Affine space in affine geometry

The most important transformation in affine space is the affine transformation, which is characterized by changing three points of a *** line into three points of a *** line. Given an affine coordinate system, the affine transformation has an explicit algebraic representation. The group of transformations consisting of all affine transformations is called the group of affine transformations. Important invariant properties and invariants under affine transformations are *** linearity, parallelism, and the ratio of lengths of parallel line segments.

If infinite points are introduced in the affine plane (or space) and they are made indistinguishable from the original points, they become projective planes (or projective spaces). By specifying a (or a) straight line l (or hyperplane π) in the projective plane (or projective space), then the transformations in the group of projective transformations that keep l (or π) immobile constitute a subgroup of transformations isomorphic to the group of affine transformations. In this sense, the group of affine transformations is a subgroup of the group of projective transformations, and affine geometry becomes a subgeometry of projective geometry.