for (i = 0; i < setOverlaps[set].length; ++i) {
                    size += setOverlaps[set][i].size * setOverlaps[set][i].weight;
                }

                mostOverlapped.push({set: set, size:size});
            }
        }

        // sort by size desc
        function sortOrder(a,b) {
            return b.size - a.size;
        }
        mostOverlapped.sort(sortOrder);

        // keep track of what sets have been laid out
        var positioned = {};
        function isPositioned(element) {
            return element.set in positioned;
        }

        // adds a point to the output
        function positionSet(point, index) {
            circles[index].x = point.x;
            circles[index].y = point.y;
            positioned[index] = true;
        }

        // add most overlapped set at (0,0)
        positionSet({x: 0, y: 0}, mostOverlapped[0].set);

        // get distances between all points. TODO, necessary?
        // answer: probably not
        // var distances = venn.getDistanceMatrices(circles, areas).distances;
        for (i = 1; i < mostOverlapped.length; ++i) {
            var setIndex = mostOverlapped[i].set,
                overlap = setOverlaps[setIndex].filter(isPositioned);
            set = circles[setIndex];
            overlap.sort(sortOrder);

            if (overlap.length === 0) {
                // this shouldn't happen anymore with addMissingAreas
                throw "ERROR: missing pairwise overlap information";
            }

            var points = [];
            for (var j = 0; j < overlap.length; ++j) {
                // get appropriate distance from most overlapped already added set
                var p1 = circles[overlap[j].set],
                    d1 = distanceFromIntersectArea(set.radius, p1.radius,
                                                   overlap[j].size);

                // sample positions at 90 degrees for maximum aesthetics
                points.push({x : p1.x + d1, y : p1.y});
                points.push({x : p1.x - d1, y : p1.y});
                points.push({y : p1.y + d1, x : p1.x});
                points.push({y : p1.y - d1, x : p1.x});

                // if we have at least 2 overlaps, then figure out where the
                // set should be positioned analytically and try those too
                for (var k = j + 1; k < overlap.length; ++k) {
                    var p2 = circles[overlap[k].set],
                        d2 = distanceFromIntersectArea(set.radius, p2.radius,
                                                       overlap[k].size);

                    var extraPoints = circleCircleIntersection(
                        { x: p1.x, y: p1.y, radius: d1},
                        { x: p2.x, y: p2.y, radius: d2});

                    for (var l = 0; l < extraPoints.length; ++l) {
                        points.push(extraPoints[l]);
                    }
                }
            }

            // we have some candidate positions for the set, examine loss
            // at each position to figure out where to put it at
            var bestLoss = 1e50, bestPoint = points[0];
            for (j = 0; j < points.length; ++j) {
                circles[setIndex].x = points[j].x;
                circles[setIndex].y = points[j].y;
                var localLoss = loss(circles, areas);
                if (localLoss < bestLoss) {
                    bestLoss = localLoss;
                    bestPoint = points[j];
                }
            }

            positionSet(bestPoint, setIndex);
        }

        return circles;
    }

    /** Given a bunch of sets, and the desired overlaps between these sets - computes
    the distance from the actual overlaps to the desired overlaps. Note that
    this method ignores overlaps of more than 2 circles */
    function lossFunction(sets, overlaps) {
        var output = 0;

        function getCircles(indices) {
            return indices.map(function(i) { return sets[i]; });
        }

        for (var i = 0; i < overlaps.length; ++i) {
            var area = overlaps[i], overlap;
            if (area.sets.length == 1) {
                continue;
            } else if (area.sets.length == 2) {
                var left = sets[area.sets[0]],
                    right = sets[area.sets[1]];
                overlap = circleOverlap(left.radius, right.radius,
                                        distance(left, right));
            } else {
                overlap = intersectionArea(getCircles(area.sets));
            }

            var weight = area.hasOwnProperty('weight') ? area.weight : 1.0;
            output += weight * (overlap - area.size) * (overlap - area.size);
        }

