projected

Projected is the past participle of the verb to project and can function as an adjective meaning planned, estimated, or shown as a projection. It is also used to describe something that has been cast forward or rendered onto a surface, such as an image on a screen.

In mathematics and computer graphics, projection refers to a mapping from a space to a subspace or image. A projection operator P typically satisfies P squared equals P, making it idempotent. Orthogonal projection maps a point to the nearest point in a subspace, while perspective projection simulates how a camera or eye perceives depth, transforming three-dimensional scenes into two-dimensional representations for display.

In statistics and data analysis, projection describes the act of projecting data onto a lower-dimensional subspace. Techniques like principal component analysis use projection matrices to summarize data with reduced dimensionality while preserving as much variance as possible. Projections are also encountered in regression and least squares, where observations are projected onto a model subspace to obtain fitted values.

In geography and cartography, a map projection transforms coordinates from the curved surface of the Earth to a flat plane. Projected coordinates depend on a chosen projection method (for example, Mercator or Lambert conformal conic), each with specific distortions of area, shape, distance, or direction. The term projected is used for coordinates or features after such a transformation.

In psychology, projection is a defense mechanism in which unwanted thoughts or impulses are attributed to others. In general use, projected often describes forecasts or estimates for future outcomes, such as projected growth, revenue, or enrollment.