From 0b8efa5c406ca3eb6383acd12601f67e54be4394 Mon Sep 17 00:00:00 2001
From: liabru <liabru@brm.io>
Date: Sun, 1 Feb 2015 22:55:24 +0000
Subject: [PATCH] added Demo.concave, added Bodies.fromVertices, added
 Vertices.isConvex, fixed compound properties

---
 demo/dev.html            |   4 +
 demo/js/Demo.js          |  22 +-
 demo/js/lib/decomp.js    | 670 +++++++++++++++++++++++++++++++++++++++
 src/body/Body.js         |   7 +-
 src/factory/Bodies.js    |  98 +++++-
 src/geometry/Vertices.js |  49 ++-
 6 files changed, 842 insertions(+), 8 deletions(-)
 create mode 100644 demo/js/lib/decomp.js

diff --git a/demo/dev.html b/demo/dev.html
index c2c43a6..3edd648 100644
--- a/demo/dev.html
+++ b/demo/dev.html
@@ -6,6 +6,9 @@
         <meta name="viewport" content="width=device-width,minimal-ui">
         <meta name="robots" content="noindex">
 
+        <!-- only required if using Bodies.fromVertices with concave vertices -->
+        <script type="text/javascript" src="./js/lib/decomp.js"></script>
+
         <!-- only required if using Matter.RenderPixi -->
         <script type="text/javascript" src="./js/lib/pixi.dev.js"></script>
 
@@ -32,6 +35,7 @@
                     <option value="mixed">Mixed Shapes</option>
                     <option value="mixedSolid">Solid Rendering</option>
                     <option value="compound">Compound Bodies</option>
+                    <option value="concave">Concave Bodies</option>
                     <option value="newtonsCradle">Newton's Cradle</option>
                     <option value="wreckingBall">Wrecking Ball</option>
                     <option value="slingshot">Slingshot Game</option>
diff --git a/demo/js/Demo.js b/demo/js/Demo.js
index 4469942..3d2e7d6 100644
--- a/demo/js/Demo.js
+++ b/demo/js/Demo.js
@@ -130,8 +130,6 @@
         });
 
         World.add(_world, compound);
-
-        _world.gravity.y = 1;
         
         var renderOptions = _engine.render.options;
         renderOptions.showCollisions = true;
@@ -141,6 +139,26 @@
         renderOptions.showConvexHulls = true;
     };
 
+    Demo.concave = function() {
+        var _world = _engine.world;
+
+        Demo.reset();
+
+        var vertices = Matter.Vertices.fromPath('354 89,336 118,310 145,302 227,322 271,375 292,490 289,539 271,540 233,549 133,526 100,552 36,601 63,633 122,628 227,594 304,505 340,426 340,327 330,265 294,246 242,246 181,256 133,283 81,346 44');
+        //var vertices = Matter.Vertices.fromPath('164 171,232 233,213 302,273 241,342 305,316 231,364 170,309 188,281 117,240 182');
+
+        var concave = Bodies.fromVertices(200, 200, vertices);
+        World.add(_world, concave);
+
+        var renderOptions = _engine.render.options;
+        renderOptions.showCollisions = true;
+        renderOptions.showBounds = true;
+        renderOptions.showAxes = true;
+        renderOptions.showPositions = true;
+        renderOptions.showConvexHulls = true;
+        //renderOptions.showVertexNumbers = true;
+    };
+
     Demo.slingshot = function() {
         var _world = _engine.world;
 
