projektive

Projektive refers to concepts in projective geometry, a branch of mathematics that studies properties invariant under projective transformations. In projective space, distance and angles are not fundamental; instead, incidence relations and alignment are primary. A key construction is the projective n-space P^n(F) over a field F, defined as the set of one-dimensional subspaces of F^(n+1). A point is represented by homogeneous coordinates [x0: x1: ... : xn], with not all xi zero and [x0: ... : xn] = [λx0: ... : λxn] for any nonzero λ in F.

Projective transformations are automorphisms of P^n(F) induced by invertible linear maps on F^(n+1), i.e., matrices in GL(n+1, F) up to a nonzero scalar. These transformations map lines to lines and preserve incidence relations. In the projective line, the cross ratio of four collinear points is an invariant under projective transformations, a fundamental tool in the study of projective geometry. Projective duality in a projective plane links points and lines, exchanging their roles while preserving incidence structures.

Historically, projective geometry originated with observations by Girard Desargues in the 17th century and matured through the development of homogeneous coordinates by Plücker and later formalizations by Cayley and Klein. The Erlangen Program of Felix Klein helped situate projective geometry within the broader context of transformation groups.

Applications of projektive concepts appear in computer graphics and vision (perspective projection and camera models), art and design (perspective drawing), and algebraic geometry (study of projective varieties). The framework provides a robust language for describing geometric relations that are independent of measurement and scale.