0
$\begingroup$

Learning Computer Graphics -

Can similarity transformation be linear transformation?

Similarity T is a rigid transformation (translation and rotations) with uniform scaling. so I guess a similarity transformation can be linear. Am I missing someting?

Thank you

$\endgroup$

1 Answer 1

2
$\begingroup$

Linear transformations fulfill two criteria:

T(A) + T(B) = T(A + B)

and

k T(A) = T(k A)  (for any scalar k)

What happens if A is zero?

 T(0) + T(B) = T(0 + B)
 T(0) + T(B) = T(B)
 T(0) = 0

 k T(0) = T(k 0)
 k T(0) = T(0)
 (k - 1) T(0) = 0
 T(0) = 0   (since k can be any scalar, not limited to k = 1)

So, a linear transformation must transform zero to zero.

In two or more dimensions, that means transforming the origin (0, 0) or (0, 0, 0) etc. to itself.

This means that a linear transformation cannot include translation. A triangle joining points (0, 0), (1, 0), and (0, 1) can't be linearly transformed to the similar triangle spanning (1, 0), (2, 0), (1, 1), because that would require mapping the origin to a different point, which linear transformations can never do.

A transformation that can include a linear transformation and a shift in the origin is called an affine transformation.

So: some similarity transforms are not linear.


In game development, we work around this by bumping up a dimension. So if it's a 3D game, we work with 4D vectors and 4x4 matrices. We set the 4th, w, component to 1 for all positions, so the whole game world sits on the 3-dimensional w=1 hyperplane within the 4D space. We can then use a matrix as a linear transformation to shear this hyperplane around in the x, y, and z dimensions - including the origin of our 3D game world at (0, 0, 0, 1), even though the "real" origin of the containing 4D space at (0, 0, 0, 0) never moves.


To cover our bases, we should probably also attack this from the opposite direction: not all linear transformations are similarity transformations either.

To see this, take the linear transformation that scales the y axis by 2 but leaves the x axis as-is. This obeys the two criteria above, but changes angles and the ratios of lengths of non-parallel line segments, so it's not a similarity transformation.


So:

  • Some similarity transformations are linear (those involving only rotation, uniform scale, and mirroring, but no translation)

  • Some linear transformations are similar (those involving only rotation, uniform scale, and mirroring, but no non-uniform scale or shear)

But in general, similarity transformations are not the same as linear transformations.

$\endgroup$
1
  • $\begingroup$ Thank you! it was very helpful $\endgroup$ Oct 24, 2021 at 10:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.