Affine transformations and their properties

An affine transformation is a function that moves points in the plane (or space) in a way that preserves straight lines and parallelism. Common examples include translations (sliding), rotations, reflections, scaling (stretching or shrinking), and shearing. In coordinate geometry, an affine transformation can be written as T(oldsymbol{x}) = Aoldsymbol{x} + oldsymbol{b}, where $A$ is a matrix and oldsymbol{b} is a vector.

Important properties

• Affine transformations map lines to lines.

• Parallel lines remain parallel after an affine transformation.

• Ratios of lengths along a line are preserved (but not necessarily lengths or angles).

• The composition of two affine transformations is also an affine transformation.