How to convert world to screen coordinates and vice versa

This is a thing every 3D developer should know: for a given point in 3D what is the position of that point in 2D, e.g. on the screen in pixel coordinates? This problem appears quite often (starting with mouse picking) and it' always handy to have the solution at hand :)

The first thing you should remember once and for all regarding screen coordinates - the upper left corner of the screen is (0,0) and lower right is (width, height) - no arguing! Now, lets say we got a 3D point in world coordinate space at (x, y, z) - this is not relative to the camera, but absolute (so the camera can have coordinates (cx, cy, cz)). The camera defines the viewMatrix, and I suppose you also have defined a projectionMatrix (you better have!). The last thing you need is the width and height of the are you are rendering on (not the whole screen!). If you have all these things, then it's pretty easy:

function point2D get2dPoint(Point3D point3D, Matrix viewMatrix,

                 Matrix projectionMatrix, int width, int height) {

     Matrix4 viewProjectionMatrix = projectionMatrix * viewMatrix;

     //transform world to clipping coordinates

     point3D = viewProjectionMatrix.multiply(point3D);

     int winX = (int) Math.round((( point3D.getX() + 1 ) / 2.0) *

                                   width );

      //we calculate -point3D.getY() because the screen Y axis is

      //oriented top->down 

     int winY = (int) Math.round((( 1 - point3D.getY() ) / 2.0) *

                                   height );

      return new Point2D(winX, winY);

}

And that's it! Depending on your development evironment, you might have a viewProjection matrix already computed, or your world coordinates are relative to the camera position - don't forget to modify the code according to that!

For the ones of you who want or need to know the way back from 2D to 3D - well, it's just doing the inverted opperations:

function Point3D get3dPoint(Point2D point2D, int width,

        int height, Matrix viewMatrix, Matrix projectionMatrix) {

       double x = 2.0 * winX / clientWidth - 1;

      double y = - 2.0 * winY / clientHeight + 1;

        Matrix4 viewProjectionInverse = inverse(projectionMatrix *

             viewMatrix);

       Point3D point3D = new Point3D(x, y, 0);

        return viewProjectionInverse.multiply(point3D);

}

The only unclear thing here is the z coordinate - I have set it to 0. The reason is that from the arguments passed to our function we can not figure out the Z coordinate exactly - because every number would be correct! Think quick - how many points from our 3D world can you map on a point of glass when you look through the window? Indefinitely many! To sum up: there is no unique mapping between a 2D and a 3D point, but there is between a 3D and a 2D point. 

Hope I made some things clearer.. you can find more info on the net, but as a more detailed overview I can recommend you this picking tutorial.