Projektio

Projektio is a term used in several languages to denote projection, referring to the act or result of projecting points, lines, or shapes from one space onto another. It commonly appears in mathematics, cartography, and computer graphics, among other fields, and carries a general sense of mapping or transforming data or geometry from a higher-dimensional or curved domain to a simpler, often planar or lower-dimensional one.

In mathematics, a projection is a linear transformation P of a vector space such that P squared equals P (P^2 = P). The image of P is a subspace, and applying P twice yields the same result as applying it once. Orthogonal projection, where P is self-adjoint with respect to an inner product, is a standard example: the projection of a vector onto a subspace U yields the closest point in U to the original vector. An oblique projection projects along a direction not necessarily orthogonal to U.

In cartography, a map projection transforms the Earth's curved surface onto a plane, inherently introducing distortions in properties such as area, shape, distance, or direction. Different projections prioritize different aspects; examples include Mercator, Lambert conformal conic, and equal-area projections, each with specific trade-offs for navigation, geography, or visualization.

In computer graphics, projection refers to converting 3D coordinates into 2D display coordinates, typically via perspective or orthographic projection. This process is implemented through projection matrices within a rendering pipeline, mapping world coordinates to clip space before rasterization.

In data analysis, projection describes dimensionality reduction, where data are projected onto a lower-dimensional subspace, as in principal component analysis, preserving as much variance as possible while simplifying interpretation.