Orthographic projection

An orthogonal projection (of gr ὀρθός just γωνία angle and Latin proiacere forward throw ), orthogonal projection or vertical projection is a map that is used in many areas of mathematics. In geometry, an orthogonal projection is the mapping of a point on a straight line or a plane, so that the connecting line between the point and its image with this line or plane forms a right angle. The image has then from all points of the straight line or plane of the shortest distance to the starting point. An orthogonal projection is therefore a special case of a parallel projection, wherein the projection direction is equal to the normal direction of a straight line or plane.

In linear algebra, this concept is extended to higher-dimensional vector spaces over the real or complex numbers and more general angle and distance concepts. An orthographic projection is then the projection of a vector onto a subspace, so that the difference vector from the image and output vector lies in the orthogonal complement. In the functional analysis, the term is even wider in infinite Skalarprodukträumen and in particular applied to functions. The existence and uniqueness of such orthogonal then ensures the projection set.

Orthogonal projections have wide range of applications within mathematics, for example, in descriptive geometry, the Gram-Schmidt orthogonalization 's, the method of least squares, the conjugate gradient method, the Fourier analysis or the best approximation. They have, among other applications in cartography, architecture, computer graphics and physics.

2.2.1 Definition
2.2.2 calculation
2.2.3 example
2.2.4 properties
2.2.5 General case
2.2.6 Alternative calculation

3.1 Algebraic representation 3.1.1 Definition
3.1.2 representation
3.1.3 Examples
3.1.4 properties
3.1.5 General case
3.1.6 Complementary representation

3.2.1 coordinates
3.2.2 representation
3.2.3 Examples
3.2.4 properties
3.2.5 General case
3.2.6 Complementary representation

4.1 Definition
4.2 Existence and Uniqueness
4.3 presentation
4.4 Example
4.5 Properties

Descriptive Geometry

In descriptive geometry and technical drawing projections are used to establish two-dimensional images of three-dimensional geometric bodies. In addition to the central projection in this case are often parallel projections used. A parallel projection is a diagram depicting the points of the three-dimensional space of a given point on the image plane, wherein the projection beams are mutually parallel. Meet the projection rays at right angles to the plane of projection, it is called an orthogonal projection.

If, instead of an image plane three projection planes used which are perpendicular to each other, then it is a three-panel projection or normal projection. In most cases, the projection planes are parallel to the axes of the used ( Cartesian ) coordinate system. Has a point in space the coordinates, the result is the orthogonal coordinates of the point on the three planes through

Passing a projection plane in parallel to the two coordinate axes, but not through the origin of the coordinate system, we obtain the projected point by replacing the value of the intersection of the plane with the third coordinate axis. In an orthogonal axonometric, for example, an isometry or a dimetry, the object being imaged before the projection is rotated in a specific way.

Analytic Geometry

Analytical geometry is concerned with the calculation and the mathematical properties of the orthogonal two - and three-dimensional space, in particular for the case where the projection plane is not parallel to the coordinate axes.

Projection onto a line

Definition

In the Euclidean plane is an orthogonal projection of the image of a point on a straight line, so that the connecting line between the point and its image is precisely forms a right angle with the. An orthogonal projection must therefore both conditions

( Projection)
( Orthogonality )

. meet The line is called solder the point on the straight line and the projected point is called the nadir point. The graphic design of the solder with ruler and compass is a standard task of Euclidean geometry, and this is referred to the cases of the solder.

Calculation

In analytical geometry points will be described in a Cartesian coordinate system by the position vectors and a straight line equation of straight line in parameter form, wherein the position vector of a point line, the direction vector of the straight line, and a real parameter is typically. Two vectors, forming a right angle, if their dot product is. If the straight line as a line through the origin, then apply through the zero point and the orthogonal projection of the point on the line has the two conditions

Meet for one. The first equation set in the second, one obtains

Which resolved after

Results. An orthogonal projection thus has the representation

If, in addition, the direction vector of the line is a unit vector, ie holds, then we obtain the simple representation

The factor then indicates how far the projected point is a straight line away from the zero point on the. In an analogous way, one point in Euclidean space may be projected onto a straight line in the space is orthogonal, it is merely expected to three instead of two-component vectors.

