Do we use 3x3 matrices in computer graphics?

Question

I've been doing some side-research on computer graphics as a hobby and came across this article on quaternions: http://www.opengl-tutorial.org/assets/faq_quaternions/index.html#Q2

In the first section on matrices, it describes 2x2, 3x3, and 4x4 matrices and their functions. Here is the entry on 3x3 matrices:

3x3 matrices are used to perform low-budget 3D animation. Operations such as rotation and multiplication can be performed using matrix
operations, but perspective depth projection is performed using
standard optimised into pure divide operations.

I'm not sure if this blurb means that 3x3 matrices aren't used anymore or if we are still using them for simpler calculations on certain animations. I understand what perspective depth projection is, but what does it mean it is performed using pure divide operations?

Answer 1

That quote is a very strange way to phrase things. We definitely use 3x3 matrices in computer graphics. They tend to be most useful for doing affine transformations of 2D objects. It allows you to have scale, rotation, shearing, and translation (in 2D), but not perspective transformations. I believe that's what the quote is trying to say.

To get a perspective projection of a 3D scene onto a 2D plane (such as your computer screen), at some point you will need to divide the x and y components of each point in your geometry by some factor that foreshortens lines in the z dimension. That's the divide that they are referring to. See this article for more details. In it, they describe:

The simplest perspective projection uses the origin as the center of projection, and z = 1 as the image plane. The functional form of this transformation is then x' = x / z; y' = y / z.