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A term related to the operation of projecting, which can be defined as follows (see Fig.): One chooses an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p0751301.png" /> of the space as the centre of projection and a plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p0751302.png" /> not passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p0751303.png" /> as the plane of projection. To project a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p0751304.png" /> (a pre-image) of the space onto the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p0751305.png" /> through the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p0751306.png" />, one draws the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p0751307.png" /> to its intersection with the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p0751308.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p0751309.png" />. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513010.png" /> (the image) is called the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513011.png" />. The projection of a figure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513012.png" /> is defined to be the collection of projections of all its points.
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A term related to the operation of projecting, which can be defined as follows (see Fig.): One chooses an arbitrary point $  S $
 +
of the space as the centre of projection and a plane $  \Pi  ^  \prime  $
 +
not passing through $  S $
 +
as the plane of projection. To project a point $  A $(
 +
a pre-image) of the space onto the plane $  \Pi  ^  \prime  $
 +
through the centre $  S $,  
 +
one draws the straight line $  S A $
 +
to its intersection with the plane $  \Pi  ^  \prime  $
 +
at a point $  A  ^  \prime  $.  
 +
The point $  A  ^  \prime  $(
 +
the image) is called the projection of $  A $.  
 +
The projection of a figure $  F $
 +
is defined to be the collection of projections of all its points.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p075130a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p075130a.gif" />
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Parallel projections are widely used in [[Descriptive geometry|descriptive geometry]] for obtaining various types of images (see, for example, [[Axonometry|Axonometry]]; [[Perspective|Perspective]]). There are special forms of projections onto the plane, sphere and other surfaces (see, for example, [[Cartographic projection|Cartographic projection]]; [[Stereographic projection|Stereographic projection]]).
 
Parallel projections are widely used in [[Descriptive geometry|descriptive geometry]] for obtaining various types of images (see, for example, [[Axonometry|Axonometry]]; [[Perspective|Perspective]]). There are special forms of projections onto the plane, sphere and other surfaces (see, for example, [[Cartographic projection|Cartographic projection]]; [[Stereographic projection|Stereographic projection]]).
 
 
  
 
====Comments====
 
====Comments====
In geometry and linear algebra one also encounters projections parallel to a subspace. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513013.png" /> is a vector space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513014.png" /> a subspace and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513015.png" /> a complementary subspace (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513017.png" />), then the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513018.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513019.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513020.png" /> parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513021.png" /> is the linear mapping that sends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513024.png" />, to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513025.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513026.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513027.png" />, and each such operator comes from a decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513028.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513030.png" />.
+
In geometry and linear algebra one also encounters projections parallel to a subspace. For instance, if $  X $
 +
is a vector space, $  V $
 +
a subspace and $  W $
 +
a complementary subspace (i.e. $  V \cap W = \{ 0 \} $
 +
and $  X = V+ W $),  
 +
then the projection $  P $
 +
from $  X $
 +
onto $  V $
 +
parallel to $  W $
 +
is the linear mapping that sends $  x = v+ w $,  
 +
$  v \in V $,  
 +
$  w \in W $,  
 +
to $  v $.  
 +
The operator $  P $
 +
satisfies $  P  ^ {2} = P $,  
 +
and each such operator comes from a decomposition $  X = V \oplus W $
 +
with $  V = P( X) $,  
 +
$  W = ( I- P)( X) $.
  
