Orthogonal Projection

In Mathematical, the orthogonal projection is a transformation of space, a Linear application:

in Géométrie planes , it is a projection such as the two lines - the line on which one projects and direction of projection - are perpendicular;
in Solid geometry, it is a projection such as the line and the plan - whatever their roles respective - is perpendicular.

Orthogonal projection is a type of Perspective very much used in Technical design (descriptive Géométrie), and in Infographie: the generation of the figures is simple, on the other hand, one cannot represent the distance (the size of the objects does not vary with the distance).

In a more general way, in Linear algebra, an orthogonal projection is a projector such as the two subspaces are orthogonal.

Orthogonal projection makes it possible to solve the problem of shortest the Distance between a point on the one hand and a line, a plan, or more generally a subspace closely connected of a Euclidean Espace on the other hand. One can then use this concept to solve problems of the type “least squares”.

The general idea, based on the Théorème of Pythagore, is that the problem of short distance brings back to a property Orthogonalité.

The Plumbline is a tool which makes it possible to visualize the orthogonal projection of a point on a plan (in first analyzes at least).

Drawing by orthogonal projection

Orthogonal projections are used for the drawing, in particular the Technical design and the video games. One typically distinguishes two types of projections used:

the descriptive Geometry: the plan of projection contains two of the axes of the direct orthonormé reference mark;
the axonometric Prospect: the plan of projection is distinct from the caused plans.

See these articles.

Orthogonal projection in geometry closely connected “elementary”

Projected orthogonal on a line, outdistances

The simplest example of projection is in the usual plan (refines Euclidean): the orthogonal projection of a point has on a line (D), is the point H pertaining to (D) such as the lines (D) and (AH) are Perpendiculaire S. One often uses the expression “to lower the perpendicular resulting from has” for the construction of H, which can be done with the rule and the compass. But also by carrying out the scalar product.

Distance AH is then lower than the distances AM for the other points M of (D), strictly safe if M=H.

This distance is called distance from the point has with the right-hand side D . explicit calculation can be made by the application of the formulas of trigonometry for the right-angled triangles.

The point has is on the line D if and only if it equal to its is projected ( A=H ), or if and only if its distance to D is null.

Orthogonal projection of a line on another line

Always in the plan refines Euclidean, one can consider two secant lines (D) and (Of) forming an angle θ. Orthogonal projection is the Application p which at each point M of (D) associates its projected orthogonal H=p (M) on (Of).

The point of intersection I is its characteristic projected: p (I) =I .

A remarkable property of projection is the way in which it transforms the distances. If M and NR is points of (D) and M'=p (M), N'=p (NR) projected to them orthogonal respective, one obtains Me N'=MN. cos θ.

In particular one will notice, by parity of the function Cosinus, that orthogonally to project the elements of (D) on (Of) multiplies all the distances by a factor cos θ, but orthogonally to project the elements of (Of) on (D) multiplies all the distances by the same factor.

Projected orthogonal on a plan, outdistances

Orthogonal projection in a vector space préhilbertien

Orthogonal projections are endomorphism S which belong to the more general class of the Projecteur S, that one can then consider, a contrario, like “oblique” projections.

One places oneself in a Espace préhilbertien E , of unspecified Dimension. One gives oneself a vectorial subspace F of E . The problem of orthogonal projection on F can be stated as follows: can one break up an unspecified vector of E into a component on F and an orthogonal component with F? The answer will depend in fact on space F considered.

Orthogonal projection on a vectorial line

If F is a vectorial line generated by the vector has , the whole of the orthogonal vectors with F is a Hyperplan called normal hyperplane with F and defined by

F^ \ perp = \ {H \ in E, (H \ cdot a)=0 \} ~

If X is an arbitrary vector of E , one can always break up it in the following way

x=x_F+x_ \ perp

with

x_F = \ frac {(has \ cdot X)}{\|has \|^2} a

And it is noted that

x_F

is in F , while

x_ \ perp=x-x_F

is in the normal hyperplane with F .

It is thus always possible to carry out an orthogonal projection on a vectorial line.

Existence of an orthogonal projection

One can give an example of space F for which the concept of orthogonal projection on F does not have a direction. Thus if one considers space $\ mathbb R$ of the Polynôme S realities provided with the usual scalar product, and F the hyperplane Vect ( $1+X, 1+X^2,…, 1+X^n,…$ ), the whole of the orthogonal vectors with F is tiny room to {0}. One cannot thus break up the elements of E , others that those of F , in an element of F and an orthogonal element.

This example is striking: whereas a line always has an orthogonal additional (single besides), a hyperplane can not have any additional orthogonal very well. It is difficult to make a convincing drawing for such a situation!

More generally there is equivalence between the following properties

it exists an orthogonal projection on F
F admits additional orthogonal a
$F^ \ perp$ is the additional orthogonal one of F

This shows in the passing that the additional orthogonal one, if there exists, is single.

Two important cases of existence

One can generalize the formula of projection on a line if F is of finished size. Indeed, by considering a orthonormal Base $(e_1,…, e_n)$ of F , one exhibe the decomposition

x=x_F+x_ \ perp

with

x_F = \ sum_ {i=1} ^n (e_i \ cdot X) e_i

Attention not to apply this formula with an unspecified base!

if E is a space of Hilbert and F a vectorial subspace closed, then one shows that orthogonal '' F '' is additional a.

Minimization of the distance

If the subspace F admits additional orthogonal and if X is a point of E , projected orthogonal the p of X on F checks the property of following minimization

\ forall F \ in F, \|x-p \|\ Leq \|x-f \|~

and there is equality only for f=p .

Thus p is the point of F nearest to X , which provides an alternative definition of p .

The distance $\|x-p \|$ is called distance from X to F .

See too

Projection (geometry)
Projector
Theorem of projection on convex closed in a space of Hilbert
Theorem of additional orthogonal of one closed in Hilbert
Method of least squares
Pseudo-solution of a linear system
Base of Hilbert

Determinant of Gram

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