Oblique Frustum Clipping

这里是例子代码：
http://www.terathon.com/code/oblique.html

具体的说明可以参见 http://www.opengl.org/discussion_boards/cgi_directory/ultimatebb.cgi?ubb=get_topic;f=6;t=000170
其重要部分摘录如下：

Let P = (Px, Py, Pz, Pw) be the eye-space plane to which you would like to clip.This will replace your ordinary near plane and should be facing away from thecamera, so Pz < 0.

Let Q be an eye-space point.  If P dot Q = 0, then Q lies on the plane P.

What we want to do is modify the projection matrix M so that points Q forwhich P dot Q = 0 get transformed in such a way that the transformed z-coordinateis the negation of the transformed w-coordinate.  This corresponds tothe projected z-coordinate that you would ordinarily get for a point lyingon the near plane.  After division by the w-coordinate, you get -1.

The projection matrix M transforms points from eye space into homogeneous clipspace.  To obtain the clip-space plane P', we need to multiply P by the inversetranspose of M.  (This is because planes are covariant vectors.)  The clip-spaceplane P' is given by

       -1 TP' = (M  )  P

Let Q' be the projection of the point Q, given by Q' = MQ.  If Q lies on the plane P,then P' dot Q' = 0, but we want it to be -Q'w so that it corresponds to the near plane.So we force it by subtracting 1 from the w-coordinate of P'.  Before doing this,however, we want to divide P' by its z-coordinate as to avoid scaling the zdirection in clip space.  Doing all of this gives us the following plane P":

       P'x    P'y       P'wP" = (-----, -----, 1, ----- - 1)       P'z    P'z       P'z

We integrate this into the projection matrix M by constructing a new projectionmatrix M' as follows.

     [  1    0    0    0  ]     [                    ]     [  0    1    0    0  ]M' = [                    ] M     [ P"x  P"y   1   P"w ]     [                    ]     [  0    0    0    1  ]

The matrices M and M' only differ in the entries of the 3rd row, so we could justcalculate replacements for 3rd-row entries of M as follows:

M'(3,1) = P" dot M_1M'(3,2) = P" dot M_2M'(3,3) = P" dot M_3M'(3,4) = P" dot M_4

where M_i means the i-th column of M and M'(i,j) means the (i,j)-th entry of M'.

Okay -- here's the short, short version of the optimal implementation of the oblique frustum.

Let C = camera-space clipping plane. Assume the camera is on the negative side of the plane: C_w < 0.

Let M be the original projection matrix; M must be invertible, but otherwise we don't care whatit is. We'll modify the third row of M so that the near plane coincides with the arbitraryclipping plane C.  We aren't allowed to modify the fourth row of M because doing so would screwup the perspective-correct vertex attribute interpolation. (The fourth row usually justmoves the z-coordinate of a camera-space point into the w-coordinate of a clip-space point.)

Given a projection matrix M, the near plane is always M_4 + M_3, and the far plane is alwaysM_4 - M_3. (M_i means the i-th row of M.)  To force the near plane to coincide with theclipping plane C, we must have M_3 = aC - M_4, where a is some positive scale factor that wecan adjust.

Why adjust a?  Because now our far plane F has been moved to F = 2*M_4 - aC, which is notgenerally parallel to C.  F intersects C on the x-y plane, so it's really not in a goodposition.  The best we can do is minimize the size of the view frustum by choosing theconstant a so that F contains the corner Q of the view frustum opposite the near plane C.This point Q is given by

Q = M^-1 * (sgn(C_x), sgn(C_y), 1, 1),

where M^-1 is the inverse of the projection matrix. (The points (+/-1, +/-1, 1, 1) are thefour corners of the view frustum on the far plane in clip space.)  The scale factor a is nowgiven by

a = (2*M_4 dot Q) / (C dot Q).

Since M_4 is usually (0, 0, -1, 0), this can be simplified a little.

All we have to do to the original projection matrix M is replace the third row with aC - M_4.

-- Eric Lengyel

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