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Edit File: _path.js

/* global a2c */
'use strict';

var regPathInstructions = /([MmLlHhVvCcSsQqTtAaZz])\s*/,
    regPathData = /[-+]?(?:\d*\.\d+|\d+\.?)([eE][-+]?\d+)?/g,
    regNumericValues = /[-+]?(\d*\.\d+|\d+\.?)(?:[eE][-+]?\d+)?/,
    transform2js = require('./_transforms').transform2js,
    transformsMultiply = require('./_transforms').transformsMultiply,
    transformArc = require('./_transforms').transformArc,
    collections = require('./_collections.js'),
    referencesProps = collections.referencesProps,
    defaultStrokeWidth = collections.attrsGroupsDefaults.presentation['stroke-width'],
    cleanupOutData = require('../lib/svgo/tools').cleanupOutData,
    removeLeadingZero = require('../lib/svgo/tools').removeLeadingZero,
    prevCtrlPoint;

/**
 * Convert path string to JS representation.
 *
 * @param {String} pathString input string
 * @param {Object} params plugin params
 * @return {Array} output array
 */
exports.path2js = function(path) {
    if (path.pathJS) return path.pathJS;

var paramsLength = { // Number of parameters of every path command
            H: 1, V: 1, M: 2, L: 2, T: 2, Q: 4, S: 4, C: 6, A: 7,
            h: 1, v: 1, m: 2, l: 2, t: 2, q: 4, s: 4, c: 6, a: 7
        },
        pathData = [],   // JS representation of the path data
        instruction, // current instruction context
        startMoveto = false;

// splitting path string into array like ['M', '10 50', 'L', '20 30']
    path.attr('d').value.split(regPathInstructions).forEach(function(data) {
        if (!data) return;
        if (!startMoveto) {
            if (data == 'M' || data == 'm') {
                startMoveto = true;
            } else return;
        }

// instruction item
        if (regPathInstructions.test(data)) {
            instruction = data;

// z - instruction w/o data
            if (instruction == 'Z' || instruction == 'z') {
                pathData.push({
                    instruction: 'z'
                });
            }
        // data item
        } else {
            data = data.match(regPathData);
            if (!data) return;

data = data.map(Number);

// Subsequent moveto pairs of coordinates are threated as implicit lineto commands
            // http://www.w3.org/TR/SVG/paths.html#PathDataMovetoCommands
            if (instruction == 'M' || instruction == 'm') {
                pathData.push({
                    instruction: pathData.length == 0 ? 'M' : instruction,
                    data: data.splice(0, 2)
                });
                instruction = instruction == 'M' ? 'L' : 'l';
            }

for (var pair = paramsLength[instruction]; data.length;) {
                pathData.push({
                    instruction: instruction,
                    data: data.splice(0, pair)
                });
            }
        }
    });

// First moveto is actually absolute. Subsequent coordinates were separated above.
    if (pathData.length && pathData[0].instruction == 'm') {
        pathData[0].instruction = 'M';
    }
    path.pathJS = pathData;

return pathData;
};

/**
 * Convert relative Path data to absolute.
 *
 * @param {Array} data input data
 * @return {Array} output data
 */
var relative2absolute = exports.relative2absolute = function(data) {
    var currentPoint = [0, 0],
        subpathPoint = [0, 0],
        i;

data = data.map(function(item) {

var instruction = item.instruction,
            itemData = item.data && item.data.slice();

if (instruction == 'M') {

set(currentPoint, itemData);
            set(subpathPoint, itemData);

} else if ('mlcsqt'.indexOf(instruction) > -1) {

for (i = 0; i < itemData.length; i++) {
                itemData[i] += currentPoint[i % 2];
            }
            set(currentPoint, itemData);

if (instruction == 'm') {
                set(subpathPoint, itemData);
            }

} else if (instruction == 'a') {

itemData[5] += currentPoint[0];
            itemData[6] += currentPoint[1];
            set(currentPoint, itemData);

} else if (instruction == 'h') {

itemData[0] += currentPoint[0];
            currentPoint[0] = itemData[0];

} else if (instruction == 'v') {

itemData[0] += currentPoint[1];
            currentPoint[1] = itemData[0];

} else if ('MZLCSQTA'.indexOf(instruction) > -1) {

set(currentPoint, itemData);

} else if (instruction == 'H') {

currentPoint[0] = itemData[0];

