Projection (mathematics)

Geometric projections

Geometric projections include orthographic (parallel) projections and perspective projections. Orthographic projection maps every point along lines parallel to a fixed direction onto a plane; it preserves lengths along directions parallel to the plane but does not model perspective foreshortening. Perspective projection maps points along lines that converge to a finite viewpoint and produces the familiar depth cues in drawings and camera images. Such constructions are basic in descriptive geometry and computer graphics and relate the solid object to its planar image.

Linear projections and operators

In linear algebra a projection is a linear operator P on a vector space V satisfying P^2 = P (idempotence). The image Im(P) is the subspace of points left unchanged by P, and the kernel Ker(P) is the set of vectors sent to zero. If V decomposes as a direct sum V = U ⊕ W one obtains the projection onto U along W. An important special case is the orthogonal projection onto a subspace U of an inner-product space: it is the unique linear map that sends every vector to the nearest point of U and is represented, in orthonormal coordinates, by a symmetric idempotent matrix. When the columns of a matrix U are orthonormal, the projection matrix can be written as U U^T; if A has full column rank a common formula is P = A (A^T A)^{-1} A^T. Eigenvalues of a linear projection are either 0 or 1.

Coordinate, topological and categorical projections

More generally, a projection can be any map that extracts part of a structured object. The coordinate projection pi_1: X × Y → X sends a pair to its first coordinate and is an elementary example of a product-space map; see the standard account of functions for background. In topology such coordinate projections are continuous surjections when the product carries the product topology, and they are used in constructions of fiber bundles and product spaces. In category theory projections are canonical morphisms from a product object to its factors. In logic and set theory projection refers to selecting components from tuples or to marginalising relations.

• Key algebraic properties: idempotence P^2 = P; decomposition V = Im(P) ⊕ Ker(P); spectrum contained in {0,1} for linear projections.
• Geometric distinctions: orthogonal versus oblique projections; parallel (orthographic) versus perspective projections, each with different invariants and distortions.

Computation and applications

Projections appear throughout applied mathematics and computing. In computer graphics, perspective and orthographic projections convert 3D scenes to 2D images. In statistics and machine learning, principal component analysis uses orthogonal projection onto a lower-dimensional subspace to capture variance. In numerical linear algebra projection matrices and subspace projections underlie least-squares solvers and Krylov methods. In database theory the relational projection operation extracts specified columns from a table. For further reading consult introductory accounts in higher mathematics texts and concrete examples of coordinate projections and geometric images.

Because the term is used in many contexts, it helps to distinguish the intended meaning by context: geometric projection when discussing images and views, linear projection when discussing operators and matrices, and coordinate or categorical projection when discussing products and factor maps. Elementary proofs of idempotence consequences and algorithms to compute projection matrices appear in standard linear algebra references; computational libraries provide routines for constructing and applying projections in practice.