OpenGL：除以零的w

Question

I'm currently trying to implement a Camera class, complete with frustum, that I can use for culling objects from the scene. To find the bounds of the frustum, I'm transforming the system coordinate space (cube with sides ranging [-1..1] each) by the inverse of the PVM matrix. This seems to be working flawlessly, except...

When transforming the origin (0, 0, 0, 1) by the inverse PVM matrix for testing purposes, I get (0, 0, 0, 0 ). I know exactly why it's happening given the particular values of the PVM matrix whilst debugging, but I'm wondering if I'm logically doing something wrong, since as far as I know, the Euclidian point should be (0, 0, 0), not a bunch of NaNs resulting from aw of 0. What am I forgetting here? Am I not supposed to divide by w? Are my matrices are wrong? Blargh...

More details:

Projection matrix is defined by the following method (using LWJGL in Java):

/** fieldOfView is in radians, aspectRatio is simply width / height. */
public static Matrix4f makeProjectionMatrix(Matrix4f dest, float fieldOfView, float aspectRatio, float nearPlane, float farPlane)
{
    float y_scale = GLUtils.cotan(fieldOfView * .5f);
    float x_scale = y_scale / aspectRatio;
    float frustumLength = farPlane - nearPlane;

    dest.setIdentity();
    dest.m00 = x_scale;
    dest.m11 = y_scale;
    dest.m22 = -(farPlane + nearPlane) / frustumLength;
    dest.m23 = -1f;
    dest.m32 = -2f * nearPlane * farPlane / frustumLength;
    dest.m33 = 0f;

    return dest;
}

*Based on http://unspecified.wordpress.com/2012/06/21/calculating-the-gluperspective-matrix-and-other-opengl-matrix-maths/

Notice that m33 is always 0, which happens to be true for my inverse as well (and I believe to be true of all inverses given the above formula).

Since the Model and View are both the identity matrix, and the Projection is unmodified from the above, a coordinate (0, 0, 0, x) transformed by the inverse PVM matrix would always yield w of 0... right?

Example matrices causing the dilemma:

Original Projection:

1.732 0     0  0
0     1.732 0  0
0     0     0 -2
0     0    -1  0

Inverted Projection:

.577 0    0   0
0    .577 0   0
0    0    0  -1
0    0   -.5  0

I'd be happy to post additional code if a miscalculation is evident.

Answer 1

Further to the comments:

Well az of 0 produces aw of 0 because thats what your projection matrix says to do ;) That said in projection space az of 0 is right at the origin of the projection. it needs to be in front of the projection to be visible. Otherwise both the center of projection AND the vertex position occupy the exact same location. This is why we have near planes to prevent any geometry with aw of 0 (or close to) getting drawn.