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Merge pull request #1167 from nockawa/earcut

Bye bye Poly2Tri, welcome Earcut
David Catuhe 9 years ago
parent
commit
13351c722f

+ 1 - 0
Tools/Gulp/config.json

@@ -180,6 +180,7 @@
       "../../src/PostProcess/babylon.displayPassPostProcess.js",
       "../../src/Mesh/babylon.meshSimplification.js",
       "../../src/Tools/babylon.sceneSerializer.js",
+      "../../src/Tools/babylon.earcut.js",
       "../../src/Mesh/babylon.csg.js",
       "../../src/PostProcess/babylon.vrDistortionCorrectionPostProcess.js",
       "../../src/Tools/babylon.virtualJoystick.js",

+ 1 - 0
dist/preview release/what's new.md

@@ -20,6 +20,7 @@
 	- Interleaved buffers are now directly supported. Create a `Buffer` object and then use `buffer.createVertexBuffer` to specify the vertex buffers ([benaadams](https://github.com/benaadams)) 
 	- Vertex buffers can be marked as instanced to allow custom instancing attributes ([benaadams](https://github.com/benaadams)) 
 	- Mesh can have `overridenInstanceCount` set to specify the number of meshes to draw when custom instancing is used ([benaadams](https://github.com/benaadams)) 
+    - Now supporting the [Earcut](https://github.com/mapbox/earcut) polygon triangulation library as part of babylon.js library. (Look for the `Earcut` module). The `PolygonMeshBuilder` class now relies on Earcut. ([nockawa](https://github.com/nockawa))	
   - **Updates**
     - Added `renderTargetTexture.useCameraPostProcesses` to control postprocesses for render targets ([deltakosh](https://github.com/deltakosh))
     - Added `mesh.toLefthanded()` to convert a mesh from right handed system ([kesshi](https://github.com/Kesshi))

+ 38 - 38
src/Mesh/babylon.polygonMesh.ts

@@ -98,7 +98,6 @@ module BABYLON {
 
     export class PolygonMeshBuilder {
 
-        private _swctx: poly2tri.SweepContext;
         private _points = new PolygonPoints();
         private _outlinepoints = new PolygonPoints();
         private _holes = [];
@@ -106,6 +105,15 @@ module BABYLON {
         private _name: string;
         private _scene: Scene;
 
+        private _epoints: number[] = new Array<number>();
+        private _eholes: number[] = new Array<number>();
+
+        private _addToepoint(points: Vector2[]) {
+            for (let p of points) {
+                this._epoints.push(p.x, p.y);
+            }
+        }
+
         constructor(name: string, contours: Path2, scene: Scene)
         constructor(name: string, contours: Vector2[], scene: Scene)
         constructor(name: string, contours: any, scene: Scene) {
@@ -123,15 +131,21 @@ module BABYLON {
                 points = (<Vector2[]>contours);
             }
 
-            this._swctx = new poly2tri.SweepContext(this._points.add(points));
-            this._outlinepoints.add(points)
+            this._addToepoint(points);
+
+            this._points.add(points);
+            this._outlinepoints.add(points);
         }
 
         addHole(hole: Vector2[]): PolygonMeshBuilder {
-            this._swctx.addHole(this._points.add(hole));
+            this._points.add(hole);
             var holepoints = new PolygonPoints();
             holepoints.add(hole); 
-            this._holes.push(holepoints) ;
+            this._holes.push(holepoints);
+
+            this._eholes.push(this._epoints.length/2);
+            this._addToepoint(hole);
+
             return this;
         }
 
@@ -151,14 +165,13 @@ module BABYLON {
 
             var indices = [];
 
-            this._swctx.triangulate();
-            this._swctx.getTriangles().forEach((triangle) => {
-                triangle.getPoints().forEach((point) => {
-                    indices.push((<IndexedVector2>point).index);
-                });
-            });
+            let res = Earcut.earcut(this._epoints, this._eholes, 2);
+
+            for (let i = 0; i < res.length; i++) {
+                indices.push(res[i]);
+            }
 