Examples

The orthogonal projection of the point on the line through the origin with direction in the Euclidean plane

The orthogonal projection of the point on the line through the origin with direction in Euclidean space

Properties

The point to be projected is already on the line, then there is a number and the orthogonal projection

Does not change the point. Otherwise, as for the square of this distance according to the Pythagorean theorem, the orthogonal projection minimizes the distance between the starting point and all straight points,

Applies to all figures. The minimum is thereby clearly accepted at the orthogonally projected point, since the first term of the sum is exactly zero. Forming the vectors, and a right angle, as is the projected point of the zero point.

General case

The orthogonal projection of a point on a line that is not a line through the origin is given by

Optionally, which can be determined by employing the general line equation to the orthogonality and solving for the free parameters. For the general case we obtain the above special cases, by the support vector will be moved straight into the zero point and its direction vector is normalized, ie is divided by its amount. In the example of the figure above, as well as and so.

Alternative calculation

Alternatively, an orthogonal projection of the two-dimensional case are calculated with the perpendicular straight line by determining the intersection of the output line. If a normal vector of the output line, then follows from the two conditions

By substituting the first equation into the second equation and solving for the free parameters for the orthogonal

A normal vector can be determined by switching of the two components of the direction vector of the straight line and by inverting the sign of one of the two components. In the above example is one. Since a straight line in three-dimensional space has no excellent normal direction, this simple approach is only possible in two dimensions.

Projection on a plane

Definition

In three-dimensional space a point can be orthogonally projected on a plane. An orthogonal projection then you have the two conditions

( Projection)
( Orthogonality )

. meet Again, one speaks of Lot and Perpendicular. The orthogonality implies here that the Lot is perpendicular to all the lines of the plane through the nadir point.

Calculation

A point in the Euclidian space will be described again by a spatial vector, and the level is given in the form of parameters, and said real parameters, as well as and the tension vectors of the plane may not be collinear. Due to the linearity of the scalar product, it is sufficient here to prove orthogonality with respect to the two tension vectors rather than with respect to all vectors of the plane. If it is at the level to an original level, that is, the orthogonal projection of the point must be on the plane of the three conditions

. meet Substituting the first equation into the other two equations, one obtains with

A linear system of two equations and two unknowns and. If the tension vectors are orthogonal to each other, that is true, then this system of equations is divided into two independent equations and its solution can be specified directly. The orthogonal projection of the point on the plane is then given by

Given. If the tension vectors even orthonormal, ie is in addition, one has the simple representation

One thus obtains the orthogonal projection of a point onto a plane by determining the orthogonal projections of the point and the two formed by the tension vectors and lines and by adding the results ( see figure).

Example

The orthogonal projection of the point to the original level, and is spanned by the orthogonal vectors,

Properties

The point to be projected is already on the plane, then there are numbers and with and the orthogonal

Does not change the point. Otherwise, the orthogonally projected point minimizes the distance between the starting point and all points in the plane, as for the square of this distance with the Pythagorean Theorem

Applies to all figures. The minimum is assumed for this unique and orthogonal to the projected point. Both forms, as well as a right angle, the projected point is the zero point.

General case

Passing a plane not passing through the origin, it may be moved by translation to the origin. Are their tension vectors are not orthogonal, they can be orthogonalized using the Gram-Schmidt orthogonalization 's. To this end, we determined (for example) to an orthogonal vector as a connecting vector of the orthogonal projection of the straight line in the direction

And thus obtains the general case of an orthogonal projection of a point on a plane through

Alternative calculation

Alternatively, an orthogonal projection can also be calculated by calculating the intersection of the perpendicular straight line with the plane. A normal vector of the plane can, if it is not given in normal form, tension vectors are calculated by using the cross product of (not necessarily orthogonal ). Then obtained as in the two-dimensional case as orthogonal

Linear Algebra

In linear algebra, the concept of orthogonal projection is generalized to general vector spaces with finite dimension over the field of real or complex numbers, as well as general scalar and thus Orthogonalitätsbegriffe. Two vectors are by definition if and only orthogonal if their dot product is.

Algebraic representation

Definition

An orthogonal projection onto a subspace of a vector space is a linear mapping that for all vectors, the two properties

( Projection)
For all ( orthogonality )

Met. The difference vector is thus in the orthogonal complement of. The orthogonal complement is itself a subspace consisting of those vectors in which to all vectors are orthogonal in.

