projections are also at the basis of the theory of perspective. The need for unifying the two kinds of projections and of defining the image by a central projection of any point different of the center of projection are at the origin of projective geometry. However, a projective transformation is a bijection of a projective space, a property not shared with the projections of this article. Definition Generally, a mapping where the domain and codomain are the same set (or mathematical structure) is a projection if the mapping is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a right inverse. Both notions are strongly related, as follows. Let p be an idempotent mapping from a set A into itself (thus p ∘ p = p) and B = p(A) be the image of p. If we denote by π the map p viewed as a map from A onto B and by i the injection of B into A (so that p = i ∘ π), then we have π ∘ i = IdB (so that π has a right inverse). Conversely, if π has a right inverse, then π ∘ i = IdB implies that i ∘ π is idempotent. Applications The original notion of projection has been extended or generalized to various mathematical situations, frequently, but not always, related to geometry, for example: In set theory: An operation typified by the jth projection map, written projj, that takes an element of the Cartesian product to the value This map is always surjective and, when each space has a topology, this map is also continuous and open. A mapping that takes an element to its equivalence class under a given equivalence relation is known as the canonical projection. The evaluation map sends a function f to the value f(x) for a fixed x. The space of functions YX can be identified with the Cartesian product , and the evaluation map is a projection map from the Cartesian product. For relational databases and query languages, the projection is a unary operation written as where is a set of attribute names. The result of such projection is defined as the set that is obtained when all tuples in R are restricted to the set . R is a database-relation. In spherical geometry, projection of a sphere