        return output;
    }

    // orientates a bunch of circles to point in orientation
    function orientateCircles(circles, orientation, orientationOrder) {
        if (orientationOrder === null) {
            circles.sort(function (a, b) { return b.radius - a.radius; });
        } else {
            circles.sort(orientationOrder);
        }

        var i;
        // shift circles so largest circle is at (0, 0)
        if (circles.length > 0) {
            var largestX = circles[0].x,
                largestY = circles[0].y;

            for (i = 0; i < circles.length; ++i) {
                circles[i].x -= largestX;
                circles[i].y -= largestY;
            }
        }

        if (circles.length == 2) {
            // if the second circle is a subset of the first, arrange so that
            // it is off to one side. hack for https://github.com/benfred/venn.js/issues/120
            var dist = distance(circles[0], circles[1]);
            if (dist < Math.abs(circles[1].radius - circles[0].radius)) {
                circles[1].x = circles[0].x + circles[0].radius - circles[1].radius - 1e-10;
                circles[1].y = circles[0].y;
            }
        }

        // rotate circles so that second largest is at an angle of 'orientation'
        // from largest
        if (circles.length > 1) {
            var rotation = Math.atan2(circles[1].x, circles[1].y) - orientation,
                c = Math.cos(rotation),
                s = Math.sin(rotation), x, y;

            for (i = 0; i < circles.length; ++i) {
                x = circles[i].x;
                y = circles[i].y;
                circles[i].x = c * x - s * y;
                circles[i].y = s * x + c * y;
            }
        }

        // mirror solution if third solution is above plane specified by
        // first two circles
        if (circles.length > 2) {
            var angle = Math.atan2(circles[2].x, circles[2].y) - orientation;
            while (angle < 0) { angle += 2* Math.PI; }
            while (angle > 2*Math.PI) { angle -= 2* Math.PI; }
            if (angle > Math.PI) {
                var slope = circles[1].y / (1e-10 + circles[1].x);
                for (i = 0; i < circles.length; ++i) {
                    var d = (circles[i].x + slope * circles[i].y) / (1 + slope*slope);
                    circles[i].x = 2 * d - circles[i].x;
                    circles[i].y = 2 * d * slope - circles[i].y;
                }
            }
        }
    }

    function disjointCluster(circles) {
        // union-find clustering to get disjoint sets
        circles.map(function(circle) { circle.parent = circle; });

        // path compression step in union find
        function find(circle) {
            if (circle.parent !== circle) {
                circle.parent = find(circle.parent);
            }
            return circle.parent;
        }

        function union(x, y) {
            var xRoot = find(x), yRoot = find(y);
            xRoot.parent = yRoot;
        }

        // get the union of all overlapping sets
        for (var i = 0; i < circles.length; ++i) {
            for (var j = i + 1; j < circles.length; ++j) {
                var maxDistance = circles[i].radius + circles[j].radius;
                if (distance(circles[i], circles[j]) + 1e-10 < maxDistance) {
                    union(circles[j], circles[i]);
                }
            }
        }

        // find all the disjoint clusters and group them together
        var disjointClusters = {}, setid;
        for (i = 0; i < circles.length; ++i) {
            setid = find(circles[i]).parent.setid;
            if (!(setid in disjointClusters)) {
                disjointClusters[setid] = [];
            }
            disjointClusters[setid].push(circles[i]);
        }

        // cleanup bookkeeping
        circles.map(function(circle) { delete circle.parent; });

