Proiettive

Proiettive is the Italian adjective used to describe concepts related to projection and projective geometry. In English-language mathematics, the corresponding term is projective. The word encompasses ideas and objects that are defined or studied through projection, as well as properties that are invariant under projective transformations.

In geometry, projective concepts arise from viewing figures up to projection from a point or line, which leads to the framework of projective space and projective planes. A projective space P^n over a field is constructed from (n+1)-tuples of coordinates by identifying nonzero scalar multiples, so that points are represented by homogeneous coordinates [x0: ... : xn]. This framework naturally adds points at infinity and emphasizes incidence relations rather than metric properties.

Projective transformations, or collineations, are the automorphisms of projective space that preserve straightness and incidence. They can be represented by invertible linear transformations on homogeneous coordinates, defined up to a common nonzero factor; equivalently, they form the projective general linear group PGL(n+1, F). These transformations map lines to lines and preserve cross ratios of four collinear points, a key invariant in projective geometry.

Related ideas include duality, where points and lines exchange roles, and various theorems that hold under any projective transformation. In Italian mathematical literature, many foundational notions in this area are described using the term proiettive, reflecting the same underlying projective concepts encountered in English texts.