-            if (depth > 0) { 
+            if (depth > 0) {
                 var positionscount = (positions.length / 3); //get the current pointcount
                
                 this._points.elements.forEach((p) => { //add the elements at the depth
@@ -167,35 +180,22 @@ module BABYLON {
                     uvs.push(1-(p.x - bounds.min.x) / bounds.width,1-(p.y - bounds.min.y) / bounds.height);
                 });
 
-                var p1: IndexedVector2;           //we need to change order of point so the triangles are made in the rigth way.
-                var p2: IndexedVector2;
-                var poscounter: number = 0;
-                this._swctx.getTriangles().forEach((triangle) => {
-                    triangle.getPoints().forEach((point) => {
-
-                        switch (poscounter) {
-                            case 0:
-                                p1 = <IndexedVector2>point;
-                                break;
-                            case 1:
-                                p2 = <IndexedVector2>point;
-                                break;
-                            case 2:
-                                indices.push((<IndexedVector2>point).index + positionscount); 
-                                indices.push(p2.index + positionscount);
-                                indices.push(p1.index + positionscount);
-                                poscounter = -1;
-                                break;
-                        }
-                        poscounter++;
-                        //indices.push((<IndexedVector2>point).index + positionscount);
-                    });
-                });
+                let totalCount = indices.length;
+                for (let i = 0; i < totalCount; i += 3) {
+                    let i0 = indices[i + 0];
+                    let i1 = indices[i + 1];
+                    let i2 = indices[i + 2];
+
+                    indices.push(i2 + positionscount);
+                    indices.push(i1 + positionscount);
+                    indices.push(i0 + positionscount);
+                }
+
                 //Add the sides
-                this.addSide(positions, normals, uvs, indices, bounds, this._outlinepoints, depth, false)
+                this.addSide(positions, normals, uvs, indices, bounds, this._outlinepoints, depth, false);
 
                 this._holes.forEach((hole) => {
-                    this.addSide(positions, normals, uvs, indices, bounds, hole, depth, true)
+                    this.addSide(positions, normals, uvs, indices, bounds, hole, depth, true);
                 });                               
             }
 