diff --git a/demo/js/lib/decomp.js b/demo/js/lib/decomp.js
new file mode 100644
index 0000000..f16bc45
--- /dev/null
+++ b/demo/js/lib/decomp.js
@@ -0,0 +1,670 @@
+!function(e){"object"==typeof exports?module.exports=e():"function"==typeof define&&define.amd?define(e):"undefined"!=typeof window?window.decomp=e():"undefined"!=typeof global?global.decomp=e():"undefined"!=typeof self&&(self.decomp=e())}(function(){var define,module,exports;
+return (function e(t,n,r){function s(o,u){if(!n[o]){if(!t[o]){var a=typeof require=="function"&&require;if(!u&&a)return a(o,!0);if(i)return i(o,!0);throw new Error("Cannot find module '"+o+"'")}var f=n[o]={exports:{}};t[o][0].call(f.exports,function(e){var n=t[o][1][e];return s(n?n:e)},f,f.exports,e,t,n,r)}return n[o].exports}var i=typeof require=="function"&&require;for(var o=0;o<r.length;o++)s(r[o]);return s})({1:[function(require,module,exports){
+var Scalar = require('./Scalar');
+
+module.exports = Line;
+
+/**
+ * Container for line-related functions
+ * @class Line
+ */
+function Line(){};
+
+/**
+ * Compute the intersection between two lines.
+ * @static
+ * @method lineInt
+ * @param  {Array}  l1          Line vector 1
+ * @param  {Array}  l2          Line vector 2
+ * @param  {Number} precision   Precision to use when checking if the lines are parallel
+ * @return {Array}              The intersection point.
+ */
+Line.lineInt = function(l1,l2,precision){
+    precision = precision || 0;
+    var i = [0,0]; // point
+    var a1, b1, c1, a2, b2, c2, det; // scalars
+    a1 = l1[1][1] - l1[0][1];
+    b1 = l1[0][0] - l1[1][0];
+    c1 = a1 * l1[0][0] + b1 * l1[0][1];
+    a2 = l2[1][1] - l2[0][1];
+    b2 = l2[0][0] - l2[1][0];
+    c2 = a2 * l2[0][0] + b2 * l2[0][1];
+    det = a1 * b2 - a2*b1;
+    if (!Scalar.eq(det, 0, precision)) { // lines are not parallel
+        i[0] = (b2 * c1 - b1 * c2) / det;
+        i[1] = (a1 * c2 - a2 * c1) / det;
+    }
+    return i;
+};
+
+/**
+ * Checks if two line segments intersects.
+ * @method segmentsIntersect
+ * @param {Array} p1 The start vertex of the first line segment.
+ * @param {Array} p2 The end vertex of the first line segment.
+ * @param {Array} q1 The start vertex of the second line segment.
+ * @param {Array} q2 The end vertex of the second line segment.
+ * @return {Boolean} True if the two line segments intersect
+ */
+Line.segmentsIntersect = function(p1, p2, q1, q2){
+   var dx = p2[0] - p1[0];
+   var dy = p2[1] - p1[1];
+   var da = q2[0] - q1[0];
+   var db = q2[1] - q1[1];
+
+   // segments are parallel
+   if(da*dy - db*dx == 0)
+      return false;
+
+   var s = (dx * (q1[1] - p1[1]) + dy * (p1[0] - q1[0])) / (da * dy - db * dx)
+   var t = (da * (p1[1] - q1[1]) + db * (q1[0] - p1[0])) / (db * dx - da * dy)
+
+   return (s>=0 && s<=1 && t>=0 && t<=1);
+};
+
+
+},{"./Scalar":4}],2:[function(require,module,exports){
+module.exports = Point;
+
+/**
+ * Point related functions
+ * @class Point
+ */
+function Point(){};
+
+/**
+ * Get the area of a triangle spanned by the three given points. Note that the area will be negative if the points are not given in counter-clockwise order.
+ * @static
+ * @method area
+ * @param  {Array} a
+ * @param  {Array} b
+ * @param  {Array} c
+ * @return {Number}
+ */
+Point.area = function(a,b,c){
+    return (((b[0] - a[0])*(c[1] - a[1]))-((c[0] - a[0])*(b[1] - a[1])));
+};
+
+Point.left = function(a,b,c){
+    return Point.area(a,b,c) > 0;
+};
+
+Point.leftOn = function(a,b,c) {
+    return Point.area(a, b, c) >= 0;
+};
+
+Point.right = function(a,b,c) {
+    return Point.area(a, b, c) < 0;
+};
+
+Point.rightOn = function(a,b,c) {
+    return Point.area(a, b, c) <= 0;
+};
+
+var tmpPoint1 = [],
+    tmpPoint2 = [];
+
+/**
+ * Check if three points are collinear
+ * @method collinear
+ * @param  {Array} a
+ * @param  {Array} b
+ * @param  {Array} c
+ * @param  {Number} [thresholdAngle=0] Threshold angle to use when comparing the vectors. The function will return true if the angle between the resulting vectors is less than this value. Use zero for max precision.
+ * @return {Boolean}
+ */
+Point.collinear = function(a,b,c,thresholdAngle) {
+    if(!thresholdAngle)
+        return Point.area(a, b, c) == 0;
+    else {
+        var ab = tmpPoint1,
+            bc = tmpPoint2;
+
+        ab[0] = b[0]-a[0];
+        ab[1] = b[1]-a[1];
+        bc[0] = c[0]-b[0];
+        bc[1] = c[1]-b[1];
+
+        var dot = ab[0]*bc[0] + ab[1]*bc[1],
+            magA = Math.sqrt(ab[0]*ab[0] + ab[1]*ab[1]),
+            magB = Math.sqrt(bc[0]*bc[0] + bc[1]*bc[1]),
+            angle = Math.acos(dot/(magA*magB));
+        return angle < thresholdAngle;
+    }
+};
+