} else if (instruction == 'V') {

currentPoint[1] = itemData[0];

} else if (instruction == 'z') {

set(currentPoint, subpathPoint);

}

return instruction == 'z' ?
            { instruction: 'z' } :
            {
                instruction: instruction.toUpperCase(),
                data: itemData
            };

});

return data;
};

/**
 * Apply transformation(s) to the Path data.
 *
 * @param {Object} elem current element
 * @param {Array} path input path data
 * @param {Object} params whether to apply transforms to stroked lines and transform precision (used for stroke width)
 * @return {Array} output path data
 */
exports.applyTransforms = function(elem, path, params) {
    // if there are no 'stroke' attr and references to other objects such as
    // gradiends or clip-path which are also subjects to transform.
    if (!elem.hasAttr('transform') || !elem.attr('transform').value ||
        elem.someAttr(function(attr) {
            return ~referencesProps.indexOf(attr.name) && ~attr.value.indexOf('url(');
        }))
        return path;

var matrix = transformsMultiply(transform2js(elem.attr('transform').value)),
        stroke = elem.computedAttr('stroke'),
        id = elem.computedAttr('id'),
        transformPrecision = params.transformPrecision,
        newPoint, scale;

if (stroke && stroke != 'none') {
        if (!params.applyTransformsStroked ||
            (matrix.data[0] != matrix.data[3] || matrix.data[1] != -matrix.data[2]) &&
            (matrix.data[0] != -matrix.data[3] || matrix.data[1] != matrix.data[2]))
            return path;

// "stroke-width" should be inside the part with ID, otherwise it can be overrided in <use>
        if (id) {
            var idElem = elem,
                hasStrokeWidth = false;

do {
                if (idElem.hasAttr('stroke-width')) hasStrokeWidth = true;
            } while (!idElem.hasAttr('id', id) && !hasStrokeWidth && (idElem = idElem.parentNode));

if (!hasStrokeWidth) return path;
        }

scale = +Math.sqrt(matrix.data[0] * matrix.data[0] + matrix.data[1] * matrix.data[1]).toFixed(transformPrecision);

if (scale !== 1) {
            var strokeWidth = elem.computedAttr('stroke-width') || defaultStrokeWidth;

if (elem.hasAttr('stroke-width')) {
                elem.attrs['stroke-width'].value = elem.attrs['stroke-width'].value.trim()
                    .replace(regNumericValues, function(num) { return removeLeadingZero(num * scale) });
            } else {
                elem.addAttr({
                    name: 'stroke-width',
                    prefix: '',
                    local: 'stroke-width',
                    value: strokeWidth.replace(regNumericValues, function(num) { return removeLeadingZero(num * scale) })
                });
            }
        }
    } else if (id) { // Stroke and stroke-width can be redefined with <use>
        return path;
    }

path.forEach(function(pathItem) {

if (pathItem.data) {

// h -> l
            if (pathItem.instruction === 'h') {

pathItem.instruction = 'l';
                pathItem.data[1] = 0;

// v -> l
            } else if (pathItem.instruction === 'v') {

pathItem.instruction = 'l';
                pathItem.data[1] = pathItem.data[0];
                pathItem.data[0] = 0;

}

// if there is a translate() transform
            if (pathItem.instruction === 'M' &&
                (matrix.data[4] !== 0 ||
                matrix.data[5] !== 0)
            ) {

// then apply it only to the first absoluted M
                newPoint = transformPoint(matrix.data, pathItem.data[0], pathItem.data[1]);
                set(pathItem.data, newPoint);
                set(pathItem.coords, newPoint);

// clear translate() data from transform matrix
                matrix.data[4] = 0;
                matrix.data[5] = 0;

} else {

if (pathItem.instruction == 'a') {

transformArc(pathItem.data, matrix.data);

// reduce number of digits in rotation angle
                    if (Math.abs(pathItem.data[2]) > 80) {
                        var a = pathItem.data[0],
                            rotation = pathItem.data[2];
                        pathItem.data[0] = pathItem.data[1];
                        pathItem.data[1] = a;
                        pathItem.data[2] = rotation + (rotation > 0 ? -90 : 90);
                    }

newPoint = transformPoint(matrix.data, pathItem.data[5], pathItem.data[6]);
                    pathItem.data[5] = newPoint[0];
                    pathItem.data[6] = newPoint[1];

