Projection
Introduction
In mathematics, a projection is a type of transformation that maps elements from a set or space onto a subset or lower-dimensional space, often characterized by idempotence in algebraic contexts or by simulating visual or geometric correspondence in spatial contexts.[1] This concept appears across branches such as linear algebra, geometry, and cartography, where it enables the representation of higher-dimensional structures in reduced forms while preserving specific properties like orthogonality, perspective, or conformality.[2][3][4]
In linear algebra, a projection is defined as a linear transformation P:V→VP: V \to VP:V→V on a vector space VVV that satisfies P2=PP^2 = PP2=P, meaning applying the transformation twice yields the same result as once, with the image of PPP being a subspace W⊆VW \subseteq VW⊆V.[1] An orthogonal projection onto WWW is the unique such map where the kernel of PPP is the orthogonal complement W⊥W^\perpW⊥, ensuring that for any vector x∈Vx \in Vx∈V, PxPxPx is the point in WWW closest to xxx in the Euclidean norm, and x−Pxx - Pxx−Px is orthogonal to every vector in WWW.[2] This decomposition x=Px+(x−Px)x = Px + (x - Px)x=Px+(x−Px) with Px∈WPx \in WPx∈W and x−Px∈W⊥x - Px \in W^\perpx−Px∈W⊥ forms the basis for applications in least-squares problems, data fitting, and dimensionality reduction.[5] For a subspace W=Col(A)W = \operatorname{Col}(A)W=Col(A) spanned by the columns of a matrix AAA with full column rank, the projection matrix is given by P=A(ATA)−1ATP = A(A^TA)^{-1}A^TP=A(ATA)−1AT, which is symmetric and idempotent.[2]
In geometry, projections describe the mapping of points in three-dimensional space onto a two-dimensional plane or line, often via families of parallel or converging lines.[6] A key type is the perspective projection, which models the convergence of rays from an object through a focal point (center of projection) onto an image plane, as in a pinhole camera, resulting in formulas such as x=fX/Zx = fX / Zx=fX/Z and y=fY/Zy = fY / Zy=fY/Z, where (X,Y,Z)(X, Y, Z)(X,Y,Z) are object coordinates, (x,y)(x, y)(x,y) are image coordinates, and fff is the focal length.[3] This projection preserves lines (mapping them to lines) but distorts distances and angles, with parallel lines converging to vanishing points, enabling realistic depictions in art and computer graphics.[3] In contrast, parallel projections, like orthographic ones, use parallel rays and preserve parallelism but not depth foreshortening.[6]
In cartography and applied mathematics, map projections are systematic transformations of the Earth's spherical surface (using latitude and longitude) onto a flat or developable surface, such as a plane, cylinder, or cone, to create usable maps.[4] These projections inevitably introduce distortions in area, shape, distance, or direction, leading to specialized types: conformal projections (e.g., Mercator) preserve angles for navigation; equal-area projections maintain proportions of regions; and compromise projections balance multiple properties.[4] The mathematical formulation involves projecting the graticule of parallels and meridians onto the target surface, with the choice depending on the map's purpose, such as polar azimuthal for high-latitude views.[4]