+ 666 - 0
src/Tools/babylon.earcut.ts

@@ -0,0 +1,666 @@
+// All the credit goes to this project and the guy who's behind it https://github.com/mapbox/earcut
+// Huge respect for a such great lib. 
+// We just make it TypeScript compiling compliant and typed the public function.
+module Earcut {
+    /**
+     * The fastest and smallest JavaScript polygon triangulation library for your WebGL apps
+     * @param data is a flat array of vertice coordinates like [x0, y0, x1, y1, x2, y2, ...].
+     * @param holeIndices is an array of hole indices if any (e.g. [5, 8] for a 12- vertice input would mean one hole with vertices 5–7 and another with 8–11).
+     * @param dim is the number of coordinates per vertice in the input array (2 by default).
+     */
+    export function earcut(data: number[], holeIndices: number[], dim: number) {
+        dim = dim || 2;
+
+        var hasHoles = holeIndices && holeIndices.length,
+            outerLen = hasHoles ? holeIndices[0] * dim : data.length,
+            outerNode = linkedList(data, 0, outerLen, dim, true),
+            triangles = [];
+
+        if (!outerNode) return triangles;
+
+        var minX, minY, maxX, maxY, x, y, size;
+
+        if (hasHoles) outerNode = eliminateHoles(data, holeIndices, outerNode, dim);
+
+        // if the shape is not too simple, we'll use z-order curve hash later; calculate polygon bbox
+        if (data.length > 80 * dim) {
+            minX = maxX = data[0];
+            minY = maxY = data[1];
+
+            for (var i = dim; i < outerLen; i += dim) {
+                x = data[i];
+                y = data[i + 1];
+                if (x < minX) minX = x;
+                if (y < minY) minY = y;
+                if (x > maxX) maxX = x;
+                if (y > maxY) maxY = y;
+            }
+
+            // minX, minY and size are later used to transform coords into integers for z-order calculation
+            size = Math.max(maxX - minX, maxY - minY);
+        }
+
+        earcutLinked(outerNode, triangles, dim, minX, minY, size, undefined);
+
+        return triangles;
+    }
+
+    // create a circular doubly linked list from polygon points in the specified winding order
+    function linkedList(data, start, end, dim, clockwise) {
+        var i, last;
+
+        if (clockwise === (signedArea(data, start, end, dim) > 0)) {
+            for (i = start; i < end; i += dim) last = insertNode(i, data[i], data[i + 1], last);
+        } else {
+            for (i = end - dim; i >= start; i -= dim) last = insertNode(i, data[i], data[i + 1], last);
+        }
+
+        if (last && equals(last, last.next)) {
+            removeNode(last);
+            last = last.next;
+        }
+
+        return last;
+    }
+
+    // eliminate colinear or duplicate points
+    function filterPoints(start, end) {
+        if (!start) return start;
+        if (!end) end = start;
+
+        var p = start,
+            again;
+        do {
+            again = false;
+
+            if (!p.steiner && (equals(p, p.next) || area(p.prev, p, p.next) === 0)) {
+                removeNode(p);
+                p = end = p.prev;
+                if (p === p.next) return null;
+                again = true;
+
+            } else {
+                p = p.next;
+            }
+        } while (again || p !== end);
+
+        return end;
+    }
+
+    // main ear slicing loop which triangulates a polygon (given as a linked list)
+    function earcutLinked(ear, triangles, dim, minX, minY, size, pass) {
+        if (!ear) return;
+
+        // interlink polygon nodes in z-order
+        if (!pass && size) indexCurve(ear, minX, minY, size);
+
+        var stop = ear,
+            prev,
+            next;
+
+        // iterate through ears, slicing them one by one
+        while (ear.prev !== ear.next) {
+            prev = ear.prev;
+            next = ear.next;
+
+            if (size ? isEarHashed(ear, minX, minY, size) : isEar(ear)) {
+                // cut off the triangle
+                triangles.push(prev.i / dim);
+                triangles.push(ear.i / dim);
+                triangles.push(next.i / dim);
+
+                removeNode(ear);
+
+                // skipping the next vertice leads to less sliver triangles
+                ear = next.next;
+                stop = next.next;
+
+                continue;
+            }
+
+            ear = next;
+
+            // if we looped through the whole remaining polygon and can't find any more ears
