Parallel projection (geometry and linear algebra)

Definition and algebraic description

Given a vector space V and a decomposition V = U ⊕ W where U is the target subspace and W is the direction (kernel) of projection, the linear map P with image U and kernel W is called a projection. Equivalently, P satisfies P(v)=u for the unique u in U with v-u in W, and P2 = P. In coordinates a parallel projection is represented by an idempotent matrix whose column space equals U and whose nullspace equals W. For finite dimensions this determines rank and trace: rank(P)=dim(U).

Types and geometric distinctions

Two common geometric types are orthographic (orthogonal) projections, where lines of projection are perpendicular to the target plane, and oblique projections, where the direction is slanted. Orthographic projections preserve perpendicularity to the plane and minimize distortion; oblique projections preserve parallelism but distort shapes and angles. Both are parallel projections because projection lines remain parallel rather than converging at a viewpoint (as in perspective projection).

Properties and examples

• Idempotence: applying the projection twice has the same effect as once (P2 = P).
• Image and kernel split the space: V = im(P) ⊕ ker(P) when a complementary subspace exists.
• Preserves all parallel relations along projection direction, but generally not lengths or angles.

Concrete examples include dropping perpendiculars to a plane (orthographic view used in engineering drawings), axonometric projections such as isometric and dimetric views (special oblique choices), and algebraic projections used in numerical linear algebra to eliminate components of a vector or to form least-squares decompositions.

History, uses, and notable facts

Parallel projection has long been used in technical and architectural drawing because it gives measurable scaled views without perspective convergence. In mathematics, the abstract idea of projection underpins direct-sum decompositions and spectral theory for operators on infinite-dimensional spaces encountered in functional analysis. Computational applications include model reduction, graphics (orthographic rendering), and splitting methods in numerical solvers. For general background on the algebraic notion of projections see linear algebra and the formal setting of a vector space.