[ferfumarma]

Can you elaborate on your point that translation is not linear? The OP agrees with you, so clearly your point is correct, but I personally just don't understand it. Isn't it true that translation is linear within the coordinate space of your model, even if the final distance traveled within a projected camera view is not?

edit to add: (I think your point relates only to the projection system, and not a pure, unprojected model; I just want to make sure I understand because it seems like an important point)

[srean]

All linear operators map origin to origin. But translation applied to the origin will shift it. So translation cannot be linear.

Let's take another approach.

Take a point p that's sum of vectors a and b, that is

p = a + b.

Now, if translation was a linear transformation, then translating p (say along x-axis by 1 unit) is equivalent to applying same translation to a and b separately and then summing them. But the latter ends up translating by twice the amount. Or in other words

p +t ≠ (a +t) + (b +t) = p + 2t.

So translation is not a linear operators in this vector space.

[andrewla]

No, with projective geometry or affine geometry you can make translation into a linear operation. But in ordinary Euclidean space translation is not a linear operation.

Most obvious case that it fails is that it doesn't map zero to itself, and you can see the contradiction there:

    T(0 + 0) = T(0) = t
    T(0) + T(0) = t + t = 2 * t