+            if (ear === stop) {
+                // try filtering points and slicing again
+                if (!pass) {
+                    earcutLinked(filterPoints(ear, undefined), triangles, dim, minX, minY, size, 1);
+
+                    // if this didn't work, try curing all small self-intersections locally
+                } else if (pass === 1) {
+                    ear = cureLocalIntersections(ear, triangles, dim);
+                    earcutLinked(ear, triangles, dim, minX, minY, size, 2);
+
+                    // as a last resort, try splitting the remaining polygon into two
+                } else if (pass === 2) {
+                    splitEarcut(ear, triangles, dim, minX, minY, size);
+                }
+
+                break;
+            }
+        }
+    }
+
+    // check whether a polygon node forms a valid ear with adjacent nodes
+    function isEar(ear) {
+        var a = ear.prev,
+            b = ear,
+            c = ear.next;
+
+        if (area(a, b, c) >= 0) return false; // reflex, can't be an ear
+
+        // now make sure we don't have other points inside the potential ear
+        var p = ear.next.next;
+
+        while (p !== ear.prev) {
+            if (pointInTriangle(a.x, a.y, b.x, b.y, c.x, c.y, p.x, p.y) &&
+                area(p.prev, p, p.next) >= 0) return false;
+            p = p.next;
+        }
+
+        return true;
+    }
+
+    function isEarHashed(ear, minX, minY, size) {
+        var a = ear.prev,
+            b = ear,
+            c = ear.next;
+
+        if (area(a, b, c) >= 0) return false; // reflex, can't be an ear
+
+        // triangle bbox; min & max are calculated like this for speed
+        var minTX = a.x < b.x ? (a.x < c.x ? a.x : c.x) : (b.x < c.x ? b.x : c.x),
+            minTY = a.y < b.y ? (a.y < c.y ? a.y : c.y) : (b.y < c.y ? b.y : c.y),
+            maxTX = a.x > b.x ? (a.x > c.x ? a.x : c.x) : (b.x > c.x ? b.x : c.x),
+            maxTY = a.y > b.y ? (a.y > c.y ? a.y : c.y) : (b.y > c.y ? b.y : c.y);
+
+        // z-order range for the current triangle bbox;
+        var minZ = zOrder(minTX, minTY, minX, minY, size),
+            maxZ = zOrder(maxTX, maxTY, minX, minY, size);
+
+        // first look for points inside the triangle in increasing z-order
+        var p = ear.nextZ;
+
+        while (p && p.z <= maxZ) {
+            if (p !== ear.prev &&
+                p !== ear.next &&
+                pointInTriangle(a.x, a.y, b.x, b.y, c.x, c.y, p.x, p.y) &&
+                area(p.prev, p, p.next) >= 0) return false;
+            p = p.nextZ;
+        }
+
+        // then look for points in decreasing z-order
+        p = ear.prevZ;
+
+        while (p && p.z >= minZ) {
+            if (p !== ear.prev &&
+                p !== ear.next &&
+                pointInTriangle(a.x, a.y, b.x, b.y, c.x, c.y, p.x, p.y) &&
+                area(p.prev, p, p.next) >= 0) return false;
+            p = p.prevZ;
+        }
+
+        return true;
+    }
+
+    // go through all polygon nodes and cure small local self-intersections
+    function cureLocalIntersections(start, triangles, dim) {
+        var p = start;
+        do {
+            var a = p.prev,
+                b = p.next.next;
+
+            if (!equals(a, b) && intersects(a, p, p.next, b) && locallyInside(a, b) && locallyInside(b, a)) {
+
+                triangles.push(a.i / dim);
+                triangles.push(p.i / dim);
+                triangles.push(b.i / dim);
+
+                // remove two nodes involved
+                removeNode(p);
+                removeNode(p.next);
+
+                p = start = b;
+            }
+            p = p.next;
+        } while (p !== start);
+
+        return p;
+    }
+
+    // try splitting polygon into two and triangulate them independently
+    function splitEarcut(start, triangles, dim, minX, minY, size) {
+        // look for a valid diagonal that divides the polygon into two
+        var a = start;
+        do {
+            var b = a.next.next;
+            while (b !== a.prev) {
+                if (a.i !== b.i && isValidDiagonal(a, b)) {
+                    // split the polygon in two by the diagonal
+                    var c = splitPolygon(a, b);
+
+                    // filter colinear points around the cuts
+                    a = filterPoints(a, a.next);
+                    c = filterPoints(c, c.next);
+
