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The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .
A square matrix is called a projection matrix if it is equal to its square, i.e. if =. [2]: p. 38 A square matrix is called an orthogonal projection matrix if = = for a real matrix, and respectively = = for a complex matrix, where denotes the transpose of and denotes the adjoint or Hermitian transpose of .
Orthographic projection (also orthogonal projection and analemma) [a] is a means of representing three-dimensional objects in two dimensions.Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal to the projection plane, [2] resulting in every plane of the scene appearing in affine transformation on the viewing surface.
Also, let Q = (x 1, y 1) be any point on this line and n the vector (a, b) starting at point Q. The vector n is perpendicular to the line, and the distance d from point P to the line is equal to the length of the orthogonal projection of on n. The length of this projection is given by:
For example, the mapping that takes a point (x, y, z) in three dimensions to the point (x, y, 0) is a projection. This type of projection naturally generalizes to any number of dimensions n for the domain and k ≤ n for the codomain of the mapping. See Orthogonal projection, Projection (linear algebra). In the case of orthogonal projections ...
is exactly a sought for orthogonal projection of onto an image of X (see the picture below and note that as explained in the next section the image of X is just a subspace generated by column vectors of X). A few popular ways to find such a matrix S are described below.
In P 2, the line between the points x 1 and x 2 may be represented as a column vector ℓ that satisfies the equations x 1 T ℓ = 0 and x 2 T ℓ = 0, or in other words a column vector ℓ that is orthogonal to x 1 and x 2. The cross product will find such a vector: the line joining two points has homogeneous coordinates given by the equation ...
Orthographic projection in cartography has been used since antiquity. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective projection in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the orthographic projection is at infinite distance.