A representation of a higher-dimensional object on a lower-dimensional surface.Projection, in the context of geometry and physics, is the mathematical operation of mapping a higher-dimensional object onto a lower-dimensional surface. This concept is fundamental in many areas, from cartography (mapping the 3D Earth onto a 2D map) to theoretical physics, where it is used to visualize or analyze higher-dimensional spaces, such as in string theory or general relativity. A projection can be orthogonal, where the mapping is perpendicular to the target surface, or perspective, where lines converge at a point. The result is a lower-dimensional representation that may preserve certain properties, like angles or areas, but generally loses some information about the original object.
The concept of projection has roots in ancient Greek geometry, with stereographic projection attributed to Hipparchus and used for star charts. The formal mathematical treatment was developed in the 19th century with projective geometry, pioneered by Jean-Victor Poncelet and others.
Mathematically, a projection is a linear transformation that is idempotent — applying it twice gives the same result as applying it once. In practical terms, it involves taking each point of the higher-dimensional object and finding its corresponding point on the lower-dimensional surface along a specified direction or from a specified viewpoint. For example, a 3D cube can be projected onto a 2D plane by drawing lines from each vertex perpendicular to the plane, resulting in a square or a hexagon depending on the orientation. In physics, projections are used to decompose vectors into components, to reduce the dimensionality of a system, or to extract observable quantities from higher-dimensional theories.
Projections are used in computer graphics to render 3D scenes on 2D screens, in medical imaging to reconstruct 3D structures from 2D scans, and in physics to analyze systems with symmetries by projecting onto invariant subspaces.
In cartography, no flat map can preserve all properties of the Earth's surface — a fact proven by the impossibility of a perfect projection, leading to the famous trade-offs in map projections like Mercator (preserves angles) and Peters (preserves areas).