+Point.sqdist = function(a,b){
+    var dx = b[0] - a[0];
+    var dy = b[1] - a[1];
+    return dx * dx + dy * dy;
+};
+
+},{}],3:[function(require,module,exports){
+var Line = require("./Line")
+,   Point = require("./Point")
+,   Scalar = require("./Scalar")
+
+module.exports = Polygon;
+
+/**
+ * Polygon class.
+ * @class Polygon
+ * @constructor
+ */
+function Polygon(){
+
+    /**
+     * Vertices that this polygon consists of. An array of array of numbers, example: [[0,0],[1,0],..]
+     * @property vertices
+     * @type {Array}
+     */
+    this.vertices = [];
+}
+
+/**
+ * Get a vertex at position i. It does not matter if i is out of bounds, this function will just cycle.
+ * @method at
+ * @param  {Number} i
+ * @return {Array}
+ */
+Polygon.prototype.at = function(i){
+    var v = this.vertices,
+        s = v.length;
+    return v[i < 0 ? i % s + s : i % s];
+};
+
+/**
+ * Get first vertex
+ * @method first
+ * @return {Array}
+ */
+Polygon.prototype.first = function(){
+    return this.vertices[0];
+};
+
+/**
+ * Get last vertex
+ * @method last
+ * @return {Array}
+ */
+Polygon.prototype.last = function(){
+    return this.vertices[this.vertices.length-1];
+};
+
+/**
+ * Clear the polygon data
+ * @method clear
+ * @return {Array}
+ */
+Polygon.prototype.clear = function(){
+    this.vertices.length = 0;
+};
+
+/**
+ * Append points "from" to "to"-1 from an other polygon "poly" onto this one.
+ * @method append
+ * @param {Polygon} poly The polygon to get points from.
+ * @param {Number}  from The vertex index in "poly".
+ * @param {Number}  to The end vertex index in "poly". Note that this vertex is NOT included when appending.
+ * @return {Array}
+ */
+Polygon.prototype.append = function(poly,from,to){
+    if(typeof(from) == "undefined") throw new Error("From is not given!");
+    if(typeof(to) == "undefined")   throw new Error("To is not given!");
+
+    if(to-1 < from)                 throw new Error("lol1");
+    if(to > poly.vertices.length)   throw new Error("lol2");
+    if(from < 0)                    throw new Error("lol3");
+
+    for(var i=from; i<to; i++){
+        this.vertices.push(poly.vertices[i]);
+    }
+};
+
+/**
+ * Make sure that the polygon vertices are ordered counter-clockwise.
+ * @method makeCCW
+ */
+Polygon.prototype.makeCCW = function(){
+    var br = 0,
+        v = this.vertices;
+
+    // find bottom right point
+    for (var i = 1; i < this.vertices.length; ++i) {
+        if (v[i][1] < v[br][1] || (v[i][1] == v[br][1] && v[i][0] > v[br][0])) {
+            br = i;
+        }
+    }
+
+    // reverse poly if clockwise
+    if (!Point.left(this.at(br - 1), this.at(br), this.at(br + 1))) {
+        this.reverse();
+    }
+};
+
+/**
+ * Reverse the vertices in the polygon
+ * @method reverse
+ */
+Polygon.prototype.reverse = function(){
+    var tmp = [];
+    for(var i=0, N=this.vertices.length; i!==N; i++){
+        tmp.push(this.vertices.pop());
+    }
+    this.vertices = tmp;
+};
+
+/**
+ * Check if a point in the polygon is a reflex point
+ * @method isReflex
+ * @param  {Number}  i
+ * @return {Boolean}
+ */
+Polygon.prototype.isReflex = function(i){
+    return Point.right(this.at(i - 1), this.at(i), this.at(i + 1));
+};
+
+var tmpLine1=[],
+    tmpLine2=[];
+
+/**
+ * Check if two vertices in the polygon can see each other
+ * @method canSee
+ * @param  {Number} a Vertex index 1
+ * @param  {Number} b Vertex index 2
+ * @return {Boolean}
+ */
+Polygon.prototype.canSee = function(a,b) {
+    var p, dist, l1=tmpLine1, l2=tmpLine2;
+
+    if (Point.leftOn(this.at(a + 1), this.at(a), this.at(b)) && Point.rightOn(this.at(a - 1), this.at(a), this.at(b))) {
+        return false;
+    }
+    dist = Point.sqdist(this.at(a), this.at(b));
+    for (var i = 0; i !== this.vertices.length; ++i) { // for each edge
+        if ((i + 1) % this.vertices.length === a || i === a) // ignore incident edges
+            continue;
+        if (Point.leftOn(this.at(a), this.at(b), this.at(i + 1)) && Point.rightOn(this.at(a), this.at(b), this.at(i))) { // if diag intersects an edge
+            l1[0] = this.at(a);
+            l1[1] = this.at(b);
+            l2[0] = this.at(i);
+            l2[1] = this.at(i + 1);
+            p = Line.lineInt(l1,l2);
+            if (Point.sqdist(this.at(a), p) < dist) { // if edge is blocking visibility to b
+                return false;
+            }
+        }
+    }
+
+    return true;
+};
+
+/**
+ * Copy the polygon from vertex i to vertex j.
+ * @method copy
+ * @param  {Number} i
+ * @param  {Number} j
+ * @param  {Polygon} [targetPoly]   Optional target polygon to save in.
+ * @return {Polygon}                The resulting copy.