+                    // run earcut on each half
+                    earcutLinked(a, triangles, dim, minX, minY, size, undefined);
+                    earcutLinked(c, triangles, dim, minX, minY, size, undefined);
+                    return;
+                }
+                b = b.next;
+            }
+            a = a.next;
+        } while (a !== start);
+    }
+
+    // link every hole into the outer loop, producing a single-ring polygon without holes
+    function eliminateHoles(data, holeIndices, outerNode, dim) {
+        var queue = [],
+            i,
+            len,
+            start,
+            end,
+            list;
+
+        for (i = 0, len = holeIndices.length; i < len; i++) {
+            start = holeIndices[i] * dim;
+            end = i < len - 1 ? holeIndices[i + 1] * dim : data.length;
+            list = linkedList(data, start, end, dim, false);
+            if (list === list.next) list.steiner = true;
+            queue.push(getLeftmost(list));
+        }
+
+        queue.sort(compareX);
+
+        // process holes from left to right
+        for (i = 0; i < queue.length; i++) {
+            eliminateHole(queue[i], outerNode);
+            outerNode = filterPoints(outerNode, outerNode.next);
+        }
+
+        return outerNode;
+    }
+
+    function compareX(a, b) {
+        return a.x - b.x;
+    }
+
+    // find a bridge between vertices that connects hole with an outer ring and and link it
+    function eliminateHole(hole, outerNode) {
+        outerNode = findHoleBridge(hole, outerNode);
+        if (outerNode) {
+            var b = splitPolygon(outerNode, hole);
+            filterPoints(b, b.next);
+        }
+    }
+
+    // David Eberly's algorithm for finding a bridge between hole and outer polygon
+    function findHoleBridge(hole, outerNode) {
+        var p = outerNode,
+            hx = hole.x,
+            hy = hole.y,
+            qx = -Infinity,
+            m;
+
+        // find a segment intersected by a ray from the hole's leftmost point to the left;
+        // segment's endpoint with lesser x will be potential connection point
+        do {
+            if (hy <= p.y && hy >= p.next.y) {
+                var x = p.x + (hy - p.y) * (p.next.x - p.x) / (p.next.y - p.y);
+                if (x <= hx && x > qx) {
+                    qx = x;
+                    if (x === hx) {
+                        if (hy === p.y) return p;
+                        if (hy === p.next.y) return p.next;
+                    }
+                    m = p.x < p.next.x ? p : p.next;
+                }
+            }
+            p = p.next;
+        } while (p !== outerNode);
+
+        if (!m) return null;
+
+        if (hx === qx) return m.prev; // hole touches outer segment; pick lower endpoint
+
+        // look for points inside the triangle of hole point, segment intersection and endpoint;
+        // if there are no points found, we have a valid connection;
+        // otherwise choose the point of the minimum angle with the ray as connection point
+
+        var stop = m,
+            mx = m.x,
+            my = m.y,
+            tanMin = Infinity,
+            tan;
+
+        p = m.next;
+
+        while (p !== stop) {
+            if (hx >= p.x &&
+                p.x >= mx &&
+                pointInTriangle(hy < my ? hx : qx, hy, mx, my, hy < my ? qx : hx, hy, p.x, p.y)) {
+
+                tan = Math.abs(hy - p.y) / (hx - p.x); // tangential
+
+                if ((tan < tanMin || (tan === tanMin && p.x > m.x)) && locallyInside(p, hole)) {
+                    m = p;
+                    tanMin = tan;
+                }
+            }
+
+            p = p.next;
+        }
+
+        return m;
+    }
+
+    // interlink polygon nodes in z-order
+    function indexCurve(start, minX, minY, size) {
+        var p = start;
+        do {
+            if (p.z === null) p.z = zOrder(p.x, p.y, minX, minY, size);
+            p.prevZ = p.prev;
+            p.nextZ = p.next;
+            p = p.next;
+        } while (p !== start);
+
+        p.prevZ.nextZ = null;
+        p.prevZ = null;
+
+        sortLinked(p);
+    }
+
+    // Simon Tatham's linked list merge sort algorithm
+    // http://www.chiark.greenend.org.uk/~sgtatham/algorithms/listsort.html
+    function sortLinked(list) {
+        var i, p, q, e, tail, numMerges, pSize, qSize, inSize = 1;
+        do {
+            p = list;
+            list = null;
