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What is a complete exhaustive classification of all geometric transforms on the $\mathbb{R}^2$ plane obtained with:

  1. 2x2 matrices$$A = \pmatrix{a & b \\ c & d}$$ applied to a point $X= (x, y)$.

  2. 3x3 matrices $$A = \pmatrix{a & b& c \\ d & e & f \\ g & h & i}$$ applied to a point $(x, y)$ noted $X = (x, y, 1)$ using homogeneous coordinates.

?

Note:

  • I've already looked at Transformation matrix Wikipedia page, which is good, but it mainly gives examples (stretching, squeezing, etc.), and doesn't state it as a full classification of all possible transforms.

  • In most lecture notes / resources I find, it usually goes this way:

    here is a list of geometric transforms => here are their representation as matrix

    In this question I'm more looking for:

    here is a random 3x3 matrix => what geometric transform is it?

I'm looking for an exhaustive classification like "All 2x2 matrices can be either a rotation matrix with parameter $\theta$, a scaling matrix of parameter $\lambda_1, \lambda_2$, a blahblah matrix of parameter $\delta$, or a composition of any 2 of them" (nonsense, just an example).

Basj
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  • What do you mean "classification"? You mean "a finite list of random types of transformations which I can sort all these matrices into"? – rschwieb May 22 '18 at 16:10
  • @rschwieb A finite list of transformations (e.g. rotation $R_\theta$ for $\theta \in \mathbb{R}$, + ... + ...), such that any transformation can be expressed as either one of them or composition of them – Basj May 22 '18 at 16:11
  • OK: the question you're asking in the second yellow box is not fruitful, but your version of it in this last comment is reasonable. I think you should replace that yellow box with the comment you just wrote. – rschwieb May 22 '18 at 16:12
  • There are many possible elementary decompositions. You need to be more specific about what you want. In some respects, all linear operators on $\mathbb R^2$ are the same type, as they transform the unit circle into a (possibly degenerate) ellipse. –  May 22 '18 at 16:13
  • @Bungo I'm looking for a classification like "All 2x2 matrices can be either a rotation matrix with parameter $\theta$, a scaling matrix of parameter $\lambda_1, \lambda_2$, a blahblah matrix of parameter $\delta$, or a composition of any 2 of them". Something like that... – Basj May 22 '18 at 16:15
  • @Basj Are you only interested in transformations that are invertible? – rschwieb May 22 '18 at 16:16
  • @rschwieb Possibly all of them, but it's ok for only invertible one first. – Basj May 22 '18 at 16:17
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    How about: all $2\times 2$ matrices can be decomposed as the product of an orthogonal matrix (rotation and/or reflection), followed by a diagonal matrix (stretch or shrink the canonical axes), followed by another orthogonal matrix"? This is essentially the singular value decomposition –  May 22 '18 at 16:18
  • This might be it for 2x2 matrices, so are there no other corner cases for 2x2 matrices that are not covered by rotation + scaling on both axes + rotation? – Basj May 22 '18 at 16:22
  • Also @Bungo what would it be for 3x3 matrices applied on 2D plane using homogeneous coordinates? – Basj May 22 '18 at 16:23
  • @Basj Rotation and/or scaling, meaning that some of the axes may be inverted/flipped in the opposite direction. That does indeed cover all cases, i.e. all real matrices (even non-square matrices) can be decomposed in this way. Complex matrices can also be decomposed this way if we substitute unitary matrices in place of orthogonal matrices. –  May 22 '18 at 16:25
  • Thank you @Bungo. If you have a few minutes, could you post it as an answer for the case 1. mentioned in the question? (with these details about "rotation and reflection" vs "rotation or reflection" , etc.) – Basj May 22 '18 at 18:14

1 Answers1

2

Here is a partial answer, that helps to see the big picture of the classification of (nonsingular) transformations (source, page 12-16):

enter image description here

Note:

  • the "Projective transformation" here is equivalent to the terminology "Homography"

  • "Euclidean" here seems to mean an Euclidean plane isometry. It can be shown that there are four types of Euclidean plane isometries:

    • rotations
    • reflection (mirror with a reflection axis)
    • translations
    • reflection followed by a translation (known as "glide reflection")
Basj
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