} else {

for (var i = 0; i < pathItem.data.length; i += 2) {
                        newPoint = transformPoint(matrix.data, pathItem.data[i], pathItem.data[i + 1]);
                        pathItem.data[i] = newPoint[0];
                        pathItem.data[i + 1] = newPoint[1];
                    }
                }

pathItem.coords[0] = pathItem.base[0] + pathItem.data[pathItem.data.length - 2];
                pathItem.coords[1] = pathItem.base[1] + pathItem.data[pathItem.data.length - 1];

}

});

// remove transform attr
    elem.removeAttr('transform');

return path;
};

/**
 * Apply transform 3x3 matrix to x-y point.
 *
 * @param {Array} matrix transform 3x3 matrix
 * @param {Array} point x-y point
 * @return {Array} point with new coordinates
 */
function transformPoint(matrix, x, y) {

return [
        matrix[0] * x + matrix[2] * y + matrix[4],
        matrix[1] * x + matrix[3] * y + matrix[5]
    ];

}

/**
 * Compute Cubic Bézie bounding box.
 *
 * @see http://processingjs.nihongoresources.com/bezierinfo/
 *
 * @param {Float} xa
 * @param {Float} ya
 * @param {Float} xb
 * @param {Float} yb
 * @param {Float} xc
 * @param {Float} yc
 * @param {Float} xd
 * @param {Float} yd
 *
 * @return {Object}
 */
exports.computeCubicBoundingBox = function(xa, ya, xb, yb, xc, yc, xd, yd) {

var minx = Number.POSITIVE_INFINITY,
        miny = Number.POSITIVE_INFINITY,
        maxx = Number.NEGATIVE_INFINITY,
        maxy = Number.NEGATIVE_INFINITY,
        ts,
        t,
        x,
        y,
        i;

// X
    if (xa < minx) { minx = xa; }
    if (xa > maxx) { maxx = xa; }
    if (xd < minx) { minx= xd; }
    if (xd > maxx) { maxx = xd; }

ts = computeCubicFirstDerivativeRoots(xa, xb, xc, xd);

for (i = 0; i < ts.length; i++) {

t = ts[i];

if (t >= 0 && t <= 1) {
            x = computeCubicBaseValue(t, xa, xb, xc, xd);
            // y = computeCubicBaseValue(t, ya, yb, yc, yd);

if (x < minx) { minx = x; }
            if (x > maxx) { maxx = x; }
        }

}

// Y
    if (ya < miny) { miny = ya; }
    if (ya > maxy) { maxy = ya; }
    if (yd < miny) { miny = yd; }
    if (yd > maxy) { maxy = yd; }

ts = computeCubicFirstDerivativeRoots(ya, yb, yc, yd);

for (i = 0; i < ts.length; i++) {

t = ts[i];

if (t >= 0 && t <= 1) {
            // x = computeCubicBaseValue(t, xa, xb, xc, xd);
            y = computeCubicBaseValue(t, ya, yb, yc, yd);

if (y < miny) { miny = y; }
            if (y > maxy) { maxy = y; }
        }

}

return {
        minx: minx,
        miny: miny,
        maxx: maxx,
        maxy: maxy
    };

};

// compute the value for the cubic bezier function at time=t
function computeCubicBaseValue(t, a, b, c, d) {

var mt = 1 - t;

return mt * mt * mt * a + 3 * mt * mt * t * b + 3 * mt * t * t * c + t * t * t * d;

}

// compute the value for the first derivative of the cubic bezier function at time=t
function computeCubicFirstDerivativeRoots(a, b, c, d) {

var result = [-1, -1],
        tl = -a + 2 * b - c,
        tr = -Math.sqrt(-a * (c - d) + b * b - b * (c + d) + c * c),
        dn = -a + 3 * b - 3 * c + d;

if (dn !== 0) {
        result[0] = (tl + tr) / dn;
        result[1] = (tl - tr) / dn;
    }

return result;

}

/**
 * Compute Quadratic Bézier bounding box.
 *
 * @see http://processingjs.nihongoresources.com/bezierinfo/
 *
 * @param {Float} xa
 * @param {Float} ya
 * @param {Float} xb
 * @param {Float} yb
 * @param {Float} xc
 * @param {Float} yc
 *
 * @return {Object}
 */
exports.computeQuadraticBoundingBox = function(xa, ya, xb, yb, xc, yc) {

var minx = Number.POSITIVE_INFINITY,
        miny = Number.POSITIVE_INFINITY,
        maxx = Number.NEGATIVE_INFINITY,
        maxy = Number.NEGATIVE_INFINITY,
        t,
        x,
        y;

