Projection

A term related to the operation of projecting, which can be defined as follows (see Fig.): One chooses an arbitrary point of the space as the centre of projection and a plane not passing through as the plane of projection. To project a point (a pre-image) of the space onto the plane through the centre , one draws the straight line to its intersection with the plane at a point . The point (the image) is called the projection of . The projection of a figure is defined to be the collection of projections of all its points.

Figure: p075130a

The projection just described is called central (or conical). A projection with centre of projection at infinity is called parallel (or cylindrical). If, moreover, the plane of projection is perpendicular to the direction of projection, then the projection is called orthogonal.

Parallel projections are widely used in descriptive geometry for obtaining various types of images (see, for example, Axonometry; Perspective). There are special forms of projections onto the plane, sphere and other surfaces (see, for example, Cartographic projection; Stereographic projection).

Comments

In geometry and linear algebra one also encounters projections parallel to a subspace. For instance, if is a vector space, a subspace and a complementary subspace (i.e. and ), then the projection from onto parallel to is the linear mapping that sends , , , to . The operator satisfies , and each such operator comes from a decomposition with , .

The orthogonal projection of a Hilbert space to a closed subspace assigns to the unique element of such that and are orthogonal. It is the parallel projection onto along the orthogonal complement . The element is the element of best approximation to in . In this case the corresponding operator is also self-adjoint, and, conversely, self-adjoint operators such that are orthogonal projections. Cf. also Projector.

References

[a1]	M. Berger, "Geometry" , I , Springer (1987) pp. Sect. 2.4.9.6
[a2]	M.S. Birman, M.Z. Solomyak, "Spectral theory of selfadjoint operators in Hilbert space" , Reidel (1987) (Translated from Russian)
[a3]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Wiley, reprint (1988)