Why are width and height divided by 2 in the perspective projection matrix?

Question

I'm using OpenGL to learn computer graphics. I'm trying to understand projection matrices. The following are shown in the OpenGL 'Red Book':

I don't understand why the width and height are divided by 2?

Answer 1

Because $x_{proj}$ doesn't vary from $0 \to width$, it varies from $-\tfrac{width}{2} \to \tfrac{width}{2}$.

What's important is not the width, but the minimum and maximum values of $x_{proj}$. Because $(0, 0)$ is in the centre of the viewport, not the corner, the min and max values are plus and minus half of the width.

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BTW, the Red Book is not a great introduction to graphics in general. It's a technical standard, not a tutorial, so the explanation of concepts is just enough to be unambiguous - it's not trying to teach the material. It's also about a scanline rendering library in particular, so there's a lot of important concepts it doesn't describe at all, such as what BRDFs are important, how they relate to real-world surface properties, what an image really is, and many higher-level techniques.

And finally, even in the area it's about - how to use a GPU to accelerate rendering - it's hideously out of date. The GL architecture is based on GPUs from 20 years ago. Although lots of features have been added to keep up with changes to hardware, the fundamental abstractions are not a good way to understand current GPU architectures.

If you want to learn about the theory of computer graphics, start with a proper textbook such as Physically Based Rendering or Computer Graphics: Principles and Practice, or an online course. If you want to learn how to make a 3D application, start with a tutorial about the kind of application you're interested in (e.g. if it's a game, try a Unity tutorial).