// X
    if (xa < minx) { minx = xa; }
    if (xa > maxx) { maxx = xa; }
    if (xc < minx) { minx = xc; }
    if (xc > maxx) { maxx = xc; }

t = computeQuadraticFirstDerivativeRoot(xa, xb, xc);

if (t >= 0 && t <= 1) {
        x = computeQuadraticBaseValue(t, xa, xb, xc);
        // y = computeQuadraticBaseValue(t, ya, yb, yc);

if (x < minx) { minx = x; }
        if (x > maxx) { maxx = x; }
    }

// Y
    if (ya < miny) { miny = ya; }
    if (ya > maxy) { maxy = ya; }
    if (yc < miny) { miny = yc; }
    if (yc > maxy) { maxy = yc; }

t = computeQuadraticFirstDerivativeRoot(ya, yb, yc);

if (t >= 0 && t <=1 ) {
        // x = computeQuadraticBaseValue(t, xa, xb, xc);
        y = computeQuadraticBaseValue(t, ya, yb, yc);

if (y < miny) { miny = y; }
        if (y > maxy) { maxy = y ; }

}

return {
        minx: minx,
        miny: miny,
        maxx: maxx,
        maxy: maxy
    };

};

// compute the value for the quadratic bezier function at time=t
function computeQuadraticBaseValue(t, a, b, c) {

var mt = 1 - t;

return mt * mt * a + 2 * mt * t * b + t * t * c;

}

// compute the value for the first derivative of the quadratic bezier function at time=t
function computeQuadraticFirstDerivativeRoot(a, b, c) {

var t = -1,
        denominator = a - 2 * b + c;

if (denominator !== 0) {
        t = (a - b) / denominator;
    }

return t;

}

/**
 * Convert path array to string.
 *
 * @param {Array} path input path data
 * @param {Object} params plugin params
 * @return {String} output path string
 */
exports.js2path = function(path, data, params) {

path.pathJS = data;

if (params.collapseRepeated) {
        data = collapseRepeated(data);
    }

path.attr('d').value = data.reduce(function(pathString, item) {
        return pathString += item.instruction + (item.data ? cleanupOutData(item.data, params) : '');
    }, '');

};

/**
 * Collapse repeated instructions data
 *
 * @param {Array} path input path data
 * @return {Array} output path data
 */
function collapseRepeated(data) {

var prev,
        prevIndex;

// copy an array and modifieds item to keep original data untouched
    data = data.reduce(function(newPath, item) {
        if (
            prev && item.data &&
            item.instruction == prev.instruction
        ) {
            // concat previous data with current
            if (item.instruction != 'M') {
                prev = newPath[prevIndex] = {
                    instruction: prev.instruction,
                    data: prev.data.concat(item.data),
                    coords: item.coords,
                    base: prev.base
                };
            } else {
                prev.data = item.data;
                prev.coords = item.coords;
            }
        } else {
            newPath.push(item);
            prev = item;
            prevIndex = newPath.length - 1;
        }

return newPath;
    }, []);

return data;

}

function set(dest, source) {
    dest[0] = source[source.length - 2];
    dest[1] = source[source.length - 1];
    return dest;
}

/**
 * Checks if two paths have an intersection by checking convex hulls
 * collision using Gilbert-Johnson-Keerthi distance algorithm
 * http://entropyinteractive.com/2011/04/gjk-algorithm/
 *
 * @param {Array} path1 JS path representation
 * @param {Array} path2 JS path representation
 * @return {Boolean}
 */
exports.intersects = function(path1, path2) {
    if (path1.length < 3 || path2.length < 3) return false; // nothing to fill

// Collect points of every subpath.
    var points1 = relative2absolute(path1).reduce(gatherPoints, []),
        points2 = relative2absolute(path2).reduce(gatherPoints, []);

