, a projection is a mapping of a set
(or other mathematical structure
) into a subset (or sub-structure), which is equal to its square for mapping composition
(or, in other words, which is idempotent
). 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 (paper sheet). The projection of a point is its shadow on the paper sheet. The shadow of a point on the paper sheet is this 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 Euclidean space
of three dimensions 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 line ''CP'' 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 with the plane of the line parallel to ''D'' passing through ''P''. See for an accurate definition, generalized to any dimension.
The concept of projection in mathematics
is a very old one, 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.
, 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 projection
s 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.
In an abstract setting we can generally say that a ''projection'' is a mapping of a set
(or of a mathematical structure
) which 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 map
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'' = Id''B''
(so that ''π'' has a right inverse). Conversely, if ''π'' has a right inverse, then ''π'' ∘ ''i'' = Id''B''
implies that ''i'' ∘ ''π'' is idempotent.
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 ''j'' th projection map
, written proj''j''
, that takes an element of the Cartesian product
to the value This map is always surjective
** A mapping that takes an element to its equivalence class
under a given equivalence relation
is known as the .
** The evaluation map sends a function ''f'' to the value ''f''(''x'') for a fixed ''x''. The space of functions ''Y''''X''
can be identified with the Cartesian product
, and the evaluation map is a projection map from the Cartesian product.
* For relational database
s and query language
s, the projection
is a unary operation
is a set of attribute names. The result of such projection is defined as the set
that is obtained when all tuple
s in ''R'' are restricted to the set
. ''R'' is a database-relation
* In spherical geometry
, projection of a sphere upon a plane was used by Ptolemy
(~150) in his Planisphaerium
. 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''. 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
* 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) in the plane is a projection. This type of projection naturally generalizes to any number of dimensions ''n'' for the source and ''k'' ≤ ''n'' for the target 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.
* In differential topology
, any fiber bundle
includes a projection map as part of its definition. Locally at least this map looks like a projection map in the sense of the product topology and is therefore open and surjective.
* In topology
, a retraction
is a continuous map ''r'': ''X'' → ''X'' which restricts to the identity map on its image. This satisfies a similar idempotency condition ''r''2
= ''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.
* The scalar projection
(or resolute) of one vector
* In category theory
, the above notion of Cartesian product of sets can be generalized to arbitrary categories
. The product
of some objects has a canonical projection morphism
to each factor. This projection will take many forms in different categories. The projection from the Cartesian product
, the product topology
of topological space
s (which is always surjective and open
), or from the direct product
, etc. Although these morphisms are often epimorphism
s and even surjective, they do not have to be.
* Thomas Craig
(1882A Treatise on Projections
from University of Michigan
Historical Math Collection.