opengl projection- where is the mapping to NDC

Question

Where is the mapping to NDC?

According to my understanding, the projection matrix does two things: First, it clips the view space to form the frustum. It only keeps the vertices that fall into the frustum but clips others that fall outside. Second, it maps the coordinates within the clipped space to NDC [-1, 1].

But I only see that we create a frustum (to use for clipping) by define a perspective matrix with glm::perspective or glm::ortho. Where is the steps to do the mapping between frustum and NDC?

Or we just need to define the frustum and OpenGL will do the two steps for us automatically?

Answer 1

According to my understanding, the projection matrix do two things: first, it clipped the view space to form the frustum. It only keeps the vertice that's fall into the frustum but clipped others fall outside. Second, it maps the coordinates within the clipped space to NDC [-1, 1].

It does neither of these things.

A 4x4 matrix is just a transformation. A transformation by itself cannot clip anything. And while a matrix can transform coordinates into a [-1, 1] space, that's not necessarily what the clip space provided by the vertex shader will be. And it's certainly not the clip space used by 3D projection matrices.

The only mandated job of the vertex shader (or the last vertex processing shader stage) is to generate a clip-space position for each vertex. Clip-space is defined by OpenGL to be the 4D space where the XYZ components of the position are within the range [-W, W] are considered "visible". Here, "W" is the W component of the position. So for each XYZW, "visible" is defined as where the XYZ components are in the range of W.

The actual clipping of primitives happens after the vertex shader. Each primitive's vertices are clipped against the previously-defined 4D clip space.

A projection frustum is created by transforming positions into a clip-space such that those 4D positions outside of the frustum are outside of the [-W, W] range, and those inside of the frustum are inside of that range.

So it is not that the transformation clips anything; it merely sets up the data so that OpenGL's clipping system will clip things correctly.

Similarly, clip-space is not NDC space (unless the W component for the position is 1). NDC space is defined by taking the XYZ of a clip-space position and dividing it by that position's W component (and if W is 1, then this obviously changes nothing). Clip and NDC space are two separate spaces, and clipping happens before NDC space. You can conceptually think of clipping as being done against the [-1, 1] range of NDC space. After all, clip-space is on the [-W, W] range, so if you divide that by W, you get the range [-1, 1].

But it's still important to remember that clip space isn't NDC space.