// Axis-aligned bounding box check.
    if (points1.maxX <= points2.minX || points2.maxX <= points1.minX ||
        points1.maxY <= points2.minY || points2.maxY <= points1.minY ||
        points1.every(function (set1) {
            return points2.every(function (set2) {
                return set1[set1.maxX][0] <= set2[set2.minX][0] ||
                    set2[set2.maxX][0] <= set1[set1.minX][0] ||
                    set1[set1.maxY][1] <= set2[set2.minY][1] ||
                    set2[set2.maxY][1] <= set1[set1.minY][1];
            });
        })
    ) return false;

// Get a convex hull from points of each subpath. Has the most complexity O(n·log n).
    var hullNest1 = points1.map(convexHull),
        hullNest2 = points2.map(convexHull);

// Check intersection of every subpath of the first path with every subpath of the second.
    return hullNest1.some(function(hull1) {
        if (hull1.length < 3) return false;

return hullNest2.some(function(hull2) {
            if (hull2.length < 3) return false;

var simplex = [getSupport(hull1, hull2, [1, 0])], // create the initial simplex
                direction = minus(simplex[0]); // set the direction to point towards the origin

var iterations = 1e4; // infinite loop protection, 10 000 iterations is more than enough
            while (true) {
                if (iterations-- == 0) {
                    console.error('Error: infinite loop while processing mergePaths plugin.');
                    return true; // true is the safe value that means “do nothing with paths”
                }
                // add a new point
                simplex.push(getSupport(hull1, hull2, direction));
                // see if the new point was on the correct side of the origin
                if (dot(direction, simplex[simplex.length - 1]) <= 0) return false;
                // process the simplex
                if (processSimplex(simplex, direction)) return true;
            }
        });
    });

function getSupport(a, b, direction) {
        return sub(supportPoint(a, direction), supportPoint(b, minus(direction)));
    }

// Computes farthest polygon point in particular direction.
    // Thanks to knowledge of min/max x and y coordinates we can choose a quadrant to search in.
    // Since we're working on convex hull, the dot product is increasing until we find the farthest point.
    function supportPoint(polygon, direction) {
        var index = direction[1] >= 0 ?
                direction[0] < 0 ? polygon.maxY : polygon.maxX :
                direction[0] < 0 ? polygon.minX : polygon.minY,
            max = -Infinity,
            value;
        while ((value = dot(polygon[index], direction)) > max) {
            max = value;
            index = ++index % polygon.length;
        }
        return polygon[(index || polygon.length) - 1];
    }
};

function processSimplex(simplex, direction) {
    /* jshint -W004 */

// we only need to handle to 1-simplex and 2-simplex
    if (simplex.length == 2) { // 1-simplex
        var a = simplex[1],
            b = simplex[0],
            AO = minus(simplex[1]),
            AB = sub(b, a);
        // AO is in the same direction as AB
        if (dot(AO, AB) > 0) {
            // get the vector perpendicular to AB facing O
            set(direction, orth(AB, a));
        } else {
            set(direction, AO);
            // only A remains in the simplex
            simplex.shift();
        }
    } else { // 2-simplex
        var a = simplex[2], // [a, b, c] = simplex
            b = simplex[1],
            c = simplex[0],
            AB = sub(b, a),
            AC = sub(c, a),
            AO = minus(a),
            ACB = orth(AB, AC), // the vector perpendicular to AB facing away from C
            ABC = orth(AC, AB); // the vector perpendicular to AC facing away from B

if (dot(ACB, AO) > 0) {
            if (dot(AB, AO) > 0) { // region 4
                set(direction, ACB);
                simplex.shift(); // simplex = [b, a]
            } else { // region 5
                set(direction, AO);
                simplex.splice(0, 2); // simplex = [a]
            }
        } else if (dot(ABC, AO) > 0) {
            if (dot(AC, AO) > 0) { // region 6
                set(direction, ABC);
                simplex.splice(1, 1); // simplex = [c, a]
            } else { // region 5 (again)
                set(direction, AO);
                simplex.splice(0, 2); // simplex = [a]
            }
        } else // region 7
            return true;
    }
    return false;
}

function minus(v) {
    return [-v[0], -v[1]];
}

function sub(v1, v2) {
    return [v1[0] - v2[0], v1[1] - v2[1]];
}

function dot(v1, v2) {
    return v1[0] * v2[0] + v1[1] * v2[1];
}

function orth(v, from) {
    var o = [-v[1], v[0]];
    return dot(o, minus(from)) < 0 ? minus(o) : o;
}

