Projecting a 3D point to 2D screen space using a perspective camera matrix

Question

Hi everyone,

I am attempting to project a series of 3D points onto the screen using a perspective camera matrix. I do not have world space (or consider it being an identity matrix) and my camera does not have camera space (or consider it an identity matrix), I do have a 4x4 matrix for my object space.

I am taking the object matrix and multiplying it by the camera perspective matrix, generated with the following method:

Matrix4 createPerspectiveMatrix( Float fov, Float aspect, Float near, Float far )
{
    Float fov2 = (fov/2) * (Math.PI/180);
    Float tan = Math.tan(fov2);
    Float f = 1 / tan;

    return new Matrix4 ( 
        f/aspect, 0, 0, 0,
        0, f, 0, 0,
        0, 0, -((near+far)/(near-far)), (2*far*near)/(near-far),
        0, 0, 1, 0 
    );
}

I am then taking my point [x, y, z, 1] and multiplying that by the resulting multiplication of the perspective matrix and object matrix.

The next part is where i'm getting confuzzled, I'm pretty sure that I need to get these points within the ranges of either -1 and 1, or 0 and 1, and, in the case of having the first set of values, I would then multiply the points by the screen width and height for the screen coordinates x and y value respectivly or multiply the values by the screen height/2 and width/2 and add the same values to the respective points.

Any step by step telling me how this might be achieved or where I might be going wrong with any of this would be massivly appreciated!! :D

Best regards everyone!

P.S. In the example of a identity/translation matrix, my matrix format in my model is:

[1, 0, 0, tx,
 0, 1, 0, ty,
 0, 0, 1, tz,
 0, 0, 0, 1 ]

Answer 1

Your problem is that you forget to perform the perspective division.

Perspective division means that you divide x, y and z component of your point by its w component. This is required for transforming your point from Homogeneous 4D Space to Normalized Device Coordinates System (NDCS) in which each component x, y or z falls between -1 and 1, or 0 and 1.

After this transformation, you can do your viewport transformation (multiply points by screen width, heigth etc).

There's a good view of this transformation pipeline in Foley's book (Computer Graphics: Principles and Practice in C), you can see it here:

http://www.ugrad.cs.ubc.ca/~cs314/notes/pipeline.html