+            tail = null;
+            numMerges = 0;
+
+            while (p) {
+                numMerges++;
+                q = p;
+                pSize = 0;
+                for (i = 0; i < inSize; i++) {
+                    pSize++;
+                    q = q.nextZ;
+                    if (!q) break;
+                }
+
+                qSize = inSize;
+
+                while (pSize > 0 || (qSize > 0 && q)) {
+
+                    if (pSize === 0) {
+                        e = q;
+                        q = q.nextZ;
+                        qSize--;
+                    } else if (qSize === 0 || !q) {
+                        e = p;
+                        p = p.nextZ;
+                        pSize--;
+                    } else if (p.z <= q.z) {
+                        e = p;
+                        p = p.nextZ;
+                        pSize--;
+                    } else {
+                        e = q;
+                        q = q.nextZ;
+                        qSize--;
+                    }
+
+                    if (tail) tail.nextZ = e;
+                    else list = e;
+
+                    e.prevZ = tail;
+                    tail = e;
+                }
+
+                p = q;
+            }
+
+            tail.nextZ = null;
+            inSize *= 2;
+
+        } while (numMerges > 1);
+
+        return list;
+    }
+
+    // z-order of a point given coords and size of the data bounding box
+    function zOrder(x, y, minX, minY, size) {
+        // coords are transformed into non-negative 15-bit integer range
+        x = 32767 * (x - minX) / size;
+        y = 32767 * (y - minY) / size;
+
+        x = (x | (x << 8)) & 0x00FF00FF;
+        x = (x | (x << 4)) & 0x0F0F0F0F;
+        x = (x | (x << 2)) & 0x33333333;
+        x = (x | (x << 1)) & 0x55555555;
+
+        y = (y | (y << 8)) & 0x00FF00FF;
+        y = (y | (y << 4)) & 0x0F0F0F0F;
+        y = (y | (y << 2)) & 0x33333333;
+        y = (y | (y << 1)) & 0x55555555;
+
+        return x | (y << 1);
+    }
+
+    // find the leftmost node of a polygon ring
+    function getLeftmost(start) {
+        var p = start,
+            leftmost = start;
+        do {
+            if (p.x < leftmost.x) leftmost = p;
+            p = p.next;
+        } while (p !== start);
+
+        return leftmost;
+    }
+
+    // check if a point lies within a convex triangle
+    function pointInTriangle(ax, ay, bx, by, cx, cy, px, py) {
+        return (cx - px) * (ay - py) - (ax - px) * (cy - py) >= 0 &&
+        (ax - px) * (by - py) - (bx - px) * (ay - py) >= 0 &&
+        (bx - px) * (cy - py) - (cx - px) * (by - py) >= 0;
+    }
+
+    // check if a diagonal between two polygon nodes is valid (lies in polygon interior)
+    function isValidDiagonal(a, b) {
+        return a.next.i !== b.i &&
+            a.prev.i !== b.i &&
+            !intersectsPolygon(a, b) &&
+            locallyInside(a, b) &&
+            locallyInside(b, a) &&
+            middleInside(a, b);
+    }
+
+    // signed area of a triangle
+    function area(p, q, r) {
+        return (q.y - p.y) * (r.x - q.x) - (q.x - p.x) * (r.y - q.y);
+    }
+
+    // check if two points are equal
+    function equals(p1, p2) {
+        return p1.x === p2.x && p1.y === p2.y;
+    }
+
+    // check if two segments intersect
+    function intersects(p1, q1, p2, q2) {
+        if ((equals(p1, q1) && equals(p2, q2)) ||
+        (equals(p1, q2) && equals(p2, q1))) return true;
+        return area(p1, q1, p2) > 0 !== area(p1, q1, q2) > 0 &&
+            area(p2, q2, p1) > 0 !== area(p2, q2, q1) > 0;
+    }
+
+    // check if a polygon diagonal intersects any polygon segments
+    function intersectsPolygon(a, b) {
+        var p = a;
+        do {
+            if (p.i !== a.i &&
+                p.next.i !== a.i &&
+                p.i !== b.i &&
+                p.next.i !== b.i &&
+                intersects(p, p.next, a, b)) return true;
+            p = p.next;
+        } while (p !== a);
+
+        return false;
+    }
+
+    // check if a polygon diagonal is locally inside the polygon
+    function locallyInside(a, b) {
+        return area(a.prev, a, a.next) < 0
+            ? area(a, b, a.next) >= 0 && area(a, a.prev, b) >= 0
+            : area(a, b, a.prev) < 0 || area(a, a.next, b) < 0;
+    }
+
+    // check if the middle point of a polygon diagonal is inside the polygon
+    function middleInside(a, b) {
+        var p = a,