        // return in more usable form
        var ret = [];
        for (setid in disjointClusters) {
            if (disjointClusters.hasOwnProperty(setid)) {
                ret.push(disjointClusters[setid]);
            }
        }
        return ret;
    }

    function getBoundingBox(circles) {
        var minMax = function(d) {
            var hi = Math.max.apply(null, circles.map(
                                    function(c) { return c[d] + c.radius; } )),
                lo = Math.min.apply(null, circles.map(
                                    function(c) { return c[d] - c.radius;} ));
            return {max:hi, min:lo};
        };

        return {xRange: minMax('x'), yRange: minMax('y')};
    }

    function normalizeSolution(solution, orientation, orientationOrder) {
        if (orientation === null){
            orientation = Math.PI/2;
        }

        // work with a list instead of a dictionary, and take a copy so we
        // don't mutate input
        var circles = [], i, setid;
        for (setid in solution) {
            if (solution.hasOwnProperty(setid)) {
                var previous = solution[setid];
                circles.push({x: previous.x,
                              y: previous.y,
                              radius: previous.radius,
                              setid: setid});
            }
        }

        // get all the disjoint clusters
        var clusters = disjointCluster(circles);

        // orientate all disjoint sets, get sizes
        for (i = 0; i < clusters.length; ++i) {
            orientateCircles(clusters[i], orientation, orientationOrder);
            var bounds = getBoundingBox(clusters[i]);
            clusters[i].size = (bounds.xRange.max - bounds.xRange.min) * (bounds.yRange.max - bounds.yRange.min);
            clusters[i].bounds = bounds;
        }
        clusters.sort(function(a, b) { return b.size - a.size; });

        // orientate the largest at 0,0, and get the bounds
        circles = clusters[0];
        var returnBounds = circles.bounds;

        var spacing = (returnBounds.xRange.max - returnBounds.xRange.min)/50;

        function addCluster(cluster, right, bottom) {
            if (!cluster) return;

            var bounds = cluster.bounds, xOffset, yOffset, centreing;

            if (right) {
                xOffset = returnBounds.xRange.max  - bounds.xRange.min + spacing;
            } else {
                xOffset = returnBounds.xRange.max  - bounds.xRange.max;
                centreing = (bounds.xRange.max - bounds.xRange.min) / 2 -
                            (returnBounds.xRange.max - returnBounds.xRange.min) / 2;
                if (centreing < 0) xOffset += centreing;
            }

            if (bottom) {
                yOffset = returnBounds.yRange.max  - bounds.yRange.min + spacing;
            } else {
                yOffset = returnBounds.yRange.max  - bounds.yRange.max;
                centreing = (bounds.yRange.max - bounds.yRange.min) / 2 -
                            (returnBounds.yRange.max - returnBounds.yRange.min) / 2;
                if (centreing < 0) yOffset += centreing;
            }

            for (var j = 0; j < cluster.length; ++j) {
                cluster[j].x += xOffset;
                cluster[j].y += yOffset;
                circles.push(cluster[j]);
            }
        }

        var index = 1;
        while (index < clusters.length) {
            addCluster(clusters[index], true, false);
            addCluster(clusters[index+1], false, true);
            addCluster(clusters[index+2], true, true);
            index += 3;

            // have one cluster (in top left). lay out next three relative
            // to it in a grid
            returnBounds = getBoundingBox(circles);
        }

        // convert back to solution form
        var ret = {};
        for (i = 0; i < circles.length; ++i) {
            ret[circles[i].setid] = circles[i];
        }
        return ret;
    }

    /** Scales a solution from venn.venn or venn.greedyLayout such that it fits in
    a rectangle of width/height - with padding around the borders. also
    centers the diagram in the available space at the same time */
    function scaleSolution(solution, width, height, padding) {
        var circles = [], setids = [];
        for (var setid in solution) {
            if (solution.hasOwnProperty(setid)) {
                setids.push(setid);
                circles.push(solution[setid]);
            }
        }

        width -= 2*padding;
        height -= 2*padding;

        var bounds = getBoundingBox(circles),
            xRange = bounds.xRange,
            yRange = bounds.yRange;

        if ((xRange.max == xRange.min) ||
            (yRange.max == yRange.min)) {
            console.log("not scaling solution: zero size detected");
            return solution;
        }

        var xScaling = width  / (xRange.max - xRange.min),
            yScaling = height / (yRange.max - yRange.min),
            scaling = Math.min(yScaling, xScaling),