Representation

Is a basis of the vector space of dimension, then each vector has a unique representation as a linear combination. Because of the scalar product Sesquilinearität therefore it suffices to detect only the orthogonality with respect to the basis vectors instead of with respect to all the vectors of the sub-vector space. An orthogonal projection must therefore the conditions

. meet Substituting the first equation into the other equations, obtained with

A linear system of equations with the equations and unknowns. Being founded on Gram matrix is regular due to the linear independence of the basis vectors and thus this system of equations is uniquely solvable. Consequently, if an orthogonal basis of, ie for, the associated Gram matrix is a diagonal matrix and the system of equations has a solution directly assignable. The orthogonal projection of the vector onto the subspace is then given by

Given. Even forms an orthonormal basis, ie, with the Kronecker delta, then the orthogonal projection has the simple representation

Examples

If you choose the vector space of the standard room and as the standard scalar scalar, a vector subspace is a linear manifold (such as a line, plane or hyperplane ) just correspond through the zero point and the orthogonal geometry of the previous section the special cases

Projection onto a line through the origin in the plane:
Projection onto a line through the origin in the space:
Projection onto a plane in space origin:

The case corresponds in each dimension of the image of a vector to the zero point and the case leaves the vector unchanged ever since an orthogonal projection then is the identity map.

Properties

An orthographic projection is a projection, that is an idempotent linear map of the vector space into itself ( called endomorphism ). If the vector to be projected namely an element of the vector space, then there are scalars, so that is, and the orthogonal

Does not change the vector, from which the idempotence follows. The linearity of the mapping follows directly from the Sesquilinearität the scalar product. In addition, the self-adjointness applies

For all vectors. The orthogonally projected vector minimizes the distance between the output vector and all vectors of the subspace with respect to the scalar product derived from the standard, as it applies to the Pythagorean Theorem for Skalarprodukträume

For everyone. The minimum is thereby clearly accepted at the orthogonally projected vector. Is the vector in the orthogonal complement of the sub-vector space, then the projected vector of the zero vector.

General case

Is not orthogonal the basis of the subspace, so they can use the Gram-Schmidt orthogonalization 's orthogonalized and be an orthogonal from the get. Furthermore, a vector can be projected onto an affine subspace with orthogonal. Then obtained by the general case of an orthogonal projection of a vector onto an affine subspace

Complementary representation

Is now an orthogonal basis of complementary, ie an orthogonal complement of, then we obtain a result of

The complementary representation of an orthogonal projection to an affine subspace as

Matrix representation

Coordinates

If we choose for the vector space is an orthonormal basis with respect to the scalar product, then each vector as a coordinate vector on

Are shown. The coordinates are exactly the lengths of the orthogonal projections of the vector onto the basis vectors. The scalar product of two vectors in coordinate representation is then the standard scalar product of the corresponding coordinate vectors, the adjoint vector to be ( in the real case the transposed vector).

Representation

Now, the coordinates of an orthogonal basis vectors of a coordinate Untervektoraums and the vector of a vector to be projected, then the coordinates of an orthogonal representation

An orthogonal coordinate representation is thus simply a matrix-vector product where the projection matrix by

If the coordinates of vectors an orthonormal basis of, the Orthogonalprojektionsmatrix has the simple representation

Each summand is the dyadic product of a coordinate vector with itself

Examples

In coordinate space the Orthogonalprojektionsmatrix is given to the origin line with direction by

The Orthogonalprojektionsmatrix to the original level, that is defined by and is in accordance with

Properties

A Orthogonalprojektionsmatrix is idempotent, ie, it applies

Furthermore, it is self-adjoint ( symmetric in the real case ), since

Is. For the rank and trace of Orthogonalprojektionsmatrix

As match for idempotent matrices rank and track, and the individual matrices each have rank one. The eigenvalues of a Orthogonalprojektionsmatrix are and, with the associated eigenspaces are just the subspace and its orthogonal complement. The spectral norm of a Orthogonalprojektionsmatrix is so, unless is the zero vector space, equal to one.