+ */
+Polygon.prototype.copy = function(i,j,targetPoly){
+    var p = targetPoly || new Polygon();
+    p.clear();
+    if (i < j) {
+        // Insert all vertices from i to j
+        for(var k=i; k<=j; k++)
+            p.vertices.push(this.vertices[k]);
+
+    } else {
+
+        // Insert vertices 0 to j
+        for(var k=0; k<=j; k++)
+            p.vertices.push(this.vertices[k]);
+
+        // Insert vertices i to end
+        for(var k=i; k<this.vertices.length; k++)
+            p.vertices.push(this.vertices[k]);
+    }
+
+    return p;
+};
+
+/**
+ * Decomposes the polygon into convex pieces. Returns a list of edges [[p1,p2],[p2,p3],...] that cuts the polygon.
+ * Note that this algorithm has complexity O(N^4) and will be very slow for polygons with many vertices.
+ * @method getCutEdges
+ * @return {Array}
+ */
+Polygon.prototype.getCutEdges = function() {
+    var min=[], tmp1=[], tmp2=[], tmpPoly = new Polygon();
+    var nDiags = Number.MAX_VALUE;
+
+    for (var i = 0; i < this.vertices.length; ++i) {
+        if (this.isReflex(i)) {
+            for (var j = 0; j < this.vertices.length; ++j) {
+                if (this.canSee(i, j)) {
+                    tmp1 = this.copy(i, j, tmpPoly).getCutEdges();
+                    tmp2 = this.copy(j, i, tmpPoly).getCutEdges();
+
+                    for(var k=0; k<tmp2.length; k++)
+                        tmp1.push(tmp2[k]);
+
+                    if (tmp1.length < nDiags) {
+                        min = tmp1;
+                        nDiags = tmp1.length;
+                        min.push([this.at(i), this.at(j)]);
+                    }
+                }
+            }
+        }
+    }
+
+    return min;
+};
+
+/**
+ * Decomposes the polygon into one or more convex sub-Polygons.
+ * @method decomp
+ * @return {Array} An array or Polygon objects.
+ */
+Polygon.prototype.decomp = function(){
+    var edges = this.getCutEdges();
+    if(edges.length > 0)
+        return this.slice(edges);
+    else
+        return [this];
+};
+
+/**
+ * Slices the polygon given one or more cut edges. If given one, this function will return two polygons (false on failure). If many, an array of polygons.
+ * @method slice
+ * @param {Array} cutEdges A list of edges, as returned by .getCutEdges()
+ * @return {Array}
+ */
+Polygon.prototype.slice = function(cutEdges){
+    if(cutEdges.length == 0) return [this];
+    if(cutEdges instanceof Array && cutEdges.length && cutEdges[0] instanceof Array && cutEdges[0].length==2 && cutEdges[0][0] instanceof Array){
+
+        var polys = [this];
+
+        for(var i=0; i<cutEdges.length; i++){
+            var cutEdge = cutEdges[i];
+            // Cut all polys
+            for(var j=0; j<polys.length; j++){
+                var poly = polys[j];
+                var result = poly.slice(cutEdge);
+                if(result){
+                    // Found poly! Cut and quit
+                    polys.splice(j,1);
+                    polys.push(result[0],result[1]);
+                    break;
+                }
+            }
+        }
+
+        return polys;
+    } else {
+
+        // Was given one edge
+        var cutEdge = cutEdges;
+        var i = this.vertices.indexOf(cutEdge[0]);
+        var j = this.vertices.indexOf(cutEdge[1]);
+
+        if(i != -1 && j != -1){
+            return [this.copy(i,j),
+                    this.copy(j,i)];
+        } else {
+            return false;
+        }
+    }
+};
+
+/**
+ * Checks that the line segments of this polygon do not intersect each other.
+ * @method isSimple
+ * @param  {Array} path An array of vertices e.g. [[0,0],[0,1],...]
+ * @return {Boolean}
+ * @todo Should it check all segments with all others?
+ */
+Polygon.prototype.isSimple = function(){
+    var path = this.vertices;
+    // Check
+    for(var i=0; i<path.length-1; i++){
+        for(var j=0; j<i-1; j++){
+            if(Line.segmentsIntersect(path[i], path[i+1], path[j], path[j+1] )){
+                return false;
+            }
+        }
+    }
+
+    // Check the segment between the last and the first point to all others
+    for(var i=1; i<path.length-2; i++){
+        if(Line.segmentsIntersect(path[0], path[path.length-1], path[i], path[i+1] )){
+            return false;
+        }
+    }
+
+    return true;
+};
+
+function getIntersectionPoint(p1, p2, q1, q2, delta){
+    delta = delta || 0;
+   var a1 = p2[1] - p1[1];
+   var b1 = p1[0] - p2[0];
+   var c1 = (a1 * p1[0]) + (b1 * p1[1]);
+   var a2 = q2[1] - q1[1];
+   var b2 = q1[0] - q2[0];
+   var c2 = (a2 * q1[0]) + (b2 * q1[1]);
+   var det = (a1 * b2) - (a2 * b1);
+
+   if(!Scalar.eq(det,0,delta))
