Comment by ferfumarma | Hacker News Reader

Comment by ferfumarma | Hacker News Reader

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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)

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.

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

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