            // while we're at it, center the diagram too
            xOffset = (width -  (xRange.max - xRange.min) * scaling) / 2,
            yOffset = (height - (yRange.max - yRange.min) * scaling) / 2;

        var scaled = {};
        for (var i = 0; i < circles.length; ++i) {
            var circle = circles[i];
            scaled[setids[i]] = {
                radius: scaling * circle.radius,
                x: padding + xOffset + (circle.x - xRange.min) * scaling,
                y: padding + yOffset + (circle.y - yRange.min) * scaling,
            };
        }

        return scaled;
    }

    /*global console:true*/

    function VennDiagram() {
        var width = 600,
            height = 350,
            padding = 15,
            duration = 1000,
            orientation = Math.PI / 2,
            normalize = true,
            wrap = true,
            styled = true,
            fontSize = null,
            orientationOrder = null,

            // mimic the behaviour of d3.scale.category10 from the previous
            // version of d3
            colourMap = {},

            // so this is the same as d3.schemeCategory10, which is only defined in d3 4.0
            // since we can support older versions of d3 as long as we don't force this,
            // I'm hackily redefining below. TODO: remove this and change to d3.schemeCategory10
            colourScheme = ["#1f77b4", "#ff7f0e", "#2ca02c", "#d62728", "#9467bd", "#8c564b", "#e377c2", "#7f7f7f", "#bcbd22", "#17becf"],
            colourIndex = 0,
            colours = function(key) {
                if (key in colourMap) {
                    return colourMap[key];
                }
                var ret = colourMap[key] = colourScheme[colourIndex];
                colourIndex += 1;
                if (colourIndex >= colourScheme.length) {
                    colourIndex = 0;
                }
                return ret;
            },
            layoutFunction = venn,
            loss = lossFunction;


        function chart(selection) {
            var data = selection.datum();

            // handle 0-sized sets by removing from input
            var toremove = {};
            data.forEach(function(datum) {
                if ((datum.size == 0) && datum.sets.length == 1) {
                    toremove[datum.sets[0]] = 1;
                }
            });
            data = data.filter(function(datum) {
                return !datum.sets.some(function(set) { return set in toremove; });
            });

            var circles = {};
            var textCentres = {};

            if (data.length > 0) {
                var solution = layoutFunction(data, {lossFunction: loss});

                if (normalize) {
                    solution = normalizeSolution(solution,
                                                orientation,
                                                orientationOrder);
                }

                circles = scaleSolution(solution, width, height, padding);
                textCentres = computeTextCentres(circles, data);
            }

            // Figure out the current label for each set. These can change
            // and D3 won't necessarily update (fixes https://github.com/benfred/venn.js/issues/103)
            var labels = {};
            data.forEach(function(datum) {
                if (datum.label) {
                    labels[datum.sets] = datum.label;
                }
            });

            function label(d) {
                if (d.sets in labels) {
                    return labels[d.sets];
                }
                if (d.sets.length == 1) {
                    return '' + d.sets[0];
                }
            }

            // create svg if not already existing
            selection.selectAll("svg").data([circles]).enter().append("svg");

            var svg = selection.select("svg")
                .attr("width", width)
                .attr("height", height);

            // to properly transition intersection areas, we need the
            // previous circles locations. load from elements
            var previous = {}, hasPrevious = false;
            svg.selectAll(".venn-area path").each(function (d) {
                var path = d3Selection.select(this).attr("d");
                if ((d.sets.length == 1) && path) {
                    hasPrevious = true;
                    previous[d.sets[0]] = circleFromPath(path);
                }
            });