+      return [((b2 * c1) - (b1 * c2)) / det, ((a1 * c2) - (a2 * c1)) / det]
+   else
+      return [0,0]
+}
+
+/**
+ * Quickly decompose the Polygon into convex sub-polygons.
+ * @method quickDecomp
+ * @param  {Array} result
+ * @param  {Array} [reflexVertices]
+ * @param  {Array} [steinerPoints]
+ * @param  {Number} [delta]
+ * @param  {Number} [maxlevel]
+ * @param  {Number} [level]
+ * @return {Array}
+ */
+Polygon.prototype.quickDecomp = function(result,reflexVertices,steinerPoints,delta,maxlevel,level){
+    maxlevel = maxlevel || 100;
+    level = level || 0;
+    delta = delta || 25;
+    result = typeof(result)!="undefined" ? result : [];
+    reflexVertices = reflexVertices || [];
+    steinerPoints = steinerPoints || [];
+
+    var upperInt=[0,0], lowerInt=[0,0], p=[0,0]; // Points
+    var upperDist=0, lowerDist=0, d=0, closestDist=0; // scalars
+    var upperIndex=0, lowerIndex=0, closestIndex=0; // Integers
+    var lowerPoly=new Polygon(), upperPoly=new Polygon(); // polygons
+    var poly = this,
+        v = this.vertices;
+
+    if(v.length < 3) return result;
+
+    level++;
+    if(level > maxlevel){
+        console.warn("quickDecomp: max level ("+maxlevel+") reached.");
+        return result;
+    }
+
+    for (var i = 0; i < this.vertices.length; ++i) {
+        if (poly.isReflex(i)) {
+            reflexVertices.push(poly.vertices[i]);
+            upperDist = lowerDist = Number.MAX_VALUE;
+
+
+            for (var j = 0; j < this.vertices.length; ++j) {
+                if (Point.left(poly.at(i - 1), poly.at(i), poly.at(j))
+                        && Point.rightOn(poly.at(i - 1), poly.at(i), poly.at(j - 1))) { // if line intersects with an edge
+                    p = getIntersectionPoint(poly.at(i - 1), poly.at(i), poly.at(j), poly.at(j - 1)); // find the point of intersection
+                    if (Point.right(poly.at(i + 1), poly.at(i), p)) { // make sure it's inside the poly
+                        d = Point.sqdist(poly.vertices[i], p);
+                        if (d < lowerDist) { // keep only the closest intersection
+                            lowerDist = d;
+                            lowerInt = p;
+                            lowerIndex = j;
+                        }
+                    }
+                }
+                if (Point.left(poly.at(i + 1), poly.at(i), poly.at(j + 1))
+                        && Point.rightOn(poly.at(i + 1), poly.at(i), poly.at(j))) {
+                    p = getIntersectionPoint(poly.at(i + 1), poly.at(i), poly.at(j), poly.at(j + 1));
+                    if (Point.left(poly.at(i - 1), poly.at(i), p)) {
+                        d = Point.sqdist(poly.vertices[i], p);
+                        if (d < upperDist) {
+                            upperDist = d;
+                            upperInt = p;
+                            upperIndex = j;
+                        }
+                    }
+                }
+            }
+
+            // if there are no vertices to connect to, choose a point in the middle
+            if (lowerIndex == (upperIndex + 1) % this.vertices.length) {
+                //console.log("Case 1: Vertex("+i+"), lowerIndex("+lowerIndex+"), upperIndex("+upperIndex+"), poly.size("+this.vertices.length+")");
+                p[0] = (lowerInt[0] + upperInt[0]) / 2;
+                p[1] = (lowerInt[1] + upperInt[1]) / 2;
+                steinerPoints.push(p);
+
+                if (i < upperIndex) {
+                    //lowerPoly.insert(lowerPoly.end(), poly.begin() + i, poly.begin() + upperIndex + 1);
+                    lowerPoly.append(poly, i, upperIndex+1);
+                    lowerPoly.vertices.push(p);
+                    upperPoly.vertices.push(p);
+                    if (lowerIndex != 0){
+                        //upperPoly.insert(upperPoly.end(), poly.begin() + lowerIndex, poly.end());
+                        upperPoly.append(poly,lowerIndex,poly.vertices.length);
+                    }
+                    //upperPoly.insert(upperPoly.end(), poly.begin(), poly.begin() + i + 1);
+                    upperPoly.append(poly,0,i+1);
+                } else {
+                    if (i != 0){
+                        //lowerPoly.insert(lowerPoly.end(), poly.begin() + i, poly.end());
+                        lowerPoly.append(poly,i,poly.vertices.length);
+                    }
+                    //lowerPoly.insert(lowerPoly.end(), poly.begin(), poly.begin() + upperIndex + 1);
+                    lowerPoly.append(poly,0,upperIndex+1);
+                    lowerPoly.vertices.push(p);
+                    upperPoly.vertices.push(p);
+                    //upperPoly.insert(upperPoly.end(), poly.begin() + lowerIndex, poly.begin() + i + 1);
