View source for Oblique projection

{{lead too short|date=February 2015}}
{{Use dmy dates|date=May 2021}}
{{short description|Type of technical drawing}}
{{about|the imaging of 3D objects|an abstract mathematical discussion|Projection (linear algebra)}}
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{{comparison_of_graphical_projections.svg}}
'''Oblique projection''' is a simple type of technical drawing of [[graphical projection]] used for producing two-dimensional (2D) [[image]]s of three-dimensional (3D) objects.

The objects are not in [[perspective (graphical)|perspective]] and so do not correspond to any view of an object that can be obtained in practice, but the technique yields somewhat convincing and useful results.

Oblique projection is commonly used in technical drawing. The cavalier projection was used by French military artists in the 18th century to depict fortifications.

Oblique projection was used almost universally by Chinese artists from the 1st or 2nd centuries to the 18th century, especially to depict rectilinear objects such as houses.<ref name=Cucker299>{{cite book |last1=Cucker |first1=Felipe |author1-link=Felipe Cucker|title=Manifold Mirrors: The Crossing Paths of the Arts and Mathematics |date=2013 |publisher=Cambridge University Press |isbn=978-0-521-72876-8 |pages=269–278}}</ref>

Various graphical projection techniques can be used in computer graphics, including in Computer Aided Design (CAD), computer games, computer generated animations, and special effects used in movies.

== Overview ==
[[File:Graphical projection comparison.png|thumb|right|Comparison of several types of [[graphical projection]]. The presence of one or more 90° angles within a pictorial image is usually a good indication that the perspective is ''oblique''.]]
[[File:Various projections of cube above plane.svg|thumb|right|Various graphical projections and how they are produced]]
[[File:Oblique projection yz.svg|thumb|right|Oblique projection of a cube with foreshortening by half, seen from the side]]
[[File:Oblique projection comparison top.svg|thumb|right|Top view of a comparison of an oblique projection (left) and an orthographic projection (right) of a unit cube (cyan) onto the projection plane (red). The [[foreshortening]] factor (1/2 in this example) is inversely proportional to the [[tangent]] of the angle (63.43° in this example) between the [[projection plane]] (colored brown) and the projection lines (dotted).]]
[[File:Oblique projection comparison front.svg|thumb|right|Front view of the same.]]

Oblique projection is a type of [[parallel projection]]:
* it projects an image by intersecting parallel rays (projectors)
* from the three-dimensional source object with the drawing surface (projection plane).
In both oblique projection and [[orthographic projection]], parallel lines of the source object produce parallel lines in the projected image. The projectors in oblique projection intersect the projection plane at an oblique angle to produce the projected image, as opposed to the perpendicular angle used in orthographic projection.

Mathematically, the parallel projection of the point <math>(x, y, z)</math> on the <math>xy</math>-plane gives <math>(x+az, y+bz, 0)</math>. The constants <math>a</math> and <math>b</math> uniquely specify a parallel projection. When <math>a = b = 0</math>, the projection is said to be "orthographic" or "orthogonal". Otherwise, it is "oblique". The constants <math>a</math> and <math>b</math> are not necessarily less than 1, and as a consequence lengths measured on an oblique projection may be either larger or shorter than they were in space. In a general oblique projection, spheres of the space are projected as ellipses on the drawing plane, and not as circles as they would appear from an orthogonal projection.

Oblique drawing is also the crudest "3D" drawing method but the easiest to master. One way to draw using an oblique view is to draw the side of the object in two dimensions, i.e. flat, and then draw the other sides at an angle of 45°, but instead of drawing the sides full size they are only drawn with half the depth creating 'forced depth' – adding an element of realism to the object. Even with this 'forced depth', oblique drawings look very unconvincing to the eye. For this reason oblique is rarely used by professional designers or engineers.

==Oblique pictorial==
In an ''[[oblique pictorial]]'' drawing, the angles displayed among the axis, as well as the [[foreshortening]] factors (scale) are arbitrary.
More precisely, any given set of three coplanar segments originating from the same point may be construed as forming some oblique perspective of three sides of a cube. This result is known as [[Pohlke's theorem]], from the German mathematician Pohlke, who published it in the early 19th century.<ref>[http://mathworld.wolfram.com/PohlkesTheorem.html Weisstein, Eric W. "Pohlke's Theorem". From MathWorld—A Wolfram Web Resource.]</ref>
 
