Transformation (function)

mapping that preserves the operations of addition and scalar multiplication

in Euclidean geometry, a function that moves every point a constant distance in a specified direction

thumb|upright=1|Homothety: Example with . corresponds to (no point is moved); an ; a thumb|upright=1|Example with . corresponds to a point reflection at point thumb|upright=1.2|Homothety of a pyramid In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point called its center and a nonzero number called its ratio, which sends point to a point by the rule, \overrightarrow{SX'}=k\overrightarrow{SX} for a fixed number . Using position vectors: \mathbf x'=\mathbf s + k(\mathbf x -\mathbf s).

automorphism of an affine space

function from a set having some geometric structure to itself or another such set

matrix representing a Euclidean rotation

central object in linear algebra; mapping vectors to vectors

function mapping a set to itself

linear transformation that describes a reflection about a plane or hyperplane containing the origin

In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation.

symmetry operation combining reflection across and translation along an axis

concept in linear algebra

geometric transformation

isometry of a metric space

transformation method within a three-dimensional space

limiting form of small transformation