Classification of all matrix transformations on $\mathbb{R}^2$ plane

Question

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).

Answer 1

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

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")