Projection

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Projection

The term projection has several different, but similar, meanings. A few examples should help clarify the term. First, suppose we are given an object O that is specified as a subset of Euclidean three-dimensional space (E³), and a plane P in E³. The map from O to P that send each point of O to the point in P closest to it, is called the projection map from O to P. For example, suppose that O is a knot - that is a loop of string with its ends glued together. If we spray paint directly at P but in front of the knot O, then the space that is not painted is the projection of O on P. The next example is called polar projection. It is a map f from the unit sphere minus the north pole to the plane z = -1 (this plane is one unit below the x-y plane) in Euclidean three-dimensional space. For any point x other than the north pole, draw the line that passes through x and the north pole. This line intersects the plane z = -1 in the point f(x). Polar projection is the reason why the sphere is sometimes referred to as the plane plus one point at infinity. This map is conformal, i.e. if two great circles on the sphere intersect at an angle of d degrees then their images under f are lines in the plane that intersect at an angle of d degrees.

In the next example, X and Y are sets. The Cartesian product of X and Y, denoted by XxY, is the set of all ordered pairs of the form (x, y) in which x is an element of X and y is an element of Y. There are two natural projection maps in this context: the map from XxY to X that maps the element (x, y) to x, and the map from XxY to Y that maps (x, y) to y. If X and Y are topological spaces, then the product topology on XxY is the smallest one for which the two projection maps are continuous. If X and Y are groups then XxY is a group under the operation (x, y)(m, n) = (xm, yn) and the two projection maps are group homomorphisms.

In the next example, V and W are Hilbert spaces and V is contained in W. It is a fact that any orthonormal basis for V can be completed to an orthonormal basis for W. This means that any element w of W has a unique decomposition w = v + t in which v is an element of V and t is an element of the orthogonal complement of V. In other words, if x is any element of V then x·t = 0. The map that sends w to v is called the projection map from W to V.

In general, a projection is a map p from a set A to a set B such that for each point b in B, p^-1(b) has some special property. For example, the map from Eⁿ to E^n-1 defined by p((x₁,..,x_n)) = (x₂,...,x_n) has the property that p^-1(b) is a line for all b. Sometimes, a projection map is as above but restricted to some subset of X. This is the case of the example with the knot. If A = XxB and p is defined by p(x,b) = b then p^-1(b) is isomorphic to B if A and B are groups for example.