Projection (mathematics)

An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency).

This rudimentary idea was refined and abstracted, first in a geometric context and later in other branches of mathematics.

Over time different versions of the concept developed, but today, in a sufficiently abstract setting, we can unify these variations.

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.

[citation needed] The original notion of projection has been extended or generalized to various mathematical situations, frequently, but not always, related to geometry, for example:

The commutativity of this diagram is the universality of the projection π , for any map f and set X .