Geometric transformation

A geometric transformation is a bijection of a set of points to itself, or to another such set, that has some geometric significance, typically preserving a property such as distance, angle, or scale ratio. Geometric transformations form a nested hierarchy defined by what they preserve: displacements preserve distances and oriented angles; isometries preserve both angles and distances; similarities preserve angles and the ratios between distances; affine transformations preserve parallelism; and projective transformations preserve collinearity, with each broader class containing all the narrower ones. Other named classes include Mobius transformations, conformal transformations that preserve angles, equiareal transformations that preserve area or volume, and the more general homeomorphisms and diffeomorphisms. Transformations are further classified by whether they act on the plane or in space, and by which of these properties they preserve.