If [x y w] are non-zero, then [ax ay aw] for a != 0 in homogeneous coordinates would correspond to the same (x/w, y/w) in Euclidean coordinates.

The divide by 3rd coordinate here looks like the depth scaling in the pinhole camera we talked about iirc

This procedure is used in machine learning (in ML it's called the bias trick) to convert a scale and a bias to a single matrix multiplication. Interesting to see what this does visually!

What is the advantage of using homogeneous coordinates over allowing affine transformations? Is it more efficient?

@harrysha1029 I guess using homogeneous coords allows us to express translation using a linear map (matrix). Thus, when we need to do more transformations, we can simply pre-compute the product of all transformation matrices into one and directly apply it to our target. This operation is better optimized and made more efficient by GPUs than an affine transformation.