+                    upperPoly.append(poly,lowerIndex,i+1);
+                }
+            } else {
+                // connect to the closest point within the triangle
+                //console.log("Case 2: Vertex("+i+"), closestIndex("+closestIndex+"), poly.size("+this.vertices.length+")\n");
+
+                if (lowerIndex > upperIndex) {
+                    upperIndex += this.vertices.length;
+                }
+                closestDist = Number.MAX_VALUE;
+
+                if(upperIndex < lowerIndex){
+                    return result;
+                }
+
+                for (var j = lowerIndex; j <= upperIndex; ++j) {
+                    if (Point.leftOn(poly.at(i - 1), poly.at(i), poly.at(j))
+                            && Point.rightOn(poly.at(i + 1), poly.at(i), poly.at(j))) {
+                        d = Point.sqdist(poly.at(i), poly.at(j));
+                        if (d < closestDist) {
+                            closestDist = d;
+                            closestIndex = j % this.vertices.length;
+                        }
+                    }
+                }
+
+                if (i < closestIndex) {
+                    lowerPoly.append(poly,i,closestIndex+1);
+                    if (closestIndex != 0){
+                        upperPoly.append(poly,closestIndex,v.length);
+                    }
+                    upperPoly.append(poly,0,i+1);
+                } else {
+                    if (i != 0){
+                        lowerPoly.append(poly,i,v.length);
+                    }
+                    lowerPoly.append(poly,0,closestIndex+1);
+                    upperPoly.append(poly,closestIndex,i+1);
+                }
+            }
+
+            // solve smallest poly first
+            if (lowerPoly.vertices.length < upperPoly.vertices.length) {
+                lowerPoly.quickDecomp(result,reflexVertices,steinerPoints,delta,maxlevel,level);
+                upperPoly.quickDecomp(result,reflexVertices,steinerPoints,delta,maxlevel,level);
+            } else {
+                upperPoly.quickDecomp(result,reflexVertices,steinerPoints,delta,maxlevel,level);
+                lowerPoly.quickDecomp(result,reflexVertices,steinerPoints,delta,maxlevel,level);
+            }
+
+            return result;
+        }
+    }
+    result.push(this);
+
+    return result;
+};
+
+/**
+ * Remove collinear points in the polygon.
+ * @method removeCollinearPoints
+ * @param  {Number} [precision] The threshold angle to use when determining whether two edges are collinear. Use zero for finest precision.
+ * @return {Number}           The number of points removed
+ */
+Polygon.prototype.removeCollinearPoints = function(precision){
+    var num = 0;
+    for(var i=this.vertices.length-1; this.vertices.length>3 && i>=0; --i){
+        if(Point.collinear(this.at(i-1),this.at(i),this.at(i+1),precision)){
+            // Remove the middle point
+            this.vertices.splice(i%this.vertices.length,1);
+            i--; // Jump one point forward. Otherwise we may get a chain removal
+            num++;
+        }
+    }
+    return num;
+};
+
+},{"./Line":1,"./Point":2,"./Scalar":4}],4:[function(require,module,exports){
+module.exports = Scalar;
+
+/**
+ * Scalar functions
+ * @class Scalar
+ */
+function Scalar(){}
+
+/**
+ * Check if two scalars are equal
+ * @static
+ * @method eq
+ * @param  {Number} a
+ * @param  {Number} b
+ * @param  {Number} [precision]
+ * @return {Boolean}
+ */
+Scalar.eq = function(a,b,precision){
+    precision = precision || 0;
+    return Math.abs(a-b) < precision;
+};
+
+},{}],5:[function(require,module,exports){
+module.exports = {
+    Polygon : require("./Polygon"),
+    Point : require("./Point"),
+};
+
+},{"./Point":2,"./Polygon":3}]},{},[5])
+(5)
+});
+;
\ No newline at end of file
diff --git a/src/body/Body.js b/src/body/Body.js
index 9f3db17..3092182 100644
--- a/src/body/Body.js
+++ b/src/body/Body.js
@@ -563,6 +563,9 @@ var Body = {};
      * @return {}
      */
     var _totalProperties = function(body) {
+        // https://ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=st&chap_sec=07.2&page=theory
+        // http://output.to/sideway/default.asp?qno=121100087
+
         var properties = {
             mass: 0,
             area: 0,
@@ -576,10 +579,10 @@ var Body = {};
             properties.mass += part.mass;
             properties.area += part.area;
             properties.inertia += part.inertia;
-            Vector.add(properties.centre, part.position, properties.centre);
+            properties.centre = Vector.add(properties.centre, Vector.mult(part.position, part.mass));
         }
 
