Hippopede

x=stage.stageWidth/2;
y=stage.stageHeight/2;
var xp:Number = 0, yp:Number = 0;
var r:Number = 0, t:Number = 0;
var speed:Number=.1;

var range:Number=.2;
var b:Number = 20;
var a:Number = b * range;
var orbit:Number = 2 * Math.PI + speed;
 
graphics.lineStyle(0,0x000000);

addEventListener(Event.ENTER_FRAME, onLoop);
function onLoop(evt:Event):void {
	var sin:Number=Math.sin(t);
	r =  Math.sqrt(4 * b * (a - b * (sin * sin)));
	xp= r * Math.cos(t);
	yp= r * sin;
	
    if (t==0) {
		graphics.moveTo(xp, yp);
	} else {
		graphics.lineTo(xp, yp);
	}

	t += speed;
	
	if (t > orbit) {
		range += .5;
		a= b * range;
		t= 0;
	}
}

The above draws a Hippopede