function gatherPoints(points, item, index, path) {

var subPath = points.length && points[points.length - 1],
        prev = index && path[index - 1],
        basePoint = subPath.length && subPath[subPath.length - 1],
        data = item.data,
        ctrlPoint = basePoint;

switch (item.instruction) {
        case 'M':
            points.push(subPath = []);
            break;
        case 'H':
            addPoint(subPath, [data[0], basePoint[1]]);
            break;
        case 'V':
            addPoint(subPath, [basePoint[0], data[0]]);
            break;
        case 'Q':
            addPoint(subPath, data.slice(0, 2));
            prevCtrlPoint = [data[2] - data[0], data[3] - data[1]]; // Save control point for shorthand
            break;
        case 'T':
            if (prev.instruction == 'Q' && prev.instruction == 'T') {
                ctrlPoint = [basePoint[0] + prevCtrlPoint[0], basePoint[1] + prevCtrlPoint[1]];
                addPoint(subPath, ctrlPoint);
                prevCtrlPoint = [data[0] - ctrlPoint[0], data[1] - ctrlPoint[1]];
            }
            break;
        case 'C':
            // Approximate quibic Bezier curve with middle points between control points
            addPoint(subPath, [.5 * (basePoint[0] + data[0]), .5 * (basePoint[1] + data[1])]);
            addPoint(subPath, [.5 * (data[0] + data[2]), .5 * (data[1] + data[3])]);
            addPoint(subPath, [.5 * (data[2] + data[4]), .5 * (data[3] + data[5])]);
            prevCtrlPoint = [data[4] - data[2], data[5] - data[3]]; // Save control point for shorthand
            break;
        case 'S':
            if (prev.instruction == 'C' && prev.instruction == 'S') {
                addPoint(subPath, [basePoint[0] + .5 * prevCtrlPoint[0], basePoint[1] + .5 * prevCtrlPoint[1]]);
                ctrlPoint = [basePoint[0] + prevCtrlPoint[0], basePoint[1] + prevCtrlPoint[1]];
            }
            addPoint(subPath, [.5 * (ctrlPoint[0] + data[0]), .5 * (ctrlPoint[1]+ data[1])]);
            addPoint(subPath, [.5 * (data[0] + data[2]), .5 * (data[1] + data[3])]);
            prevCtrlPoint = [data[2] - data[0], data[3] - data[1]];
            break;
        case 'A':
            // Convert the arc to bezier curves and use the same approximation
            var curves = a2c.apply(0, basePoint.concat(data));
            for (var cData; (cData = curves.splice(0,6).map(toAbsolute)).length;) {
                addPoint(subPath, [.5 * (basePoint[0] + cData[0]), .5 * (basePoint[1] + cData[1])]);
                addPoint(subPath, [.5 * (cData[0] + cData[2]), .5 * (cData[1] + cData[3])]);
                addPoint(subPath, [.5 * (cData[2] + cData[4]), .5 * (cData[3] + cData[5])]);
                if (curves.length) addPoint(subPath, basePoint = cData.slice(-2));
            }
            break;
    }
    // Save final command coordinates
    if (data && data.length >= 2) addPoint(subPath, data.slice(-2));
    return points;

function toAbsolute(n, i) { return n + basePoint[i % 2] }

// Writes data about the extreme points on each axle
    function addPoint(path, point) {
        if (!path.length || point[1] > path[path.maxY][1]) {
            path.maxY = path.length;
            points.maxY = points.length ? Math.max(point[1], points.maxY) : point[1];
        }
        if (!path.length || point[0] > path[path.maxX][0]) {
            path.maxX = path.length;
            points.maxX = points.length ? Math.max(point[0], points.maxX) : point[0];
        }
        if (!path.length || point[1] < path[path.minY][1]) {
            path.minY = path.length;
            points.minY = points.length ? Math.min(point[1], points.minY) : point[1];
        }
        if (!path.length || point[0] < path[path.minX][0]) {
            path.minX = path.length;
            points.minX = points.length ? Math.min(point[0], points.minX) : point[0];
        }
        path.push(point);
    }
}

/**
 * Forms a convex hull from set of points of every subpath using monotone chain convex hull algorithm.
 * http://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain
 *
 * @param points An array of [X, Y] coordinates
 */
function convexHull(points) {
    /* jshint -W004 */

points.sort(function(a, b) {
        return a[0] == b[0] ? a[1] - b[1] : a[0] - b[0];
    });

