# Parametric UV Surface

Actionscript:
1. var matrix:Matrix3D = new Matrix3D();
2. var verts:Vector.<Number> = new Vector.<Number>();
3. var pVerts:Vector.<Number> = new Vector.<Number>();
4. var uvts:Vector.<Number> = new Vector.<Number>();
5. var sqrt2:Number = Math.sqrt(2)
6. var pi:Number = Math.PI;
7. var pi23:Number= 2 * Math.PI / 3;
8. var step:Number= pi / 50;
9. for (var u:Number = -pi; u<pi; u+=step) {
10.     for (var v:Number = -pi; v<pi; v+=step) {
11.         // from here http://local.wasp.uwa.edu.au/~pbourke/geometry/hexatorus/
12.         var px = Math.sin(u) / Math.abs(sqrt2+ Math.cos(v))
13.         var py = Math.sin(u+pi23) / Math.abs(sqrt2 +Math.cos(v + pi23))
14.         var pz = Math.cos(u-pi23) / Math.abs(sqrt2 +Math.cos(v - pi23))
15.         verts.push(px * 50);
16.         verts.push(py * 50);
17.         verts.push(pz * 50);
18.     }
19. }
20. var brush:BitmapData=new BitmapData(3,2,true,0x41FFFFFF);
21. var canvas:BitmapData=new BitmapData(400,400,false,0x000000);
23. var dx:Number=0;
24. var dy:Number=0;
26. function onLoop(evt:Event):void {
27.     dx += (mouseX - dx)/4;
28.     dy += (mouseY - dy)/4;
29.     matrix.identity();
30.     matrix.appendRotation(dy,Vector3D.X_AXIS);
31.     matrix.appendRotation(dx,Vector3D.Y_AXIS);
32.     matrix.appendTranslation(200, 200, 0);
33.     Utils3D.projectVectors(matrix, verts, pVerts, uvts);
34.     canvas.lock();
35.     canvas.fillRect(canvas.rect, 0x000000);
36.     var p = new Point();
37.     for (var i:int = 0; i<pVerts.length; i+=2) {
38.         p.x = pVerts[i];
39.         p.y = pVerts[i+1];
40.         canvas.copyPixels(brush, brush.rect, p, null, null, true);
41.     }
42.     canvas.unlock();
43. }

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.