Projection

projection, in psychology: see defense mechanism.

---

projection

See rear-projection TV, front-projection TV and LCD panel.

---

projection

1. the representation of a line, figure, or solid on a given plane as it would be seen from a particular direction or in accordance with an accepted set of rules

2. 

a. the process of showing film on a screen

b. the image or images shown

3. Psychol

a. the belief, esp in children, that others share one's subjective mental life

b. the process of projecting one's own hidden desires and impulses

4. the mixing by alchemists of powdered philosopher's stone with molten base metals in order to transmute them into gold

---

projection [prə′jek·shən]

(mapping)

A system for presenting on a plane surface the spherical surface of the earth or the celestial sphere; some of these systems are conic, cylindrical, gnomonic, Mercator, orthographic, and stereographic. Also known as map projection.

(mathematics)

The continuous map for a fiber bundle.

Geometrically, the image of a geometric object or vector superimposed on some other.

A linear mapPfrom a linear space to itself such thatP°Pis equal toP.

(psychology)

Ascribing one's motives to someone else to disguise a source of conflict in oneself.

---

projection

1. In masonry, stones which are set forward of the general wall surface to provide a rugged or rustic appearance.

2. Any component, member, or part which juts out from a building.

---

(theory)

projection - In domain theory, a function, f, which is (a) idempotent, i.e. f(f(x))=f(x) and (b) whose result is no more defined than its argument. E.g. F(x)=bottom or F(x)=x. In reduction systems, a function which returns some component of its argument. E.g. head, tail, \ (x,y) . x. In a graph reduction system the function can just return a pointer to part of its argument and does not need to build any new graph.

---

Projection 

a term in geometry used to refer to the following operation. Suppose an arbitrary point S in space (see Figure 1) is selected as the center of projection and a plane Π′ not passing through S is selected as the plane of projection, or image plane. In order to project the point A, the so-called preimage, on Π′ through the center of projection S, the line SA is extended to its intersection with Π′ at the point A′. The image point A′ is called the projection of A. The projection of a figure F is the set of the projections of all the figure’s points. A line not passing through the center of projection is projected into a line.

Figure 1

In the described type of projection, which is called a central projection, an important role is played by the choice of the center of projection S. A number of difficulties arise when points of a given plane Π are projected on the plane Π′ as in Figure 2. Π contains points that have no image in Π′. Such is the case for the point B when the projection line SB is parallel to Π′. To eliminate this difficulty, which is due to the properties of Euclidean space, elements at infinity, also called ideal elements, are

Figure 2

added to the space. In other words, the parallel lines BS and PA′ are assumed to intersect at a point at infinity B′. This point may then be considered as the image of the point B in Π′. Similarly, the point at infinity C is the preimage of the point C (see Figure 2). Thus, one-to-one correspondence defined by means of central projection can be established between the points of Π and the points of IT by introducing elements at infinity. Such a correspondence is called a perspective collineation.

The type of projection in which the center of projection is the point at infinity S_∞ (Figure 3) is of great practical importance. In this case, all the projection lines are parallel, and the projection is called a parallel projection. The one-to-one correspondence between the points of Π and the points of Π′ established by a parallel projection is called a perspective affinity.

The special type of parallel projection in which the plane Π is perpendicular to the direction of projection is widely used in drawing. Such a projection is called orthogonal.

Figure 3

Central and parallel—in particular, orthogonal—projections are widely used in descriptive geometry, and such different types of images as perspective images and axonometric images are obtained. Special types of projections on a plane, a sphere, or other surfaces are used in, for example, geography, astronomy, crystallography, and topography. Thus, cartographic projections include such types as gnomonic and stereographic projections.

Orthogonal projection of directed line segments is discussed in VECTOR CALCULUS.

N. F. CHETVERUKHIN

---

Projection 

in psychology, the perception of one’s own mental processes as those of an external object, resulting from the unconscious transfer of internal impulses and feelings to that object. Projection plays an important role in the formation of the psyche in early childhood, when a child cannot clearly differentiate between himself and the external world. It is also the basis of archaic and anthropomorphic ideas about the world that characterize the early stages of development of human consciousness.

The onset of a number of mental diseases (paranoia, phobia, mania) is associated with pathological forms of projection. In these cases, perception of the external world is severely distorted, while the illusion of control over one’s own behavior is preserved. The mechanism of projection is used diagnostically in projective tests, such as the Rorschach test, to detect hidden motivations and stimuli.