# Projective Texture / Shadow Mapping -- Why is the perspective division performed in the fragment shader?

I've just worked my way through this OpenGL shadow mapping tutorial. While I understand the basic algorithm, one thing puzzles me: During the 2nd render pass all vertices are transformed into the clip space of the light source. This is done by multiplying them with the light's view-projection matrix in the vertex shader:

vs_out.FragPosLightSpace = lightSpaceMatrix * vec4(vs_out.FragPos, 1.0);


However, for texture lookup into the shadow map a perspective division is needed. This is done in the fragment shader:

float ShadowCalculation(vec4 fragPosLightSpace)
{
// perform perspective divide
vec3 projCoords = fragPosLightSpace.xyz / fragPosLightSpace.w;
//continue w. texture lookup
[...]
}


So my question is - why can't I perform the perspective division in the vertex shader? I did try moving the division from fragment to vertex shader in my otherwise finished shadow mapping code, and ended up with some really weird artifacts. So I guess it has something to do with the interpolation performed by the rasterizer, but I would like a more detailed explanation if possible.

## tl;dr: perspective-correct interpolation of NDCs doesn't work; they need to be linearly interpolated in screen-space instead.

This answer uses math notation from this paper, section 3: interpolating vertex attributes. (I highly recommend reading the entire paper to understand how perspective-correct interpolation of vertex attributes works; the following discussion is based on it.)

## Why does the code in the tutorial work?

The hardware does perspective-correct interpolation of vertex clip-space coordinates to yield a pixel's clip-space coordinates. Then the perspective divide (by the interpolated w-coordinate) in the pixel shader transforms that into NDC.

But let's dive a bit deeper and see what the interpolation really does under the hood:

Let $$I_1$$ and $$I_2$$ be clip-space coordinates of two vertices, and let $$Z_1$$ and $$Z_2$$ be their respective view-space z-coordinate (or clip-space w-coordinate). We want to find a pixel's NDCs, or mathematically: $$I_t/Z_t$$ where $$I_t$$ is its clip-space coordinates and $$Z_t$$ is the view-space z-coordinate.

Now recall that NDCs are coordinates of a 3D object after projecting it on the 2D projection window (screen-space). So to find a pixel's NDCs in screen-space, it suffices to linearly interpolate the vertices' NDCs in screen-space. Mathematically:

$$\tag{*}\label{*} \frac{I_t}{Z_t}=\frac{I_1}{Z_1}+s\left(\frac{I_2}{Z_2}-\frac{I_1}{Z_1}\right)$$

($$s$$ is a screen-space barycentric coordinate, see in the paper linked above).

To get this, we first let the hardware do perspective-correct interpolation of the vertex clip-space coordinates. This is equation (16) in the paper and it gives us $$I_t$$ - the pixel's clip-space coords: $$I_t=\left[\frac{I_1}{Z_1}+s\left(\frac{I_2}{Z_2}-\frac{I_1}{Z_1}\right)\right]Z_t$$ Then in the pixel shader we divide by w which is simply the $$Z_t$$ of a pixel (remember that w gets interpolated as well), which gives us $$I_t/Z_t$$ (NDC) as desired.

So the hardware does the perspective divide and interpolation for us, which gives the pixel's NDCs, but it also multiplies by $$Z_t$$ which transforms from NDC back to clip-space. And what the perspective divide in the pixel shader really does is transform back to NDC by cancelling out the $$Z_t$$.

## What if we do the division in the vertex shader instead?

The vertex NDCs are $$I_1/Z_1$$ and $$I_2/Z_2$$ and the perspective-correct interpolation equation gives: $$\left[\frac{I_1}{Z_1^2}+s\left(\frac{I_2}{Z_2^2}-\frac{I_1}{Z_1^2}\right)\right]Z_t$$ which is clearly not $$I_t/Z_t$$ as defined in $$\eqref{*}$$. The reason this equation doesn't work in this case is that its derivation is based on the fact that the attribute varies linearly across the triangle in 3D space (view space). But screen-space NDC coords don't vary linearly in 3D space because of the perspective divide, so the premise is broken. (Clip-space coords do vary linearly in view-space because they are the result of a linear transform.) Perspective-correct interpolation of values that have undergone perspective divide doesn't work. They are not in 3D space anymore, rather in 2D projection space and thus require linear interpolation in screen-space to get correct results. In practice it's accomplished using perspective-correct interpolation of clip-space coords, plus perspective divide per-pixel.