General case

Forming the coordinate vectors, although a base, but no orthogonal basis of the subspace, we can orthogonalize them to calculate an orthogonal or solving a corresponding system of linear equations. Summing up the basis vectors column-wise into a matrix together, then this system of equations has the form of the normal equations

By the coefficient vector. The matrix representation of an orthogonal projection is then the basis of given

An orthogonal projection to an affine subspace in matrix representation then the affine

With the unit matrix and as the coordinate vector of. Using homogeneous coordinates can be any orthogonal projection also be represented as a simple matrix - vector product.

Complementary representation

An orthogonal projection to an affine subspace has the complementary matrix representation

With the Orthogonalprojektionsmatrix on the complementary space given by

Forming the coordinate vectors of an orthogonal complementary space, the complementary Orthogonalprojektionsmatrix has the representation

Functional Analysis

In the functional analysis, the concept of orthogonal projection is generalized to infinite-dimensional Skalarprodukträume over the real or complex numbers, in particular applied to function spaces.

Definition

Is a scalar product and is a subspace of with orthogonal complement, then an orthogonal projection is an operator (also called orthogonal projection ), with the two properties

( Projection)
( Orthogonality )

The image and the kernel of the operator is. The complementary operator has then as an image and as a core.

Existence and uniqueness

In order for such orthogonal projections also exist and are unique, but the environments being considered must be restricted. Is a Hilbert space, ie a complete inner product, and is a closed subspace of, then, the projection theorem, the existence and uniqueness of orthogonal projections safe. Are available for each unique vector vectors and then, so this vector, the representation

Possesses. This form and an orthogonal decomposition of, ie, the entire space can be represented as an orthogonal sum. A finite-dimensional vector space is always complete and the completeness of can then be dispensed with.

Representation

Every Hilbert space has an orthonormal basis, which however can not be always explicitly specify. However, if a separable Hilbert space, then such an orthonormal basis is countable as a Schauder basis, so that each vector in a series

Can be developed. Such an orthonormal basis can always be obtained from a linearly independent subset of using the Gram-Schmidt orthogonalization 's. Is now a ( also countable ) orthonormal basis of a closed subspace, then an orthogonal projection has the series representation

This representation can be generalized to non - separable, that is uncountable -dimensional Hilbert spaces. Is a closed subspace of a general Hilbert space and with an arbitrary index set is an orthonormal basis of this subspace, then an orthogonal projection has the corresponding representation

With only countably many members of this total sum are nonzero. These series are according to Bessel's inequality absolutely convergent and by the Parseval equation while each element of is actually mapped to itself.

Example

If the space L2 is the square integrable real functions in the interval with the L2 scalar product

For this room for the Legendre polynomials form a complete orthogonal system. Wanted is now the orthogonal projection of the exponential function on the sub-vector space of linear functions. For this subspace, the two monomials form an orthogonal basis, which after normalization, the orthonormal basis

Results. The orthogonal projection of the subspace of the linear functions is then given by

Properties

Is a Hilbert space, and is a closed space of, then a steady linear operator with the following properties:

Is a projection, that is.
Is self-adjoint, that is, the adjoint operator.
Is normal, ie.
Is positive, that is to say in particular for all of them.
Is a partial isometric view where the isometric portion is the identity.
Is compact if it is finite-dimensional.
Is in the best approximation Skalarproduktnorm, that is.
If, and, if ( in the operator norm ).

Conversely, a continuous linear projection, which is self-adjoint or normal or positive or normalized to one, an orthogonal projection onto the image space.

Applications

Orthogonal projections have a variety of applications, from which only a few are highlighted:

In analytic geometry in distance calculations and reflections in planes
In cartography at orthogonal plan projections and orthophotos
In physics at the resolution of forces into their components
In the computer graphics in the calculation of reflections and shadows

In the construction of orthonormal bases with the Gram-Schmidt orthogonalization 's
In solving linear least squares problems with the method of least squares
In the iterative solution of linear systems of equations using the conjugate gradient method and general Krylov subspace method

In approximation theory in the best approximation of functions
In the Fourier analysis and wavelet analysis signals
In the Sobolev theory for solving partial differential equations
In quantum mechanics to describe quantum states by means of the Bra- Ket notation

In the principal component analysis, multivariate data
With conditional expectation values

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