What is "projection"

Projection \Pro*jec"tion\, n. [L. projectio: cf. F. projection.]

1. The act of throwing or shooting forward.

2. A jutting out; also, a part jutting out, as of a building; an extension beyond something else.

3. The act of scheming or planning; also, that which is planned; contrivance; design; plan.
   --Davenant.

4. (Persp.) The representation of something; delineation; plan; especially, the representation of any object on a perspective plane, or such a delineation as would result were the chief points of the object thrown forward upon the plane, each in the direction of a line drawn through it from a given point of sight, or central point; as, the projection of a sphere. The several kinds of projection differ according to the assumed point of sight and plane of projection in each.

5. (Geog.) Any method of representing the surface of the earth upon a plane.

   Conical projection, a mode of representing the sphere, the spherical surface being projected upon the surface of a cone tangent to the sphere, the point of sight being at the center of the sphere.

   Cylindric projection, a mode of representing the sphere, the spherical surface being projected upon the surface of a cylinder touching the sphere, the point of sight being at the center of the sphere.

   Globular, Gnomonic, Orthographic, projection,etc. See under Globular, Gnomonic, etc.

   Mercator's projection, a mode of representing the sphere in which the meridians are drawn parallel to each other, and the parallels of latitude are straight lines whose distance from each other increases with their distance from the equator, so that at all places the degrees of latitude and longitude have to each other the same ratio as on the sphere itself.

   Oblique projection, a projection made by parallel lines drawn from every point of a figure and meeting the plane of projection obliquely.

   Polar projection, a projection of the sphere in which the point of sight is at the center, and the plane of projection passes through one of the polar circles.

   Powder of projection (Alchemy.), a certain powder cast into a crucible or other vessel containing prepared metal or other matter which is to be thereby transmuted into gold.

   Projection of a point on a plane (Descriptive Geom.), the foot of a perpendicular to the plane drawn through the point.

   Projection of a straight line of a plane, the straight line of the plane connecting the feet of the perpendiculars let fall from the extremities of the given line.

   Syn: See Protuberance.

late 15c., in alchemy, "transmutation by casting a powder on molten metal; 1550s in the cartographical sense "drawing of a map or chart according to scale," from Middle French projection, from Latin proiectionem (nominative proiectio), from past participle stem of proicere (see project (n.)). From 1590s as "action of projecting."

n. 1 Something which projects, protrudes, juts out, sticks out, or stands out. 2 The action of project or throw or propel something. 3 The display of an image by devices such as movie projector, video projector, overhead projector or slide projector. 4 A forecast or prognosis obtained by extrapolation 5 (context psychology English) A belief or assumption that others have similar thoughts and experiences as oneself 6 (context photography English) The image that a translucent object casts onto another object. 7 (context cartography English) Any of several systems of intersecting lines that allow the curved surface of the earth to be represented on a flat surface. The set of mathematics used to calculate coordinate positions. 8 (context geometry English) An image of an object on a surface of fewer dimensions. 9 (context linear algebra English) An idempotent linear transformation which maps vectors from a vector space onto a subspace. 10 (context mathematics English) A transformation which extracts a fragment of a mathematical object. 11 (context category theory English) A morphism from a categorical product to one of its (two) components.

1. n. a prediction made by extrapolating from past observations

2. the projection of an image from a film onto a screen

3. a planned undertaking [syn: project]

4. any structure that branches out from a central support

5. any solid convex shape that juts out from something

6. (psychiatry) a defense mechanism by which your own traits and emotions are attributed to someone else

7. the acoustic phenomenon that gives sound a penetrating quality; "our ukuleles have been designed to have superior sound and projection"; "a prime ingredient of public speaking is projection of the voice" [syn: acoustic projection, sound projection]

8. the representation of a figure or solid on a plane as it would look from a particular direction

9. the act of projecting out from something [syn: protrusion, jut, jutting]

10. the act of expelling or projecting or ejecting [syn: expulsion, ejection, forcing out]

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that . That is, whenever P is applied twice to any value, it gives the same result as if it were applied once ( idempotent). It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.

Projection was the ultimate goal of Western alchemy. Once the Philosopher's stone or powder of projection had been created, the process of projection would be used to transmute a lesser substance into a higher form, often lead into gold.

Typically, the process is described as casting a small portion of the Stone into a molten base metal.

Projection, projector, or projective may refer to:

• The display of an image by devices such a
  • Movie projector
  • Video projector
  • Overhead projector
  • Slide projector
  • Camera obscura
  • Projection screen
• Map projection, reducing the surface of a three-dimensional planet to a flat map (geography)
• Graphical projection, the production of a two-dimensional image of a three-dimensional object
  • Parallel projection
    • Orthographic projection, including:
      • Plan, or floor plan view
      • Elevation, usually a side view of an exterior
      • Section, a view of the interior at a particular cutting plane
    • Axonometric projection, including:
      • Isometric projection
      • Dimetric projection
      • Trimetric projection
    • Oblique projection, including:
      • Cabinet projection
      • Cavalier projection
  • Perspective projection

In relational algebra, a projection is a unary operation written as Π(R) where a, ..., a is a set of attribute names. The result of such projection is defined as the set obtained when the components of the tuple R are restricted to the set {a, ..., a} – it discards (or excludes) the other attributes.

In practical terms, it can be roughly thought of as picking a sub-set of all available columns. For example, if the attributes are (name, age), then projection of the relation {(Alice, 5), (Bob, 8)} onto attribute list (age) yields {5,8} – we have discarded the names, and only know what ages are present.

In addition, projection can be used to modify an attribute's value: if relation R has attributes a, b, and c, and b is a number, then Π(R) will return a relation nearly the same as R, but with all values for 'b' shrunk by half.

In set theory, a projection is one of two closely related types of functions or operations, namely:

• A set-theoretic operation typified by the j projection map, written proj, that takes an element x⃗ = (x,  …,  x,  …,  x) of the Cartesian product (X × ⋯ × X × ⋯ × X) to the value proj(x⃗) = x.
• A function that sends an element x to its equivalence class under a specified equivalence relation E, or, equivalently, a surjection from a set to another set. The function from elements to equivalence classes is a surjection, and every surjection corresponds to an equivalence relation under which two elements are equivalent when they have the same image. The result of the mapping is written as [x] when E is understood, or written as [x] when it is necessary to make E explicit.

In mathematics, 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 (idempotence). 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 do 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: 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 having 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 differing versions of the concept developed, but today, in a sufficiently abstract setting, we can unify these variations.

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.

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.