            // interpolate intersection area paths between previous and
            // current paths
            var pathTween = function(d) {
                return function(t) {
                    var c = d.sets.map(function(set) {
                        var start = previous[set], end = circles[set];
                        if (!start) {
                            start = {x : width/2, y : height/2, radius : 1};
                        }
                        if (!end) {
                            end = {x : width/2, y : height/2, radius : 1};
                        }
                        return {'x' : start.x * (1 - t) + end.x * t,
                                'y' : start.y * (1 - t) + end.y * t,
                                'radius' : start.radius * (1 - t) + end.radius * t};
                    });
                    return intersectionAreaPath(c);
                };
            };

            // update data, joining on the set ids
            var nodes = svg.selectAll(".venn-area")
                .data(data, function(d) { return d.sets; });

            // create new nodes
            var enter = nodes.enter()
                .append('g')
                .attr("class", function(d) {
                    return "venn-area venn-" +
                        (d.sets.length == 1 ? "circle" : "intersection");
                })
                .attr("data-venn-sets", function(d) {
                    return d.sets.join("_");
                });

            var enterPath = enter.append("path"),
                enterText = enter.append("text")
                .attr("class", "label")
                .text(function (d) { return label(d); } )
                .attr("text-anchor", "middle")
                .attr("dy", ".35em")
                .attr("x", width/2)
                .attr("y", height/2);


            // apply minimal style if wanted
            if (styled) {
                enterPath.style("fill-opacity", "0")
                    .filter(function (d) { return d.sets.length == 1; } )
                    .style("fill", function(d) { return colours(d.sets); })
                    .style("fill-opacity", ".25");

                enterText
                    .style("fill", function(d) { return d.sets.length == 1 ? colours(d.sets) : "#444"; });
            }

            // update existing, using pathTween if necessary
            var update = selection;
            if (hasPrevious) {
                update = selection.transition("venn").duration(duration);
                update.selectAll("path")
                    .attrTween("d", pathTween);
            } else {
                update.selectAll("path")
                    .attr("d", function(d) {
                        return intersectionAreaPath(d.sets.map(function (set) { return circles[set]; }));
                    });
            }

            var updateText = update.selectAll("text")
                .filter(function (d) { return d.sets in textCentres; })
                .text(function (d) { return label(d); } )
                .attr("x", function(d) { return Math.floor(textCentres[d.sets].x);})
                .attr("y", function(d) { return Math.floor(textCentres[d.sets].y);});

            if (wrap) {
                if (hasPrevious) {
                    // d3 4.0 uses 'on' for events on transitions,
                    // but d3 3.0 used 'each' instead. switch appropiately
                    if ('on' in updateText) {
                        updateText.on("end", wrapText(circles, label));
                    } else {
                        updateText.each("end", wrapText(circles, label));
                    }
                } else {
                    updateText.each(wrapText(circles, label));
                }
            }

            // remove old
            var exit = nodes.exit().transition('venn').duration(duration).remove();
            exit.selectAll("path")
                .attrTween("d", pathTween);

            var exitText = exit.selectAll("text")
                .attr("x", width/2)
                .attr("y", height/2);

            // if we've been passed a fontSize explicitly, use it to
            // transition
            if (fontSize !== null) {
                enterText.style("font-size", "0px");
                updateText.style("font-size", fontSize);
                exitText.style("font-size", "0px");
            }


            return {'circles': circles,
                    'textCentres': textCentres,
                    'nodes': nodes,
                    'enter': enter,
                    'update': update,
                    'exit': exit};
        }

        chart.wrap = function(_) {
            if (!arguments.length) return wrap;
            wrap = _;
            return chart;
        };

        chart.width = function(_) {
            if (!arguments.length) return width;
            width = _;
            return chart;
        };

        chart.height = function(_) {
            if (!arguments.length) return height;
            height = _;
            return chart;
        };

        chart.padding = function(_) {
            if (!arguments.length) return padding;
            padding = _;
            return chart;
        };

