Editing Orthographic projection

{{Short description|Means of projecting three-dimensional objects in two dimensions}}
{{For multi|the orthographic projection as a map projection|Orthographic map projection|mathematical discussion in terms of linear algebra|Projection (linear algebra)}}

{{Views}}

'''Orthographic projection''' (also '''orthogonal projection''' and '''analemma'''){{efn|This usage is obsolete; the common meaning of "[[analemma]]" is a diagram of the position of the Sun from the Earth.<ref>Sawyer, F., [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.142.3875&rep=rep1&type=pdf Of Analemmas, Mean Time and the Analemmatic Sundial]</ref>}} is a means of representing [[Three-dimensional space|three-dimensional]] objects in [[Plane (mathematics)|two dimensions]]. Orthographic projection is a form of [[parallel projection]] in which all the projection lines are [[orthogonal]] to the [[projection plane]],<ref name="maynard">{{Cite book | last = Maynard | first = Patric | title = Drawing Distinctions: The Varieties of Graphic Expression | publisher = Cornell University Press | year = 2005 | pages = 22 | url = https://books.google.com/books?id=4Y_YqOlXoxMC&q=axonometric%20orthographic&pg=PA22 | isbn = 0-8014-7280-6}}</ref> resulting in every plane of the scene appearing in [[affine transformation]] on the viewing surface. The obverse of an orthographic projection is an [[oblique projection]], which is a parallel projection in which the projection lines are ''not'' orthogonal to the projection plane.

The term ''orthographic'' sometimes means a technique in [[multiview projection]] in which principal axes or the planes of the subject are also parallel with the projection plane to create the ''primary views''.<ref name="maynard"/>  If the principal planes or axes of an object in an orthographic projection are ''not'' parallel with the projection plane, the depiction is called ''axonometric'' or an ''auxiliary views''. (''Axonometric projection'' is synonymous with ''parallel projection''.) Sub-types of ''primary views'' include ''plans'', ''elevations'', and ''sections''; sub-types of ''auxiliary views'' include ''isometric'', ''dimetric'', and ''trimetric projections''. 

A lens that provides an orthographic projection is an [[Telecentric lens#Object-space telecentric lenses|object-space telecentric lens]].

== Geometry ==
[[File:Graphical projection comparison.png|thumb|right|Comparison of several types of [[graphical projection]]]]
[[File:Various projections of cube above plane.svg|thumb|Various projections and how they are produced]]
[[Image:Axonometric projections.png|thumb|The three views. The percentages show the amount of foreshortening.]]

A simple orthographic [[projection (linear algebra)|projection]] onto the [[plane (mathematics)|plane]] ''z'' = 0 can be defined by the following matrix:
:<math>
P = 
\begin{bmatrix}
1 & 0 & 0  \\
0 & 1 & 0  \\
0 & 0 & 0  \\
\end{bmatrix}
</math>

For each point ''v'' = (''v''<sub>''x''</sub>, ''v''<sub>''y''</sub>, ''v''<sub>''z''</sub>), the transformed point ''Pv'' would be
:<math>
Pv = 
\begin{bmatrix}
1 & 0 & 0  \\
0 & 1 & 0  \\
0 & 0 & 0  \\
\end{bmatrix}
\begin{bmatrix}
v_x \\ v_y \\ v_z
\end{bmatrix}
=
\begin{bmatrix}
v_x \\ v_y \\ 0
\end{bmatrix}
</math>

Often, it is more useful to use [[homogeneous coordinates]].  The transformation above can be represented for homogeneous coordinates as 
:<math>
P = 
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
</math>

For each homogeneous vector ''v'' = (''v''<sub>''x''</sub>, ''v''<sub>''y''</sub>, ''v''<sub>''z''</sub>, 1), the transformed vector ''Pv'' would be
:<math>
Pv = 
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
v_x \\ v_y \\ v_z \\ 1
\end{bmatrix}
=
\begin{bmatrix}
v_x \\ v_y \\ 0 \\ 1
\end{bmatrix}
</math>