The resulting distortions make the [[wikt:technique|technique]] unsuitable for formal, working drawings. Nevertheless, the distortions are partially overcome by aligning one plane of the image parallel to the plane of projection. Doing so creates a true shape image of the chosen plane. 
This specific category of oblique projections, whereby lengths along the directions <math>x</math> and <math>y</math> are preserved, but lengths along direction <math>z</math> are drawn at angle using a reduction factor is very much in use for industrial drawings.
* ''Cavalier projection'' is the name of such a projection, where the length along the <math>z</math> axis remains unscaled.<ref name="pp">[http://www.mtsu.edu/~csjudy/planeview3D/tutorial-parallel.html Parallel Projections] {{webarchive|url=https://web.archive.org/web/20070423160654/http://www.mtsu.edu/~csjudy/planeview3D/tutorial-parallel.html |date=23 April 2007 }} from ''PlaneView3D Online''</ref>
* ''Cabinet projection'', popular in furniture illustrations, is an example of such a technique, where in the receding axis is scaled to half-size<ref name="pp"/> (sometimes instead two-thirds the original).<ref>{{citation|title=Basic Engineering|series=Butterworth-Heinemann GNVQ Engineering Series|first=William|last=Bolton|publisher=BH Newnes|year=1995|isbn=9780750625845|page=140}}.</ref>

==Cavalier projection==
{{further|Mathematics and art}}

In '''cavalier projection''' (sometimes '''cavalier perspective''' or '''high view point''') a point of the object is represented by three coordinates, ''x'', ''y'' and ''z''. On the drawing, it is represented by only two coordinates, ''x″'' and ''y″''. On the flat drawing, two axes, ''x'' and ''z'' on the figure, are [[perpendicular]] and the length on these axes are drawn with a 1:1 scale; it is thus similar to the [[dimetric projection]]s, although it is not an [[axonometric projection]], as the third axis, here ''y'', is drawn in diagonal, making an arbitrary angle with the ''x″'' axis, usually 30 or 45°. The length of the third axis is not scaled.<ref>{{cite web |url=http://www.tpub.com:80/ |title=Repair and Maintenance Manuals - Integrated Publishing |access-date=22 August 2010 |url-status=dead |archive-url=https://web.archive.org/web/20100822152816/http://www.tpub.com/ |archive-date=22 August 2010 }} from {{cite web |url=http://www.tpub.com:80/ |title=Repair and Maintenance Manuals - Integrated Publishing |access-date=22 August 2010 |url-status=dead |archive-url=https://web.archive.org/web/20100822152816/http://www.tpub.com/ |archive-date=22 August 2010 }}</ref><ref>Ingrid Carlbom, Joseph Paciorek, Planar Geometric Projections and Viewing Transformations, [[ACM Computing Surveys]], v.10 n.4, pp.&nbsp;465–502, Dec. 1978</ref>

It is very easy to draw, especially with pen and paper. It is thus often used when a figure must be drawn by hand, e.g. on a black board (lesson, oral examination).

The representation was initially used for military [[fortification]]s. In French, the "cavalier" (literally ''rider, horseman'', see ''[[Cavalry]]'') is an artificial hill behind the walls that allows sighting of the enemy above the walls.<ref>[http://trucsmaths.free.fr/etymologie.htm#C Etymologie des maths, letter C] (French)</ref> The cavalier perspective was the way the things were seen from this high point. Some also explain the name by the fact that it was the way a rider could see a small object on the ground from his horseback.<ref>[http://mapage.noos.fr/r.ferreol/langage/notations/notations.htm DES QUESTIONS D'ORIGINES] (French)</ref>

==Cabinet projection==
The term ''cabinet projection'' stems from its use in illustrations by the furniture industry.<ref>{{citation|title=Design Drawing|first1=Francis D. K.|last1=Ching|first2=Steven P.|last2=Juroszek|edition=2nd|publisher=John Wiley & Sons|year=2011|isbn=9781118007372|page=205|url=https://books.google.com/books?id=T7TJYHdhgw8C&pg=PA205}}.</ref> Like cavalier perspective, one face of the projected object is parallel to the viewing plane, and the third axis is projected as going off at an angle (typically {{mono|atan(2)}} or about ~63.4°). Unlike cavalier projection, where the third axis keeps its length, with cabinet projection the length of the receding lines is cut in half.

=== Mathematical formula ===
As a formula, if the plane facing the viewer is ''xy'', and the receding axis is ''z'', then a point ''P'' is projected like this:
:<math> P \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix}
x + \frac12 z \cos \alpha \\ y + \frac12 z \sin \alpha \\  0 \end{pmatrix}</math>
Where <math>\alpha</math> is the mentioned angle.