        chart.colours = function(_) {
            if (!arguments.length) return colours;
            colours = _;
            return chart;
        };

        chart.fontSize = function(_) {
            if (!arguments.length) return fontSize;
            fontSize = _;
            return chart;
        };

        chart.duration = function(_) {
            if (!arguments.length) return duration;
            duration = _;
            return chart;
        };

        chart.layoutFunction = function(_) {
            if (!arguments.length) return layoutFunction;
            layoutFunction = _;
            return chart;
        };

        chart.normalize = function(_) {
            if (!arguments.length) return normalize;
            normalize = _;
            return chart;
        };

        chart.styled = function(_) {
            if (!arguments.length) return styled;
            styled = _;
            return chart;
        };

        chart.orientation = function(_) {
            if (!arguments.length) return orientation;
            orientation = _;
            return chart;
        };

        chart.orientationOrder = function(_) {
            if (!arguments.length) return orientationOrder;
            orientationOrder = _;
            return chart;
        };

        chart.lossFunction = function(_) {
          if (!arguments.length) return loss;
          loss = _;
          return chart;
        };

        return chart;
    }
    // sometimes text doesn't fit inside the circle, if thats the case lets wrap
    // the text here such that it fits
    // todo: looks like this might be merged into d3 (
    // https://github.com/mbostock/d3/issues/1642),
    // also worth checking out is
    // http://engineering.findthebest.com/wrapping-axis-labels-in-d3-js/
    // this seems to be one of those things that should be easy but isn't
    function wrapText(circles, labeller) {
        return function() {
            var text = d3Selection.select(this),
                data = text.datum(),
                width = circles[data.sets[0]].radius || 50,
                label = labeller(data) || '';

                var words = label.split(/\s+/).reverse(),
                maxLines = 3,
                minChars = (label.length + words.length) / maxLines,
                word = words.pop(),
                line = [word],
                joined,
                lineNumber = 0,
                lineHeight = 1.1, // ems
                tspan = text.text(null).append("tspan").text(word);

            while (true) {
                word = words.pop();
                if (!word) break;
                line.push(word);
                joined = line.join(" ");
                tspan.text(joined);
                if (joined.length > minChars && tspan.node().getComputedTextLength() > width) {
                    line.pop();
                    tspan.text(line.join(" "));
                    line = [word];
                    tspan = text.append("tspan").text(word);
                    lineNumber++;
                }
            }

            var initial = 0.35 - lineNumber * lineHeight / 2,
                x = text.attr("x"),
                y = text.attr("y");

            text.selectAll("tspan")
                .attr("x", x)
                .attr("y", y)
                .attr("dy", function(d, i) {
                     return (initial + i * lineHeight) + "em";
                });
        };
    }

    function circleMargin(current, interior, exterior) {
        var margin = interior[0].radius - distance(interior[0], current), i, m;
        for (i = 1; i < interior.length; ++i) {
            m = interior[i].radius - distance(interior[i], current);
            if (m <= margin) {
                margin = m;
            }
        }

        for (i = 0; i < exterior.length; ++i) {
            m = distance(exterior[i], current) - exterior[i].radius;
            if (m <= margin) {
                margin = m;
            }
        }
        return margin;
    }

    // compute the center of some circles by maximizing the margin of
    // the center point relative to the circles (interior) after subtracting
    // nearby circles (exterior)
    function computeTextCentre(interior, exterior) {
        // get an initial estimate by sampling around the interior circles
        // and taking the point with the biggest margin
        var points = [], i;
        for (i = 0; i < interior.length; ++i) {
            var c = interior[i];
            points.push({x: c.x, y: c.y});
            points.push({x: c.x + c.radius/2, y: c.y});
            points.push({x: c.x - c.radius/2, y: c.y});
            points.push({x: c.x, y: c.y + c.radius/2});
            points.push({x: c.x, y: c.y - c.radius/2});
        }
        var initial = points[0], margin = circleMargin(points[0], interior, exterior);
        for (i = 1; i < points.length; ++i) {
            var m = circleMargin(points[i], interior, exterior);
            if (m >= margin) {
                initial = points[i];
                margin = m;
            }
        }

