Algorithms: Transformation without rotation

Question

Is there a name for 2D transformation having the following parameters:

• shift_x,
• shift_y,
• scale.

Transformation does not use any rotation... Thanks for your help.

Answer 1

Shift_x and Shift_y count as a translation. I don't know that there is a specific term for a transformation that involves both a translation and scaling at the same time... particularly since the order in which these transformations are carried out can affect the result (depending on how the scaling is done, this might not be true).

Answer 2

Both of the other answers are correct. I am going to add a terminological difference.

Scaling, shift, and rotation are the three transformations that are the most frequent cases of affine transformation of data. Reflection, shearing, and others are seen, but not as commonly mentioned.

These three may go by several names independently or in conjunction:

• Scaling: Scaling, re-scaling, normalization, dilation
• Shift: Shift, centering, re-centering
• Scaling + shift: Normalization
• Rotation: Rotation, projection
• Scaling + shift + rotation: steps seen in PCA, SVD, or called "whitening" or "sphering" in some contexts.

Unfortunately, these may be interpreted more or less loosely in different contexts. For instance, I generally interpret normalization to address centering and scaling (usually leading to "z-scores"), others may assume it is just scaling. I prefer to never use "sphering" or "whitening" as terms, because these are imprecise and not used in more than a few disciplines.

In statistics, shift or translation may occur when one "centers" data to have a mean of 0. Scaling occurs when one desires, say, unit variance (or a standard deviation of 1), for the sample. Rotation often occurs in order to project onto orthogonal dimensions. Because of the scaling and centering, this often utilizes orthonormal projections.

Update 1: The OP asked only about 2 dimensions, but one should note that these transformations are all allowed to be in many dimensions. There are no restrictions to 1, 2, or any number of other dimensions, nor any special terms for small #s of dimensions.