In [[computer graphics]],  one of the most common matrices used for orthographic [[projection (linear algebra)|projection]] can be defined by a [[n-tuple|6-tuple]], (''left'', ''right'', ''bottom'', ''top'', ''near'', ''far''), which defines the [[clipping (computer graphics)|clipping]] planes.  These planes form a box with the minimum corner at (''left'', ''bottom'', -''near'') and the maximum corner at (''right'', ''top'', -''far'').<ref>{{Cite web |last=Thormählen |first=Thorsten |date=November 26, 2021 |title=Graphics Programming – Cameras: Parallel Projection – Part 6, Chapter 2 |url=https://www.mathematik.uni-marburg.de/~thormae/lectures/graphics1/graphics_6_2_eng_web.html#10 |access-date=2022-04-22 |website=Mathematik Uni Marburg |pages=8 ff}}</ref>

The box is translated so that its center is at the origin, then it is scaled to the unit cube which is defined by having a minimum corner at (−1,−1,−1) and a maximum corner at (1,1,1).

The orthographic transform can be given by the following matrix:
:<math>
P = 
\begin{bmatrix}
\frac{2}{\text{right}-\text{left}} & 0 & 0 & -\frac{\text{right}+\text{left}}{\text{right}-\text{left}} \\
0 & \frac{2}{\text{top}-\text{bottom}} & 0 & -\frac{\text{top}+\text{bottom}}{\text{top}-\text{bottom}} \\
0 & 0 & \frac{-2}{\text{far}-\text{near}} & -\frac{\text{far}+\text{near}}{\text{far}-\text{near}} \\
0 & 0 & 0 & 1
\end{bmatrix}
</math>
which can be given as a [[scaling (geometry)|scaling]] ''S'' followed by a [[translation (geometry)|translation]] ''T'' of the form
:<math>
P = ST = 
\begin{bmatrix}
\frac{2}{\text{right}-\text{left}} & 0 & 0 & 0 \\
0 & \frac{2}{\text{top}-\text{bottom}} & 0 & 0 \\
0 & 0 & \frac{2}{\text{far}-\text{near}} & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 & -\frac{\text{left}+\text{right}}{2} \\
0 & 1 & 0 & -\frac{\text{top}+\text{bottom}}{2} \\
0 & 0 & -1 & -\frac{\text{far}+\text{near}}{2} \\
0 & 0 & 0 & 1
\end{bmatrix}
</math>

The inversion of the projection matrix ''P<sup>−1</sup>'', which can be used as the unprojection matrix is defined:

<math>
P^{-1} = 
\begin{bmatrix}
\frac{\text{right}-\text{left}}{2} & 0 & 0 & \frac{\text{left}+\text{right}}{2} \\
0 & \frac{\text{top}-\text{bottom}}{2} & 0 & \frac{\text{top}+\text{bottom}}{2} \\
0 & 0 & \frac{\text{far}-\text{near}}{-2} &  -\frac{\text{far}+\text{near}}{2} \\
0 & 0 & 0 & 1
\end{bmatrix}
</math>

== Types ==
{{comparison_of_graphical_projections.svg}}

Three sub-types of orthographic projection are ''[[isometric projection]]'', ''dimetric projection'', and ''trimetric projection'', depending on the exact angle at which the view deviates from the orthogonal.<ref name="maynard"/><ref name="mcreynolds">{{Cite book | last = McReynolds | first = Tom |author2= David Blythe | title = Advanced graphics programming using openGL | publisher = Elsevier | year = 2005 | pages = 502 | url = https://books.google.com/books?id=bmv2HRpG1bUC&q=axonometric | isbn = 1-55860-659-9}}</ref> Typically in axonometric drawing, as in other types of pictorials, one axis of space is shown to be vertical.

In '''isometric projection''', the most commonly used form of axonometric projection in engineering drawing,<ref name="godse">{{Cite book | title = Computer graphics | publisher = Technical Publications | year = 1984 | pages = 29 | url = https://books.google.com/books?id=YkVp-2ZrmyMC&q=axonometric+orthographic&pg=PT224 | isbn = 81-8431-558-9 | last = Godse | first = Atul P.}}</ref> the direction of viewing is such that the three axes of space appear equally [[Perspective (graphical)#Foreshortening|foreshortened]], and there is a common angle of 120° between them.  As the distortion caused by foreshortening is uniform, the proportionality between lengths is preserved, and the axes share a common scale; this eases one's ability to take measurements directly from the drawing.  Another advantage is that 120° angles are easily constructed using only a [[Compass and straightedge constructions|compass and straightedge]].

In '''dimetric projection''', the direction of viewing is such that two of the three axes of space appear equally foreshortened, of which the attendant scale and angles of presentation are determined according to the angle of viewing; the scale of the third direction is determined separately.