The [[transformation matrix]] is:
: <math> P = \begin{bmatrix}
1 & 0 & \frac12 \cos \alpha \\
0 & 1 & \frac12 \sin \alpha \\
0 & 0 & 0
\end{bmatrix}</math>

Alternatively one could remove one third from the leading arm projected off the starting face, thus giving the same result.

==Military projection==
In the ''military projection'', the angles of the ''x'' and ''z''-axis and ''y'' and ''z'' -axis are at 45°, meaning that the angle between the ''x''-axis and the ''y''-axis is 90°. That is, the ''xy''-plane is not skewed. It is rotated over 45°, though.<ref>{{cite web | url=http://www.compuphase.com/axometr.htm | title=The Geometry of Perspective Drawing on the Computer | access-date=24 April 2015}}</ref>

==Examples==
Besides technical drawing and illustrations, [[video game]]s (especially those preceding the advent of 3D games) also often use a form of oblique projection. Examples include ''[[SimCity (1989 video game)|SimCity]]'', ''[[Ultima VII]]'', ''[[Ultima Online]]'', ''[[EarthBound]]'', ''[[Paperboy (video game)|Paperboy]]'' and, more recently, ''[[Tibia (video game)|Tibia]]''.
<gallery widths="200px" align="center" style="padding:0px;">
Image:Perspective cavaliere exemple piece revolution.svg|The figures to the left are [[orthographic projection]]s. The figure to the right is an '''oblique projection''' with an angle of 30° and a ratio of {{frac|2}}.
Image:Potting-bench-cabinet-view.png|[[Potting bench]] drawn in '''cabinet projection''' with an angle of 45° and a ratio of 2/3.
Image:Perspective cavaliere fortification.jpg|Pieces of fortification in '''cavalier perspective''' (''[[Cyclopaedia, or an Universal Dictionary of Arts and Sciences|Cyclopaedia]]'' vol. 1, 1728).
Image:Perspective cavaliere report coordonnees 90deg.svg|How the coordinates are used to place a point on a '''cavalier perspective'''.
File:Militärperspektive.PNG|Stone arch drawn in '''military perspective'''.
Image:Cabinet perspective 45.svg|Stone arch drawn in '''cabinet perspective'''.
File:Korean art-Donggwoldo-Changdeokgung and Changgyeonggung-Dong-A University-01.jpg|A representative [[Korea]]n painting depicting the two royal palaces, [[Changdeokgung]] and [[Changgyeonggung]] located in the east of the main palace, [[Gyeongbokgung]].
File:Xu Yang - Entrance and yard of a yamen.jpg|''Entrance and yard of a yamen''. Detail of scroll about [[Suzhou]] by Xu Yang, ordered by the [[Qianlong Emperor]]. 18th century
File:Plan Port-Royal-des-Champs.jpg|18th century plan of [[Port-Royal-des-Champs]] drawn in '''military projection'''
File:SimCity-Indigo.gif|A variation of '''military projection''' is used in the video game ''SimCity''
File:Arteria lusoria MRA MIP-03 - Annotated.jpg|A [[3D rendering|3D rendered]] [[magnetic resonance angiography]], shown in an oblique projection in order to distinguish the [[aberrant subclavian artery]]
</gallery>

==See also==
* [[Space-oblique Mercator projection]]
* [[Oblique Mercator projection]]
* [[Hatsusaburō Yoshida]]
* [[List of art techniques]]

==References==
{{reflist}}

==Further reading==
{{Commons category|Oblique projection}}
{{Commons category|Cabinet projection}}
{{commons category|Cavalier perspective}}
* {{cite book | last = Foley | first = James | title = Computer Graphics | publisher=Addison-Wesley | location = Boston | year = 1997 | isbn = 0-201-84840-6 }}
* Ingrid Carlbom, Joseph Paciorek, Planar Geometric Projections and Viewing Transformations, [[ACM Computing Surveys]], v.10 n.4, p.&nbsp;465–502, Dec. 1978
* Alpha et al. 1988, [https://pubs.usgs.gov/imap/1799/report.pdf ''Atlas of Oblique Maps, A Collection of Landform Portrayals of Selected Areas of the World''] ([[US Geological Survey]])

==External links==
* [https://web.archive.org/web/20100822152816/http://www.tpub.com//content/draftsman/14276/css/14276_308.htm Illustrator Draftsman 3 & 2 – Volume 2 Standard Practices and Theory, page 68] from https://web.archive.org/web/20100822152816/http://www.tpub.com:80/

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[[Category:Graphical projections]]