// last points are equal to starting points of the other part
    upper.pop();
    lower.pop();

var hull = lower.concat(upper);

hull.minX = 0; // by sorting
    hull.maxX = lower.length;
    hull.minY = bottom;
    hull.maxY = (lower.length + top) % hull.length;

return hull;
}

function cross(o, a, b) {
    return (a[0] - o[0]) * (b[1] - o[1]) - (a[1] - o[1]) * (b[0] - o[0]);
}

/* Based on code from Snap.svg (Apache 2 license). http://snapsvg.io/
 * Thanks to Dmitry Baranovskiy for his great work!
 */

// jshint ignore: start
function a2c(x1, y1, rx, ry, angle, large_arc_flag, sweep_flag, x2, y2, recursive) {
    // for more information of where this Math came from visit:
    // http://www.w3.org/TR/SVG11/implnote.html#ArcImplementationNotes
    var _120 = Math.PI * 120 / 180,
        rad = Math.PI / 180 * (+angle || 0),
        res = [],
        rotateX = function(x, y, rad) { return x * Math.cos(rad) - y * Math.sin(rad) },
        rotateY = function(x, y, rad) { return x * Math.sin(rad) + y * Math.cos(rad) };
    if (!recursive) {
        x1 = rotateX(x1, y1, -rad);
        y1 = rotateY(x1, y1, -rad);
        x2 = rotateX(x2, y2, -rad);
        y2 = rotateY(x2, y2, -rad);
        var x = (x1 - x2) / 2,
            y = (y1 - y2) / 2;
        var h = (x * x) / (rx * rx) + (y * y) / (ry * ry);
        if (h > 1) {
            h = Math.sqrt(h);
            rx = h * rx;
            ry = h * ry;
        }
        var rx2 = rx * rx,
            ry2 = ry * ry,
            k = (large_arc_flag == sweep_flag ? -1 : 1) *
                Math.sqrt(Math.abs((rx2 * ry2 - rx2 * y * y - ry2 * x * x) / (rx2 * y * y + ry2 * x * x))),
            cx = k * rx * y / ry + (x1 + x2) / 2,
            cy = k * -ry * x / rx + (y1 + y2) / 2,
            f1 = Math.asin(((y1 - cy) / ry).toFixed(9)),
            f2 = Math.asin(((y2 - cy) / ry).toFixed(9));

f1 = x1 < cx ? Math.PI - f1 : f1;
        f2 = x2 < cx ? Math.PI - f2 : f2;
        f1 < 0 && (f1 = Math.PI * 2 + f1);
        f2 < 0 && (f2 = Math.PI * 2 + f2);
        if (sweep_flag && f1 > f2) {
            f1 = f1 - Math.PI * 2;
        }
        if (!sweep_flag && f2 > f1) {
            f2 = f2 - Math.PI * 2;
        }
    } else {
        f1 = recursive[0];
        f2 = recursive[1];
        cx = recursive[2];
        cy = recursive[3];
    }
    var df = f2 - f1;
    if (Math.abs(df) > _120) {
        var f2old = f2,
            x2old = x2,
            y2old = y2;
        f2 = f1 + _120 * (sweep_flag && f2 > f1 ? 1 : -1);
        x2 = cx + rx * Math.cos(f2);
        y2 = cy + ry * Math.sin(f2);
        res = a2c(x2, y2, rx, ry, angle, 0, sweep_flag, x2old, y2old, [f2, f2old, cx, cy]);
    }
    df = f2 - f1;
    var c1 = Math.cos(f1),
        s1 = Math.sin(f1),
        c2 = Math.cos(f2),
        s2 = Math.sin(f2),
        t = Math.tan(df / 4),
        hx = 4 / 3 * rx * t,
        hy = 4 / 3 * ry * t,
        m = [
            - hx * s1, hy * c1,
            x2 + hx * s2 - x1, y2 - hy * c2 - y1,
            x2 - x1, y2 - y1
        ];
    if (recursive) {
        return m.concat(res);
    } else {
        res = m.concat(res);
        var newres = [];
        for (var i = 0, n = res.length; i < n; i++) {
            newres[i] = i % 2 ? rotateY(res[i - 1], res[i], rad) : rotateX(res[i], res[i + 1], rad);
        }
        return newres;
    }
}
// jshint ignore: end

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