        // maximize the margin numerically
        var solution = nelderMead(
                    function(p) { return -1 * circleMargin({x: p[0], y: p[1]}, interior, exterior); },
                    [initial.x, initial.y],
                    {maxIterations:500, minErrorDelta:1e-10}).x;
        var ret = {x: solution[0], y: solution[1]};

        // check solution, fallback as needed (happens if fully overlapped
        // etc)
        var valid = true;
        for (i = 0; i < interior.length; ++i) {
            if (distance(ret, interior[i]) > interior[i].radius) {
                valid = false;
                break;
            }
        }

        for (i = 0; i < exterior.length; ++i) {
            if (distance(ret, exterior[i]) < exterior[i].radius) {
                valid = false;
                break;
            }
        }

        if (!valid) {
            if (interior.length == 1) {
                ret = {x: interior[0].x, y: interior[0].y};
            } else {
                var areaStats = {};
                intersectionArea(interior, areaStats);

                if (areaStats.arcs.length === 0) {
                    ret = {'x': 0, 'y': -1000, disjoint:true};

                } else if (areaStats.arcs.length == 1) {
                    ret = {'x': areaStats.arcs[0].circle.x,
                           'y': areaStats.arcs[0].circle.y};

                } else if (exterior.length) {
                    // try again without other circles
                    ret = computeTextCentre(interior, []);

                } else {
                    // take average of all the points in the intersection
                    // polygon. this should basically never happen
                    // and has some issues:
                    // https://github.com/benfred/venn.js/issues/48#issuecomment-146069777
                    ret = getCenter(areaStats.arcs.map(function (a) { return a.p1; }));
                }
            }
        }

        return ret;
    }

    // given a dictionary of {setid : circle}, returns
    // a dictionary of setid to list of circles that completely overlap it
    function getOverlappingCircles(circles) {
        var ret = {}, circleids = [];
        for (var circleid in circles) {
            circleids.push(circleid);
            ret[circleid] = [];
        }
        for (var i  = 0; i < circleids.length; i++) {
            var a = circles[circleids[i]];
            for (var j = i + 1; j < circleids.length; ++j) {
                var b = circles[circleids[j]],
                    d = distance(a, b);

                if (d + b.radius <= a.radius + 1e-10) {
                    ret[circleids[j]].push(circleids[i]);

                } else if (d + a.radius <= b.radius + 1e-10) {
                    ret[circleids[i]].push(circleids[j]);
                }
            }
        }
        return ret;
    }

    function computeTextCentres(circles, areas) {
        var ret = {}, overlapped = getOverlappingCircles(circles);
        for (var i = 0; i < areas.length; ++i) {
            var area = areas[i].sets, areaids = {}, exclude = {};
            for (var j = 0; j < area.length; ++j) {
                areaids[area[j]] = true;
                var overlaps = overlapped[area[j]];
                // keep track of any circles that overlap this area,
                // and don't consider for purposes of computing the text
                // centre
                for (var k = 0; k < overlaps.length; ++k) {
                    exclude[overlaps[k]] = true;
                }
            }

            var interior = [], exterior = [];
            for (var setid in circles) {
                if (setid in areaids) {
                    interior.push(circles[setid]);
                } else if (!(setid in exclude)) {
                    exterior.push(circles[setid]);
                }
            }
            var centre = computeTextCentre(interior, exterior);
            ret[area] = centre;
            if (centre.disjoint && (areas[i].size > 0)) {
                console.log("WARNING: area " + area + " not represented on screen");
            }
        }
        return  ret;
    }

