In
mathematics, a projection is an
idempotentmapping of a
set (or other
mathematical structure) into a
subset (or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. The
restriction to a subspace of a projection is also called a projection, even if the idempotence property is lost.
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). The shadow of a three-dimensional sphere is a closed disk. Originally, the notion of projection was introduced in
Euclidean geometry to denote the projection of the three-dimensional
Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are:
The projection from a point onto a plane or central projection: If C is a point, called the center of projection, then the projection of a point P different from C onto a plane that does not contain C is the intersection of the
lineCP with the plane. The points P such that the line CP is
parallel to the plane does not have any image by the projection, but one often says that they project to a point at infinity of the plane (see
Projective geometry for a formalization of this terminology). The projection of the point C itself is not defined.
The projection parallel to a direction D, onto a plane or parallel projection: The image of a point P is the intersection of the plane with the line parallel to D passing through P. See
Affine space § Projection for an accurate definition, generalized to any dimension.[citation needed]
The concept of projection in mathematics is a very old one, and most likely has its roots in the phenomenon of the shadows cast by real-world objects on the ground. 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.[citation needed]
In
cartography, a
map projection is a map of a part of the surface of the Earth onto a plane, which, in some cases, but not always, is the restriction of a projection in the above meaning. The
3D projections are also at the basis of the theory of
perspective.[citation needed]
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.[citation needed]
Definition
The commutativity of this diagram is the universality of the projection π, for any map f and set X.
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 i, then π ∘ i = IdB implies that i ∘ π ∘ i ∘ π = i ∘ IdB ∘ π = i ∘ π; that is, p = i ∘ π is idempotent.[citation needed]
Applications
The original notion of projection has been extended or generalized to various mathematical situations, frequently, but not always, related to geometry, for example:
An operation typified by the j-thprojection map, written projj, that takes an element x = (x1, ..., xj, ..., xn) of the
Cartesian productX1 × ⋯ × Xj × ⋯ × Xn to the value projj(x) = xj.[1] This map is always
surjective and, when each space Xk has a
topology, this map is also
continuous and
open.[2]
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.[citation needed]
In
spherical geometry, projection of a sphere upon a plane was used by
Ptolemy (~150) in his
Planisphaerium.[7] The method is called
stereographic projection and uses a plane
tangent to a sphere and a pole C diametrically opposite the point of tangency. Any point P on the sphere besides C determines a line CP intersecting the plane at the projected point for P.[8] The correspondence makes the sphere a
one-point compactification for the plane when a
point at infinity is included to correspond to C, which otherwise has no projection on the plane. A common instance is the
complex plane where the compactification corresponds to the
Riemann sphere. Alternatively, a
hemisphere is frequently projected onto a plane using the
gnomonic projection.[citation needed]
In
linear algebra, a
linear transformation that remains unchanged if applied twice: p(u) = p(p(u)). In other words, an
idempotent operator. For example, the mapping that takes a point (x, y, z) in three dimensions to the point (x, y, 0) is a projection. This type of projection naturally generalizes to any number of dimensions n for the domain and k ≤ n for the codomain of the mapping. See
Orthogonal projection,
Projection (linear algebra). In the case of orthogonal projections, the space admits a decomposition as a product, and the projection operator is a projection in that sense as well.[9][10][verification needed]
In
topology, a
retraction is a
continuous mapr: X → X which restricts to the
identity map on its image.[11] This satisfies a similar idempotency condition r2 = r and can be considered a generalization of the projection map. The image of a retraction is called a retract of the original space. A retraction which is
homotopic to the identity is known as a
deformation retraction. This term is also used in
category theory to refer to any split epimorphism.[citation needed]