transformations

Transformations are rules that assign to each element of a set another element of a (possibly different) set. In mathematics, a transformation is typically described as a function that preserves some structure or relation of interest between objects.

Geometric transformations map figures in a plane or space to new positions or shapes. Rigid motions (isometries) such as translations, rotations, reflections, and glide reflections preserve distances and angles. Similarity transformations preserve shapes up to a common scale, combining isometries with uniform scaling. Affine transformations preserve straight lines and parallelism and include translation, rotation, scaling, and shear; they map lines to lines and preserve ratios of parallel segments but not angles or lengths in general. Projective transformations preserve lines and cross-ratios and are used to model perspective.

In linear algebra, a linear transformation between vector spaces assigns to every vector a linear combination of input vectors and is completely determined by its matrix relative to chosen bases. Linear transformations preserve vector addition and scalar multiplication. They have kernel (vectors mapped to zero) and image (set of attainable outputs). Important concepts include rank, nullity, and invertibility; invertible linear transformations correspond to nonsingular matrices and establish isomorphisms between spaces.

Transformations are central in computer graphics, where transformation matrices are combined to move, rotate, scale, and project points in three-dimensional scenes. They also arise in data analysis, physics, chemistry, and many areas of geometry and algebra, serving to model change of position, orientation, scale, or state.