Inverting rotation in 3D, to make an object always face the camera?

Question

i have lots of sprites arranged in 3D space, and their parent container has rotations applied. How do i reverse the sprites 3D rotation, that they always face the camera (Actionscript 3)?

heres a code to test it:

package{
import flash.display.Sprite;
import flash.events.Event;
public class test extends Sprite{

var canvas:Sprite = new Sprite();
var sprites:Array = []

public function test(){
    addChild(canvas)
    for (var i:int=0;i<20;i++){
        var sp:Sprite = new Sprite();
        canvas.addChild(sp);
        sp.graphics.beginFill(0xFF0000);
        sp.graphics.drawCircle(0,0,4);
        sp.x = Math.random()*400-200;
        sp.y = Math.random()*400-200;
        sp.z = Math.random()*400-200;
        sprites.push(sp);
    }
    addEventListener(Event.ENTER_FRAME,function():void{
        canvas.rotationX++;
        canvas.rotationY = canvas.rotationY+Math.random()*2;
        canvas.rotationZ++;
        for (var i:int=0;i<20;i++){
            //this is not working...
            sprites[i].rotationX = -canvas.rotationX
            sprites[i].rotationY = -canvas.rotationY
            sprites[i].rotationZ = -canvas.rotationZ
        }
    })
}
}
}

I am guessing i have to do some magic with the rotation3D matrices of the sprites... I've tried to implement this script: http://ughzoid.wordpress.com/2011/02/03/papervision3d-sprite3d/ , but had so success
Thanks for help.

Answer 1

The easiest way to do this is "clearing" the rotational part of the transform matrix. Your typical homogenous transformation looks like this

| xx xy xz xw |
| yx yy yz yw |
| zx zy zz zw |
| wx wy wz ww | 

with wx = wy = wz = 0, ww = 1. If you take a closer look you'll see that in fact this matrix is composed of a 3x3 submatrix defining the rotation, a 3 subvector for the translation and a homogenous row 0 0 0 1

| R       T |
| (0,0,0) 1 |

For a billboard/sprite you want to keep the translation, but get rid of the rotation, i.e. R = I. In case some scaleing was applied the identity needs to be scaled as well.

This gives the following recipie:

d = sqrt( xx² + yx² + zx² )

| d 0 0 T.x |
| 0 d 0 T.y |
| 0 0 d T.z |
| 0 0 0   1 |

Loading this matrix allows you to draw camera aligned sprites.