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For linear functions (in the article's second, linear algebra sense), the image of zero must be zero, and that doesn't hold for (most) affine functions, so makes sense.

Note that he shows the trick of making affine functions linear by tacking on one more dimension.

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> the image of zero must be zero, and that doesn't hold for (most) affine functions

It does hold, always, in the higher dimension. And I feel like that's the most important thing to clearly understand about affine transforms. The entire beauty of the augmented matrix is that you get a class of non-linear transforms in 3d by using linear transforms in 4d. The article was nice, I'm nit picking something that was nearly there. It'd just be nice to be one teensy bit more explicit about what's going on here.

But affine transformations are indeed not linear. The augmentation trick creates a new, linear transform in n+1 dimension, which is related but different to the affine transform in question.

The categorization in the article seems correct.

Yes, exactly, you're right. The article is correct, it's just not telling quite the whole story. Affine transforms aren't linear, and their augmented matrices that do the same thing are linear. In practice, it's both, and it's precisely cool because it's both.

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