A 2d affine transformation is characterized by the following formula: $$ x' = a x + b y + c, \\ y' = d x + e y + f. $$
Hence it has 6 degrees of freedom. I can name the following 5 special types:
- translate parallel to the $x$-axis
- translate parallel to the $y$-axis
- rotate inside $xy$-plane about the origin $O$
- scale in the direction of $x$-axis
- scale in the direction of $y$-axis
The above types are basic, intuitive (has clear geometric meaning) and 'independent'. What is the other basic and independent type? I can think of 2 other basic types:
- shear parallel to the $x$-axis
- shear parallel to the $y$-axis
The problem is we need one more, not two more! Can we find ONE extra intuitive type that are independent of the above 5 types?
NOTE: Edited after reading EricTowers' answer.
A x-translate and a 90 degrees rotation will generate a y-translate. A x-scale and a 90 degrees rotation will generate a y-scale. A x-shear and a 90 degrees rotation will generate a y-shear.
It is common to think of the two translates, two scaling and 1 rotation in describing affines. Each of the above operation requires 1 continuous parameter. So if we combine them in 6 different orders (repetition of same type allowed); we will obtain all possible affines.
I don't want to include another two shears, partly because I want one only and don't want to be forced to single out the x-shear or y-shear. A total of 7 affines to describe affines when we should only need 6 ways to describe. I am curious if it is possible to do it with exactly 6 different types.
On the other hand, we can go minimum and just do the following 3 and obtain all affines (repetition of same type allowed): x-translation, x-scale and rotation