-        properties.centre = Vector.div(properties.centre, body.parts.length - 1);
+        properties.centre = Vector.div(properties.centre, properties.mass);
 
         return properties;
     };
diff --git a/src/factory/Bodies.js b/src/factory/Bodies.js
index 25d2aa2..bc6813e 100644
--- a/src/factory/Bodies.js
+++ b/src/factory/Bodies.js
@@ -93,7 +93,7 @@ var Bodies = {};
      * @param {number} y
      * @param {number} radius
      * @param {object} [options]
-     * @param {number} maxSides
+     * @param {number} [maxSides]
      * @return {body} A new circle body
      */
     Bodies.circle = function(x, y, radius, options, maxSides) {
@@ -161,4 +161,100 @@ var Bodies = {};
         return Body.create(Common.extend({}, polygon, options));
     };
 
+    /**
+     * Creates a body using the supplied vertices.
+     * If the vertices are not convex, they will be decomposed if [poly-decomp.js](https://github.com/schteppe/poly-decomp.js) is available.
+     * If the vertices can not be decomposed, the function will use the convex hull.
+     * By default the decomposition will discard collinear edges (to improve performance).
+     * The options parameter is an object that specifies any properties you wish to override the defaults.
+     * See the properties section of the `Matter.Body` module for detailed information on what you can pass via the `options` object.
+     * @method fromVertices
+     * @param {number} x
+     * @param {number} y
+     * @param [vector] vertices
+     * @param {object} [options]
+     * @param {bool} [removeCollinear=true]
+     * @return {body}
+     */
+    Bodies.fromVertices = function(x, y, vertices, options, removeCollinear) {
+        var canDecompose = true,
+            body,
+            i;
+
+        options = options || {};
+        removeCollinear = typeof removeCollinear !== 'undefined' ? removeCollinear : true;
+
+        if (Vertices.isConvex(vertices)) {
+            // vertices are convex, so just create a body normally
+            body = {
+                position: { x: x, y: y },
+                vertices: vertices
+            };
+
+            return Body.create(Common.extend({}, body, options));
+        }
+
+        // check for poly-decomp.js
+        if (!window.decomp) {
+            Common.log('Bodies.fromVertices: poly-decomp.js required. Could not decompose vertices. Fallback to convex hull.', 'warn');
+            canDecompose = false;
+        }
+
+        // initialise a decomposition
+        var concave = new decomp.Polygon();
+        for (i = 0; i < vertices.length; i++) {
+            concave.vertices.push([vertices[i].x, vertices[i].y]);
+        }
+
+        // check for complexity
+        if (!concave.isSimple()) {
+            Common.log('Bodies.fromVertices: Non-simple polygons are not supported. Could not decompose vertices. Fallback to convex hull.', 'warn');
+            canDecompose = false;
+        }
+
+        // try to decompose
+        if (canDecompose) {
+            // vertices are concave and simple, we can decompose into parts
+            concave.makeCCW();
+            if (removeCollinear)
+                concave.removeCollinearPoints(0.001);
+
+            var decomposed = concave.quickDecomp(),
+                parts = [];
+
+            // for each decomposed chunk
+            for (i = 0; i < decomposed.length; i++) {
+                var chunk = decomposed[i],
+                    chunkVertices = [];
+
+                // convert vertices into the correct structure
+                for (var j = 0; j < chunk.vertices.length; j++) {
+                    chunkVertices.push({ x: chunk.vertices[j][0], y: chunk.vertices[j][1] });
+                }
+
+                // create a compound part
+                parts.push(
+                    Body.create({
+                        position: Vertices.centre(chunkVertices),
+                        vertices: chunkVertices
+                    })
+                );
+            }
+
+            // create the parent body to be returned, that contains generated compound parts
+            body = Body.create(Common.extend({}, { parts: parts }, options));
+            Body.setPosition(body, { x: x, y: y });
+
+            return body;
+        } else {
+            // fallback to convex hull when decomposition is not possible
+            body = {
+                position: { x: x, y: y },
+                vertices: Vertices.hull(vertices)
+            };
+
+            return Body.create(Common.extend({}, body, options));
+        }
+    };
+
 })();
diff --git a/src/geometry/Vertices.js b/src/geometry/Vertices.js
index bfa7cf5..0c06a78 100644
--- a/src/geometry/Vertices.js
+++ b/src/geometry/Vertices.js
@@ -56,7 +56,7 @@ var Vertices = {};
      * @return {vertices} vertices
      */
     Vertices.fromPath = function(path, body) {
-        var pathPattern = /L\s*([\-\d\.]*)\s*([\-\d\.]*)/ig,
+        var pathPattern = /L?\s*([\-\d\.]+)\s*([\-\d\.]+)\s*,?/ig,
             points = [];
 
