Affine geometry (other)

Affine geometry studies properties of figures that remain unchanged under affine transformations (such as translation, scaling, rotation, and shearing). Beyond basic transformations and combinations, affine geometry also explores concepts like parallelism, ratios along parallel lines, and affine invariants (properties that do not change under affine transformations).

Important properties

• Parallel lines remain parallel after any affine transformation.

• The midpoint of a line segment may not remain the midpoint after a general affine transformation, unless the transformation is a similarity.

• Ratios of lengths along a line are preserved (e.g., if a point divides a segment in a ratio $k:1$, this ratio is preserved).

• Collinearity of points is preserved under affine transformations.