Oblique View Frustum Depth Projection and Clipping (Eric Lengyel)

Question

Trying to understand oblique clipping method i've got some problem with theory. According to this article written by Eric Lengyel at the end of 2 chapter we get clipping spaces:

• Near <0,0,1,1>
• Far <0,0,-1,1> ...

And it is said that:

each camera-space plane is expressed as a sum or difference of two rows of the projection matrix

THIS moment i can not understand. For example, if it's said that Near plane value is "M4 + M3" (where M4 and M3 are the fourth and third rows of projection matrix), and other values ​​are calculated similarly, then the conclusion follows that projection matrix MUST be Identity (to get <0,0,1,1> result from M4 + M3). But we know that it's different. So, can someone explain, what matrix we use and and what is the connection with the projection matrix?

Answer 1

THIS moment i can not understand. For example, if it's said that Near plane value is "M4 + M3" (where M4 and M3 are the fourth and third rows of projection matrix), and other values ​​are calculated similarly, then the conclusion follows that projection matrix MUST be Identity (to get <0,0,1,1> result from M4 + M3).

First of all, your logic is very flawed here. To get a vector c=(0,0,1,1) out of the sum of two vectors a+b, you can find an infinte amount of vectors a and b fulfilling this, for example (7,-2pi,0,42) + (-7, 2pi, 1, -41) = (0,0,1,1).

However, this is completely besides the point, because you misunderstood crucial parts of that article. The clip planes you specified here are in clip space (for the special case that w = 1, as explained in the article). If we would want to find the equations for the clip planes in clip space, there would be absolutely no need for doing any calculations at all because the clip planes are defined in clip space as fixed equations. There is no point in calculating M4+M3 if we already know it would yield (0,0,1,1).

The whole article talks about efficiently calculating the clip planes in eye space. And table 1 of that paper makes this extremely clear: