Let’s pretend you have an application that lets users create shapes to be used in a physics simulation and that the user must click on the screen to set the vertices of the shape. Many physics engines only support convex polygons, or shapes that don’t have inlets, bites, or coves, basically shapes that don’t have inward facing edges. With this limitation we have to be able to restrict [read as "guide"] the user to make only convex polygons. For this we are going to need an algorithm that takes an unordered set of points and finds the convex hull that encloses those points. This way the user can click and add points at random, if desired, and your program will keep track of what points create a convex polygon, while the others are thrown away [or dealt with however you see fit].
The Graham Scan Algorithm is a process of ordering a random set of points and then calculating jumps to the points in that set that constitute a convex polygon. In this algorithm there are three steps. First is to find a corner point, usually the topmost, leftmost point in the set. The second step is to order all other points by the polar angle between the corner point and the point in question. The last step is to traverse the set, taking each proceeding subset of three points (n, n-1, n-2) to determine whether the angle made by these three points is a left turn, right turn, or a straight line. If the turn made is our desired turn [which is usually left - but in Flash it's right, due to the flipped y-axis] then we add that point to the convex hull. If the turn is not our desired turn, we get rid of that point and move on.
Here is an example that shows first the data set drawn from point to point. Each successive line gets progressively whiter. In the second step we find the corner point, order the other points and then show the outer polygon.
Here’s the code for the class:
* Use this class freely - 2009 blog.efnx.com
public class GrahamScan extends Object
* The Graham scan is a method of computing the convex hull of a finite set of points
* in the plane with time complexity O(n log n). It is named after Ronald Graham, who
* published the original algorithm in 1972. The algorithm finds all vertices of
* the convex hull ordered along its boundary. It may also be easily modified to report
* all input points that lie on the boundary of their convex hull.
public function GrahamScan()
* Returns a convex hull given an unordered array of points.
public static function convexHull(data:Array):Array
return findHull( order(data) );
* Orders an array of points counterclockwise.
public static function order(data:Array):Array
// first run through all the points and find the upper left [lower left]
var p:Point = data;
var n:int = data.length;
for (var i:int = 1; i < n; i++)
if(data[i].y < p.y)
//trace(" d.y < p.y / d is new p.");
p = data[i];
else if(data[i].y == p.y && data[i].x < p.x)
//trace(" d.y == p.y, d.x < p.x / d is new p.");
p = data[i];
// next find all the cotangents of the angles made by the point P and the
// other points
var sorted :Array = new Array();
// we need arrays for positive and negative values, because Array.sort
// will put sort the negatives backwards.
var pos :Array = new Array();
var neg :Array = new Array();
// add points back in order
for (i = 0; i < n; i++)
var a :Number = data[i].x - p.x;
var b :Number = data[i].y - p.y;
var cot :Number = b/a;
if(cot < 0)
// sort the arrays
pos.sortOn("cotangent", Array.NUMERIC | Array.DESCENDING);
neg.sortOn("cotangent", Array.NUMERIC | Array.DESCENDING);
sorted = neg.concat(pos);
var ordered :Array = new Array();
for (i = 0; i < n; i++)
if(p == sorted[i].point)
* Given an array of points ordered counterclockwise, findHull will
* filter the points and return an array containing the vertices of a
* convex polygon that envelopes those points.
public static function findHull(data:Array):Array
var n :int = data.length;
var hull:Array = new Array();
hull.push(data); // add the pivot
hull.push(data); // makes first vector
for (var i:int = 2; i < n; i++)
while(direction(hull[hull.length - 2], hull[hull.length - 1], data[i]) >= 0)
private static function direction(p1:Point, p2:Point, p3:Point):Number
// > 0 is right turn
// == 0 is collinear
// < 0 is left turn
// we only want right turns, usually we want right turns, but
// flash's grid is flipped on y.
return (p2.x - p1.x) * (p3.y - p1.y) - (p2.y - p1.y) * (p3.x - p1.x);