    // sorts all areas in the venn diagram, so that
    // a particular area is on top (relativeTo) - and
    // all other areas are so that the smallest areas are on top
    function sortAreas(div, relativeTo) {

        // figure out sets that are completly overlapped by relativeTo
        var overlaps = getOverlappingCircles(div.selectAll("svg").datum());
        var exclude = {};
        for (var i = 0; i < relativeTo.sets.length; ++i) {
            var check = relativeTo.sets[i];
            for (var setid in overlaps) {
                var overlap = overlaps[setid];
                for (var j = 0; j < overlap.length; ++j) {
                    if (overlap[j] == check) {
                        exclude[setid] = true;
                        break;
                    }
                }
            }
        }

        // checks that all sets are in exclude;
        function shouldExclude(sets) {
            for (var i = 0; i < sets.length; ++i) {
                if (!(sets[i] in exclude)) {
                    return false;
                }
            }
            return true;
        }

        // need to sort div's so that Z order is correct
        div.selectAll("g").sort(function (a, b) {
            // highest order set intersections first
            if (a.sets.length != b.sets.length) {
                return a.sets.length - b.sets.length;
            }

            if (a == relativeTo) {
                return shouldExclude(b.sets) ? -1 : 1;
            }
            if (b == relativeTo) {
                return shouldExclude(a.sets) ? 1 : -1;
            }

            // finally by size
            return b.size - a.size;
        });
    }

    function circlePath(x, y, r) {
        var ret = [];
        ret.push("\nM", x, y);
        ret.push("\nm", -r, 0);
        ret.push("\na", r, r, 0, 1, 0, r *2, 0);
        ret.push("\na", r, r, 0, 1, 0,-r *2, 0);
        return ret.join(" ");
    }

    // inverse of the circlePath function, returns a circle object from an svg path
    function circleFromPath(path) {
        var tokens = path.split(' ');
        return {'x' : parseFloat(tokens[1]),
                'y' : parseFloat(tokens[2]),
                'radius' : -parseFloat(tokens[4])
                };
    }

    /** returns a svg path of the intersection area of a bunch of circles */
    function intersectionAreaPath(circles) {
        var stats = {};
        intersectionArea(circles, stats);
        var arcs = stats.arcs;

        if (arcs.length === 0) {
            return "M 0 0";

        } else if (arcs.length == 1) {
            var circle = arcs[0].circle;
            return circlePath(circle.x, circle.y, circle.radius);

        } else {
            // draw path around arcs
            var ret = ["\nM", arcs[0].p2.x, arcs[0].p2.y];
            for (var i = 0; i < arcs.length; ++i) {
                var arc = arcs[i], r = arc.circle.radius, wide = arc.width > r;
                ret.push("\nA", r, r, 0, wide ? 1 : 0, 1,
                         arc.p1.x, arc.p1.y);
            }
            return ret.join(" ");
        }
    }

    exports.intersectionArea = intersectionArea;
    exports.circleCircleIntersection = circleCircleIntersection;
    exports.circleOverlap = circleOverlap;
    exports.circleArea = circleArea;
    exports.distance = distance;
    exports.venn = venn;
    exports.greedyLayout = greedyLayout;
    exports.scaleSolution = scaleSolution;
    exports.normalizeSolution = normalizeSolution;
    exports.bestInitialLayout = bestInitialLayout;
    exports.lossFunction = lossFunction;
    exports.disjointCluster = disjointCluster;
    exports.distanceFromIntersectArea = distanceFromIntersectArea;
    exports.VennDiagram = VennDiagram;
    exports.wrapText = wrapText;
    exports.computeTextCentres = computeTextCentres;
    exports.computeTextCentre = computeTextCentre;
    exports.sortAreas = sortAreas;
    exports.circlePath = circlePath;
    exports.circleFromPath = circleFromPath;
    exports.intersectionAreaPath = intersectionAreaPath;

    Object.defineProperty(exports, '__esModule', { value: true });

})));