Projection

Projection is a process that maps points from one space to another, often reducing dimensionality or aligning data with a subspace. In mathematics, geography, computer graphics, and psychology, projection describes various ways of representing objects by expressing them through a lower-dimensional or alternative reference frame.

In linear algebra, a projection is a linear operator P satisfying P^2 = P. The image of P is a subspace, and any vector x decomposes into x = Px + (I−P)x, with Px lying in the target subspace. An orthogonal projection uses the inner product to minimize the length of (I−P)x, giving Px as the component of x in the subspace. If the subspace has an orthonormal basis, the projection operator can be written as P = UU^T when U contains the basis vectors as columns.

In computer graphics, projections convert 3D coordinates to 2D screen coordinates. Orthographic projection preserves size along depth, while perspective projection makes distant objects appear smaller. Projection matrices encode these transformations and are applied in the graphics pipeline before rendering.

In cartography, a map projection describes the method for representing the Earth's curved surface on a plane. Projections introduce distortions in area, shape, distance, or direction. Common examples include the Mercator projection (angle-preserving), the Peters projection (area-preserving), and the Robinson projection (a compromise).

In psychology, projection is a defense mechanism in which a person attributes their own unacceptable thoughts or feelings to others, effectively externalizing them.