         path.replace(pathPattern, function(match, x, y) {
@@ -330,15 +330,58 @@ var Vertices = {};
      * @return {vertices} vertices
      */
     Vertices.clockwiseSort = function(vertices) {
-        var mean = Vertices.mean(vertices);
+        var centre = Vertices.mean(vertices);
 
         vertices.sort(function(vertexA, vertexB) {
-            return Vector.angle(mean, vertexA) - Vector.angle(mean, vertexB);
+            return Vector.angle(centre, vertexA) - Vector.angle(centre, vertexB);
         });
 
         return vertices;
     };
 
+    /**
+     * Returns true if the vertices form a convex shape (vertices must be in clockwise order).
+     * @method isConvex
+     * @param {vertices} vertices
+     * @return {bool} `true` if the `vertices` are convex, `false` if not (or `null` if not computable).
+     */
+    Vertices.isConvex = function(vertices) {
+        // http://paulbourke.net/geometry/polygonmesh/
+
+        var flag = 0,
+            n = vertices.length,
+            i,
+            j,
+            k,
+            z;
+
+        if (n < 3)
+            return null;
+
+        for (i = 0; i < n; i++) {
+            j = (i + 1) % n;
+            k = (i + 2) % n;
+            z = (vertices[j].x - vertices[i].x) * (vertices[k].y - vertices[j].y);
+            z -= (vertices[j].y - vertices[i].y) * (vertices[k].x - vertices[j].x);
+
+            if (z < 0) {
+                flag |= 1;
+            } else if (z > 0) {
+                flag |= 2;
+            }
+
+            if (flag === 3) {
+                return false;
+            }
+        }
+
+        if (flag !== 0){
+            return true;
+        } else {
+            return null;
+        }
+    };
+
     /**
      * Returns the convex hull of the input vertices as a new array of points.
      * @method hull