Category: Math

var matrix:Matrix3D = new Matrix3D();
var verts:Vector.<Number> = new Vector.<Number>();
var pVerts:Vector.<Number> = new Vector.<Number>();
var uvts:Vector.<Number> = new Vector.<Number>();
var sqrt2:Number = Math.sqrt(2)
var pi:Number = Math.PI;
var pi23:Number= 2 * Math.PI / 3;
var step:Number= pi / 50;
for (var u:Number = -pi; u<pi; u+=step) {
	for (var v:Number = -pi; v<pi; v+=step) {
		// from here http://local.wasp.uwa.edu.au/~pbourke/geometry/hexatorus/
		var px = Math.sin(u) / Math.abs(sqrt2+ Math.cos(v))
        var py = Math.sin(u+pi23) / Math.abs(sqrt2 +Math.cos(v + pi23))
        var pz = Math.cos(u-pi23) / Math.abs(sqrt2 +Math.cos(v - pi23))
		verts.push(px * 50);
		verts.push(py * 50);
		verts.push(pz * 50);
	}
}
var brush:BitmapData=new BitmapData(3,2,true,0x41FFFFFF);
var canvas:BitmapData=new BitmapData(400,400,false,0x000000);
addChild(new Bitmap(canvas));
var dx:Number=0;
var dy:Number=0;
addEventListener(Event.ENTER_FRAME, onLoop);
function onLoop(evt:Event):void {
	dx += (mouseX - dx)/4;
	dy += (mouseY - dy)/4;
	matrix.identity();
	matrix.appendRotation(dy,Vector3D.X_AXIS);
	matrix.appendRotation(dx,Vector3D.Y_AXIS);
	matrix.appendTranslation(200, 200, 0);
	Utils3D.projectVectors(matrix, verts, pVerts, uvts);
	canvas.lock();
	canvas.fillRect(canvas.rect, 0x000000);
	var p = new Point();
	for (var i:int = 0; i<pVerts.length; i+=2) {
		p.x = pVerts[i];
		p.y = pVerts[i+1];
		canvas.copyPixels(brush, brush.rect, p, null, null, true);
	}
	canvas.unlock();
}

More 3D shapes inspired by (taken from) Paul Bourke’s website. I figured I should probably add a parametric 3D surface to this set of posts – so I chose to plot the Triaxial Hexatorus. It’s important to note that U and V go from -PI to PI.

I googled Triaxial Hexatorus and stumbled upon a great papervision demo that creates meshes using parametric equations – it also morphs between shapes.

I also noticed that I don’t need to populate the uvts or pVerts Vectors with zeros – seems to work fine without doing that.

var matrix:Matrix3D = new Matrix3D();
var verts:Vector.<Number> = new Vector.<Number>();
var pVerts:Vector.<Number> = new Vector.<Number>();
var uvts:Vector.<Number> = new Vector.<Number>();
for (var i:Number = -10; i<10; i+=.04) {
	for (var j:Number = -5; j<8; j+=.04) {
		for (var k:Number = -10; k<10; k+=.04) {
			var s:Number =  i * i * j + j * k * k;
			if (s <10 && s > 9.95) {
				verts.push(i * 20);
				verts.push(j * 20);
				verts.push(k * 20);
				pVerts.push(0),pVerts.push(0);
				uvts.push(0),uvts.push(0),uvts.push(0);
			}
		}
	}
}
var brush:BitmapData=new BitmapData(3,2,true,0x41FFFFFF);
var canvas:BitmapData=new BitmapData(400,400,false,0x000000);
addChild(new Bitmap(canvas));
var dx:Number=0;
var dy:Number=0;
addEventListener(Event.ENTER_FRAME, onLoop);
function onLoop(evt:Event):void {
	dx += (mouseX - dx)/4;
	dy += (mouseY - dy)/4;
	matrix.identity();
	matrix.appendRotation(dy,Vector3D.X_AXIS);
	matrix.appendRotation(dx,Vector3D.Y_AXIS);
	matrix.appendTranslation(200, 200, 0);
	Utils3D.projectVectors(matrix, verts, pVerts, uvts);
	canvas.lock();
	canvas.fillRect(canvas.rect, 0x000000);
	var p = new Point();
	for (var i:int = 0; i<pVerts.length; i+=2) {
		p.x = pVerts[i];
		p.y = pVerts[i+1];
		canvas.copyPixels(brush, brush.rect, p, null, null, true);
	}
	canvas.unlock();
}

Yet another variation on the post from the last two days. This one plots something resembling a wormhole – I was randomly tweaking the equation for a sphere and stumbled upon this…