In '''trimetric projection''', the direction of viewing is such that all of the three axes of space appear unequally foreshortened. The scale along each of the three axes and the angles among them are determined separately as dictated by the angle of viewing. Trimetric perspective is seldom used in technical drawings.<ref name="mcreynolds"/>

==Multiview projection==
[[Image:Convention placement vues dessin technique.svg|thumb|right|Symbols used to define whether a ''multiview projection'' is either third-angle (right) or first-angle (left)]]
{{main|Multiview projection}}
In ''multiview projection'', up to six pictures of an object are produced, called ''primary views'', with each projection plane parallel to one of the coordinate axes of the object. The views are positioned relative to each other according to either of two schemes: ''first-angle'' or ''third-angle'' projection. In each, the appearances of views may be thought of as being ''projected'' onto planes that form a six-sided box around the object. Although six different sides can be drawn, usually three views of a drawing give enough information to make a three-dimensional object. These views are known as [[front view]] (also ''elevation''), [[top view]] (also ''plan'') and [[end view]] (also ''section''). When the plane or axis of the object depicted is ''not'' parallel to the projection plane, and where multiple sides of an object are visible in the same image, it is called an ''auxiliary view''. Thus ''isometric projection'', ''dimetric projection'' and ''trimetric projection'' would be considered ''auxiliary views'' in multiview projection. A typical characteristic of multiview projection is that one axis of space is usually displayed as vertical.

== Cartography ==
{{main|Orthographic projection in cartography}}
[[File:Orthographic projection SW.jpg|thumb|right|Orthographic projection (equatorial aspect) of eastern hemisphere 30°W&ndash;150°E]]
An orthographic projection map is a [[map projection]] of [[cartography]]. Like the [[stereographic projection]] and [[gnomonic projection]], orthographic projection is a [[perspective projection|perspective (or azimuthal) projection]], in which the [[sphere]] is projected onto a [[tangent plane]] or [[secant plane]].  The ''point of perspective'' for the orthographic projection is at [[Infinity|infinite]] distance.  It depicts a [[Sphere|hemisphere]] of the [[globe]] as it appears from [[outer space]], where the [[horizon]] is a [[great circle]].  The shapes and areas are [[Distortion#Map projections|distorted]], particularly near the edges.<ref name="SnyderWorkingManual">{{Cite book | author=Snyder, J. P.| title=Map Projections—A Working Manual (US Geologic Survey Professional Paper 1395) | publisher=US Government Printing Office | location=Washington, D.C.| year=1987 | pages=145–153}}</ref><ref name="Snyder16">Snyder, John P. (1993). ''Flattening the Earth: Two Thousand Years of Map Projections'' pp. 16–18. Chicago and London: The University of Chicago Press. {{ISBN|0-226-76746-9}}.</ref>

The orthographic projection has been known since antiquity, with its cartographic uses being well documented. [[Hipparchus]] used the projection in the 2nd century BC to determine the places of star-rise and star-set. In about 14 BC, Roman engineer [[Vitruvius|Marcus Vitruvius Pollio]] used the projection to construct sundials and to compute sun positions.<ref name="Snyder16"/>

Vitruvius also seems to have devised the term orthographic – from the Greek ''orthos'' ("straight") and ''graphē'' ("drawing") – for the projection. However, the name ''[[analemma]]'', which also meant a sundial showing latitude and longitude, was the common name until [[François d'Aguilon]] of Antwerp promoted its present name in 1613.<ref name="Snyder16"/>

The earliest surviving maps on the projection appear as woodcut drawings of terrestrial globes of 1509 (anonymous), 1533 and 1551 (Johannes Schöner), and 1524 and 1551 (Apian).<ref name="Snyder16"/>

== Notes ==
{{notelist}}

== References ==
{{Reflist}}

==External links==
{{Commons category|Orthographic projections}}
*[http://www.3doro.de/axon.htm Normale (orthogonale) Axonometrie] {{in lang|de}}
*[https://web.archive.org/web/20160402021438/http://www.globmaritime.com/martech/marine-navigation/general-concepts/6573-orthographic-projection Orthographic Projection Video and mathematics]

{{Structural geology}}

[[Category:Euclidean geometry]]
[[Category:Graphical projections]]
[[Category:Linear algebra]]

[[de:Orthogonale Projektion#Orthogonale Projektion]]