Catmull-Rom Spline

var canvas:BitmapData = new BitmapData(400,400,false, 0x000000);
addChild(new Bitmap(canvas));

var p0:Sprite = dot(100, 100);
var p1:Sprite = dot(120, 180);
var p2:Sprite = dot(220, 180);
var p3:Sprite = dot(250, 250);

addEventListener(Event.ENTER_FRAME, onLoop);
function onLoop(evt:Event):void {
	canvas.lock();   
	canvas.fillRect(canvas.rect, 0x000000);

	curve(p0, p1, p2, p3);
	canvas.unlock();
}


// all math from http://en.wikipedia.org/wiki/Cubic_Hermite_spline
function curve(p0:Sprite, p1:Sprite, p2:Sprite, p3:Sprite, rez:Number=.02):void{
	var px:Number = 0;
    var py:Number = 0;
	var m0:Point = tangent(p1, p0);
	var m1:Point = tangent(p2, p0);
	var m2:Point = tangent(p3, p1);
	var m3:Point = tangent(p3, p2);
	
	for (var t:Number = 0; t < 1; t+=rez){
		 var t_2:Number = t * t;
		 var _1_t:Number = 1 - t;
		 var _2t:Number = 2 * t;
		 
		 var h00:Number =  (1 + _2t) * (_1_t) * (_1_t);
		 var h10:Number =  t  * (_1_t) * (_1_t);
		 var h01:Number =  t_2 * (3 - _2t);
		 var h11:Number =  t_2 * (t - 1);
		 
		 px = h00 * p0.x + h10 * m0.x + h01 * p1.x + h11 * m1.x;
		 py = h00 * p0.y + h10 * m0.y + h01 * p1.y + h11 * m1.y;
		 canvas.setPixel(px, py, 0xFFFFFF);
		 
		 px = h00 * p1.x + h10 * m1.x + h01 * p2.x + h11 * m2.x;
		 py = h00 * p1.y + h10 * m1.y + h01 * p2.y + h11 * m2.y;
		 canvas.setPixel(px, py, 0xFFFFFF);
		 
		 px = h00 * p2.x + h10 * m2.x + h01 * p3.x + h11 * m3.x;
		 py = h00 * p2.y + h10 * m2.y + h01 * p3.y + h11 * m3.y;
		 canvas.setPixel(px, py, 0xFFFFFF);
	}
}

function tangent(pk1:Sprite, pk_1:Sprite){
	return new Point((pk1.x - pk_1.x) / 2, (pk1.y - pk_1.y) / 2);
}

// draggable dot
function dot(xp:Number, yp:Number, col:uint = 0x507399, rad:Number=5):Sprite {
	var s:Sprite = Sprite(addChild(new Sprite));
	s.x = xp;
	s.y = yp;
	with(s.graphics) beginFill(col), drawCircle(0,0,rad);
	s.buttonMode = true;
	s.addEventListener(MouseEvent.MOUSE_DOWN, onDrag);
	return s;
}
function onDrag(evt:MouseEvent):void {
	evt.currentTarget.startDrag()
}
stage.addEventListener(MouseEvent.MOUSE_UP, onUp);
function onUp(evt:MouseEvent):void{
	stopDrag();
}

This is snippet draws a Catmull-Rom spline based on 4 control points. A nice feature of Catmull-Rom splines is that they always pass through each of their control points.

have a look at the swf here:

I first learned about Catmull-Rom splines somewhere in the processing forums.

I got all my info for this snippet from this wikipedia page.

There is a good deal of room for optimization here… I may post an optimized version in the future.