+            inside = false,
+            px = (a.x + b.x) / 2,
+            py = (a.y + b.y) / 2;
+        do {
+            if (((p.y > py) !== (p.next.y > py)) && (px < (p.next.x - p.x) * (py - p.y) / (p.next.y - p.y) + p.x))
+                inside = !inside;
+            p = p.next;
+        } while (p !== a);
+
+        return inside;
+    }
+
+    // link two polygon vertices with a bridge; if the vertices belong to the same ring, it splits polygon into two;
+    // if one belongs to the outer ring and another to a hole, it merges it into a single ring
+    function splitPolygon(a, b) {
+        var a2 = new Node(a.i, a.x, a.y),
+            b2 = new Node(b.i, b.x, b.y),
+            an = a.next,
+            bp = b.prev;
+
+        a.next = b;
+        b.prev = a;
+
+        a2.next = an;
+        an.prev = a2;
+
+        b2.next = a2;
+        a2.prev = b2;
+
+        bp.next = b2;
+        b2.prev = bp;
+
+        return b2;
+    }
+
+    // create a node and optionally link it with previous one (in a circular doubly linked list)
+    function insertNode(i, x, y, last) {
+        var p = new Node(i, x, y);
+
+        if (!last) {
+            p.prev = p;
+            p.next = p;
+
+        } else {
+            p.next = last.next;
+            p.prev = last;
+            last.next.prev = p;
+            last.next = p;
+        }
+        return p;
+    }
+
+    function removeNode(p) {
+        p.next.prev = p.prev;
+        p.prev.next = p.next;
+
+        if (p.prevZ) p.prevZ.nextZ = p.nextZ;
+        if (p.nextZ) p.nextZ.prevZ = p.prevZ;
+    }
+
+    function Node(i, x, y) {
+        // vertice index in coordinates array
+        this.i = i;
+
+        // vertex coordinates
+        this.x = x;
+        this.y = y;
+
+        // previous and next vertice nodes in a polygon ring
+        this.prev = null;
+        this.next = null;
+
+        // z-order curve value
+        this.z = null;
+
+        // previous and next nodes in z-order
+        this.prevZ = null;
+        this.nextZ = null;
+
+        // indicates whether this is a steiner point
+        this.steiner = false;
+    }
+
+    /**
+     * return a percentage difference between the polygon area and its triangulation area;
+     * used to verify correctness of triangulation
+     */
+    export function deviation(data: number[], holeIndices: number[], dim: number, triangles: number[]) {
+        var hasHoles = holeIndices && holeIndices.length;
+        var outerLen = hasHoles ? holeIndices[0] * dim : data.length;
+
+        var polygonArea = Math.abs(signedArea(data, 0, outerLen, dim));
+        if (hasHoles) {
+            for (var i = 0, len = holeIndices.length; i < len; i++) {
+                var start = holeIndices[i] * dim;
+                var end = i < len - 1 ? holeIndices[i + 1] * dim : data.length;
+                polygonArea -= Math.abs(signedArea(data, start, end, dim));
+            }
+        }
+
+        var trianglesArea = 0;
+        for (i = 0; i < triangles.length; i += 3) {
+            var a = triangles[i] * dim;
+            var b = triangles[i + 1] * dim;
+            var c = triangles[i + 2] * dim;
+            trianglesArea += Math.abs(
+            (data[a] - data[c]) * (data[b + 1] - data[a + 1]) -
+            (data[a] - data[b]) * (data[c + 1] - data[a + 1]));
+        }
+
+        return polygonArea === 0 && trianglesArea === 0 ? 0 : Math.abs((trianglesArea - polygonArea) / polygonArea);
+    };
+
+    function signedArea(data, start, end, dim) {
+        var sum = 0;
+        for (var i = start, j = end - dim; i < end; i += dim) {
+            sum += (data[j] - data[i]) * (data[i + 1] + data[j + 1]);
+            j = i;
+        }
+        return sum;
+    }
+
+    /**
+     *  turn a polygon in a multi-dimensional array form (e.g. as in GeoJSON) into a form Earcut accepts
+     */
+    export function flatten(data: number[][][]) {
+        var dim = data[0][0].length,
+            result = { vertices: [], holes: [], dimensions: dim },
+            holeIndex = 0;
+
+        for (var i = 0; i < data.length; i++) {
+            for (var j = 0; j < data[i].length; j++) {
+                for (var d = 0; d < dim; d++) result.vertices.push(data[i][j][d]);
+            }
+            if (i > 0) {
+                holeIndex += data[i - 1].length;
+                result.holes.push(holeIndex);
+            }
+        }
+        return result;
+    };
+}