mathdirectxdirect3ddirect3d11homogenous-transformation

DirectX negative W

I really was trying to find an answer on this very basic (at first sight) question.

For simplicity depth test is disabled during further discussion (it doesn’t have a big deal).

For example, we have triangle (after transformation) with next float4 coordinates.

top CenterPoint: (0.0f, +0.6f, 0.6f, 1f)

basic point1: (+0.4f, -0.4f, 0.4f, 1f),

basic point2: (-0.4f, -0.4f, 0.4f, 1f),

I’m sending float4 for input and use straight VertexShader (without transforms), so I’m sure about input. And we have result is reasonable:

img

But what we will get if we'll start to move CenterPoint to point of camera position. In our case we don’t have camera so will move this point to minus infinity. I'm getting quite reasonable results as long as w (with z) is positive. For example, (0.0f, +0.006f, 0.006f, .01f) – look the same.

But what if I'll use next coordinates (0.0f, -0.6f, -1f, -1f). (Note: we have to switch points or change rasterizer for culling preventing).

According to huge amount of resource I'll have test like: -w < z < w, so GPU should cut of that point. And yes, in principle, I don’t see point. But triangle still visible! OK, according to huge amount of other resource (and my personal understanding) we'll have division like (x/w, y/w, z/w) so result should be (0, 0.6, 1). But I'm getting

img

And even if that result have some sense (one point is somewhere far away behind as), how really DirectX (I think it is rather GPU) works in such cases (in case of infinite points and negative W)?

It seems that I don't know something very basic, but it seems that nobody know that.

[Added]: I want to note that point w < 0 - is not a real input. In real life such points are result of transformation by matrices and according to the math (math that are used in standard Direct sdk and other places) corresponds to the point that appears behind the camera position. And yes, that point is clipped, but questions is rather about strange triangle that contains such point.

[Brief answer]: Clipping is essentially not just z/w checking and division (see details below).

Solution

  • Theoretically, NDC depth is divided into two distinct areas. The following diagram shows these areas for znear = 1, zfar = 3. The horizontal axis shows view-space z and the vertical axis shows the resulting NDC depth for a standard projective transform:

    NDC Depth

    We can see that the part between view-space z of 1 and 3 (znear, zmax) gets mapped to NDC depth 0 to 1. This is the part that we are actually interested in. However, the part where view-space z is negative also produces positive NDC depth. However, those are parts that result from fold-overs. I.e., if you take a corner of your triangle and slowly decrease z (along with w), starting in the area between znear and zfar, you would observe the following:

    However, this last part is the area behind the camera. So, homogeneous clipping is made, such that this part is also clipped away by znear clipping.

    Check the old D3D9 documentation for the formulas and some more illustrative explanations here.