The orthogonal projection of a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513031.png" /> to a closed subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513032.png" /> assigns to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513033.png" /> the unique element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513035.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513037.png" /> are orthogonal. It is the parallel projection onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513038.png" /> along the orthogonal complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513039.png" />. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513040.png" /> is the element of best approximation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513041.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513042.png" />. In this case the corresponding operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513043.png" /> is also self-adjoint, and, conversely, self-adjoint operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513044.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075130/p07513045.png" /> are orthogonal projections. Cf. also [[Projector|Projector]].
+
The orthogonal projection of a Hilbert space $  H $
 +
to a closed subspace $  F $
 +
assigns to $  x \in H $
 +
the unique element $  y $
 +
of $  F $
 +
such that $  x - y $
 +
and $  F $
 +
are orthogonal. It is the parallel projection onto $  F $
 +
along the orthogonal complement $  F  ^  \perp  = \{ {x \in H } : {\langle  x, y \rangle = 0 \textrm{ for  all  }  y \in F } \} $.  
 +
The element $  y $
 +
is the element of best approximation to $  x $
 +
in $  F $.  
 +
In this case the corresponding operator $  P $
 +
is also self-adjoint, and, conversely, self-adjoint operators $  P $
 +
such that $  P  ^ {2} = P $
 +
are orthogonal projections. Cf. also [[Projector|Projector]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''I''' , Springer  (1987)  pp. Sect. 2.4.9.6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.S. Birman,  M.Z. Solomyak,  "Spectral theory of selfadjoint operators in Hilbert space" , Reidel  (1987)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Wiley, reprint  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''I''' , Springer  (1987)  pp. Sect. 2.4.9.6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.S. Birman,  M.Z. Solomyak,  "Spectral theory of selfadjoint operators in Hilbert space" , Reidel  (1987)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Wiley, reprint  (1988)</TD></TR></table>

Latest revision as of 08:08, 6 June 2020


A term related to the operation of projecting, which can be defined as follows (see Fig.): One chooses an arbitrary point $ S $ of the space as the centre of projection and a plane $ \Pi ^ \prime $ not passing through $ S $ as the plane of projection. To project a point $ A $( a pre-image) of the space onto the plane $ \Pi ^ \prime $ through the centre $ S $, one draws the straight line $ S A $ to its intersection with the plane $ \Pi ^ \prime $ at a point $ A ^ \prime $. The point $ A ^ \prime $( the image) is called the projection of $ A $. The projection of a figure $ F $ is defined to be the collection of projections of all its points.

Figure: p075130a

The projection just described is called central (or conical). A projection with centre of projection at infinity is called parallel (or cylindrical). If, moreover, the plane of projection is perpendicular to the direction of projection, then the projection is called orthogonal.

Parallel projections are widely used in descriptive geometry for obtaining various types of images (see, for example, Axonometry; Perspective). There are special forms of projections onto the plane, sphere and other surfaces (see, for example, Cartographic projection; Stereographic projection).

Comments

In geometry and linear algebra one also encounters projections parallel to a subspace. For instance, if $ X $ is a vector space, $ V $ a subspace and $ W $ a complementary subspace (i.e. $ V \cap W = \{ 0 \} $ and $ X = V+ W $), then the projection $ P $ from $ X $ onto $ V $ parallel to $ W $ is the linear mapping that sends $ x = v+ w $, $ v \in V $, $ w \in W $, to $ v $. The operator $ P $ satisfies $ P ^ {2} = P $, and each such operator comes from a decomposition $ X = V \oplus W $ with $ V = P( X) $, $ W = ( I- P)( X) $.

The orthogonal projection of a Hilbert space $ H $ to a closed subspace $ F $ assigns to $ x \in H $ the unique element $ y $ of $ F $ such that $ x - y $ and $ F $ are orthogonal. It is the parallel projection onto $ F $ along the orthogonal complement $ F ^ \perp = \{ {x \in H } : {\langle x, y \rangle = 0 \textrm{ for all } y \in F } \} $. The element $ y $ is the element of best approximation to $ x $ in $ F $. In this case the corresponding operator $ P $ is also self-adjoint, and, conversely, self-adjoint operators $ P $ such that $ P ^ {2} = P $ are orthogonal projections. Cf. also Projector.

References

[a1] M. Berger, "Geometry" , I , Springer (1987) pp. Sect. 2.4.9.6
[a2] M.S. Birman, M.Z. Solomyak, "Spectral theory of selfadjoint operators in Hilbert space" , Reidel (1987) (Translated from Russian)
[a3] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Wiley, reprint (1988)
How to Cite This Entry:
Projection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projection&oldid=14898
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article