Projeccions

Projeccions refer to the process of mapping points from a higher‑dimensional space to a lower‑dimensional space or onto a chosen subspace. The term is used across disciplines such as geometry, linear algebra, cartography and data analysis. It derives from the Latin projectionem, meaning throwing forward or projecting outward.

In mathematics, a projection is a linear transformation P that maps a space to itself or to a subspace and satisfies P squared equals P (P^2 = P). If P preserves angles and orthogonality, it is an orthogonal projection; if the range and kernel are not orthogonal, it is an oblique projection. Projections are described by projection operators or projection matrices and are used to isolate components along specified directions or subspaces.

In cartography, map projections convert the curved surface of the Earth to a flat plane. All such projections introduce some distortion, trading accuracy in area, shape, distance, or direction for a usable representation. Common classes include conformal projections that preserve local angles, equal‑area projections that preserve surface area, and true‑direction projections that preserve certain directions. Examples include the Mercator projection (conformal), the Lambert azimuthal equal‑area projection (equal‑area), and the Peters projection (area‑focused).

In data analysis, projections reduce dimensionality or aid visualization by projecting high‑dimensional data into lower dimensions. Linear projections include principal component analysis (PCA), which projects data onto directions of maximal variance. Nonlinear projections such as t‑SNE or UMAP aim to preserve local structure while reducing dimensionality.

Applications of projeccions span engineering, computer graphics, geography, and statistics. Limitations arise from inherent distortions that depend on the chosen projection, requiring trade‑offs between competing properties.