Oblique projection is a simple type of technical drawing of
graphical projection
A 3D projection (or graphical projection) is a design technique used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These projections rely on visual perspective and aspect analysis to project a complex object f ...
used for producing two-dimensional (2D)
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
s of three-dimensional (3D) objects.
The objects are not in
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.
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

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
Orthographic projection (also orthogonal projection and analemma) 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 ...
, 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
on the
-plane gives
. The constants
and
uniquely specify a parallel projection. When
, the projection is said to be "orthographic" or "orthogonal". Otherwise, it is "oblique". The constants
and
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
Linear or point-projection perspective () is one of two types of 3D projection, graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, generally on a fla ...
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 Pohlke's theorem is the fundamental theorem of axonometry. It was established 1853 by the German painter and teacher of descriptive geometry Karl Wilhelm Pohlke. The first proof of the theorem was published 1864 by the German mathematician Hermann A ...
, from the German mathematician Pohlke, who published it in the early 19th century.
The resulting distortions make the
technique
Technique or techniques may refer to:
Music
* The Techniques, a Jamaican rocksteady vocal group of the 1960s
* Technique (band), a British female synth pop band in the 1990s
* ''Technique'' (album), by New Order, 1989
* ''Techniques'' (album), by ...
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
and
are preserved, but lengths along direction
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
axis remains unscaled.
[Parallel Projections](_blank)
from ''PlaneView3D Online''
* ''Cabinet projection'', popular in furniture illustrations, is an example of such a technique, where in the receding axis is scaled to half-size
(sometimes instead two-thirds the original).
Cavalier projection
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
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
and the length on these axes are drawn with a 1:1 scale; it is thus similar to the
dimetric projection
Axonometric projection is a type of orthographic projection used for creating a pictorial drawing of an object, where the object is rotated around one or more of its axes to reveal multiple sides.Gary R. Bertoline et al. (2002) ''Technical Graph ...
s, although it is not an
axonometric projection
Axonometric projection is a type of orthographic projection used for creating a pictorial drawing of an object, where the object is rotated around one or more of its axes to reveal multiple sides.Gary R. Bertoline et al. (2002) ''Technical Graph ...
, 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.
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
A fortification (also called a fort, fortress, fastness, or stronghold) is a military construction designed for the defense of territories in warfare, and is used to establish rule in a region during peacetime. The term is derived from Lati ...
s. In French, the "cavalier" (literally ''rider, horseman'', see ''
Cavalry
Historically, cavalry (from the French word ''cavalerie'', itself derived from ''cheval'' meaning "horse") are groups of soldiers or warriors who Horses in warfare, fight mounted on horseback. Until the 20th century, cavalry were the most mob ...
'') is an artificial hill behind the walls that allows sighting of the enemy above the walls. 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.
Cabinet projection
The term ''cabinet projection'' stems from its use in illustrations by the furniture industry. 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 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:
:
Where
is the mentioned angle.
The
transformation matrix
In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then there exists an m \times n matrix A, called the transfo ...
is:
:
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.
Examples
Besides technical drawing and illustrations,
video game
A video game or computer game is an electronic game that involves interaction with a user interface or input device (such as a joystick, game controller, controller, computer keyboard, keyboard, or motion sensing device) to generate visual fe ...
s (especially those preceding the advent of 3D games) also often use a form of oblique projection. Examples include ''
SimCity
''SimCity'' is an open-ended city-building video game franchise originally designed by Will Wright. The first game in the series, '' SimCity'', was published by Maxis in 1989 and was followed by several sequels and many other spin-off ''S ...
'', ''
Ultima VII'', ''
Ultima Online'', ''
EarthBound
''EarthBound'', originally released in Japan as is a 1994 role-playing video game, role-playing video game developed by Ape, Inc., Ape Inc. and HAL Laboratory and published by Nintendo for the Super Nintendo Entertainment System as the second e ...
'', ''
Paperboy'' and, more recently, ''
Tibia
The tibia (; : tibiae or tibias), also known as the shinbone or shankbone, is the larger, stronger, and anterior (frontal) of the two Leg bones, bones in the leg below the knee in vertebrates (the other being the fibula, behind and to the outsi ...
''.
Image:Perspective cavaliere exemple piece revolution.svg, The figures to the left are orthographic projection
Orthographic projection (also orthogonal projection and analemma) 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 ...
s. The figure to the right is an oblique projection with an angle of 30° and a ratio of .
Image:Potting-bench-cabinet-view.png, Potting bench Potting may refer to:
* Potting, in pottery, the making of pots, generally on the potter's wheel
* Potting (electronics), the encapsulation of electronic components
* In farming and gardening, potting is planting a plant in a pot, such as a flowe ...
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'' 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
Korea is a peninsular region in East Asia consisting of the Korean Peninsula, Jeju Island, and smaller islands. Since the end of World War II in 1945, it has been politically Division of Korea, divided at or near the 38th parallel north, 3 ...
n painting depicting the two royal palaces, Changdeokgung
Changdeokgung () is a former royal palace in Seoul, South Korea. A UNESCO World Heritage Site and a Historic Sites (South Korea), Historic Site of South Korea, it is among the best preserved of all Korean palaces. It and its neighboring palace Cha ...
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
Suzhou is a major prefecture-level city in southern Jiangsu province, China. As part of the Yangtze Delta megalopolis, it is a major economic center and focal point of trade and commerce.
Founded in 514 BC, Suzhou rapidly grew in size by the ...
by Xu Yang, ordered by the Qianlong Emperor
The Qianlong Emperor (25 September 17117 February 1799), also known by his temple name Emperor Gaozong of Qing, personal name Hongli, was the fifth Emperor of China, emperor of the Qing dynasty and the fourth Qing emperor to rule over China pr ...
. 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''
See also
*
Space-oblique Mercator projection
*
Oblique Mercator projection
*
Hatsusaburō Yoshida
was a Japanese cartographer and artist, known by his bird's-eye view maps of cities and towns. Known as the "Hiroshige of the Taisho Era," Yoshida created over 3,000 maps in his lifetime.
Biography
Yoshida was born in Kyoto in 1884 as Hatsusabu ...
*
List of art techniques There is no exact definition of what constitutes art. Artists have explored many styles and have used many different techniques to create art.
Art techniques
A
* Airbrush#Technique, Airbrushing technique
* Aerial_perspective#In_art, Aerial pe ...
References
Further reading
*
* Ingrid Carlbom, Joseph Paciorek, Planar Geometric Projections and Viewing Transformations,
ACM Computing Surveys
''ACM Computing Surveys'' is peer-reviewed quarterly scientific journal and is published by the Association for Computing Machinery. It publishes survey articles and tutorials related to computer science and computing. The journal was established i ...
, v.10 n.4, p. 465–502, Dec. 1978
* Alpha et al. 1988
''Atlas of Oblique Maps, A Collection of Landform Portrayals of Selected Areas of the World''(
US Geological Survey
The United States Geological Survey (USGS), founded as the Geological Survey, is an agency of the U.S. Department of the Interior whose work spans the disciplines of biology, geography, geology, and hydrology. The agency was founded on March ...
)
External links
Illustrator Draftsman 3 & 2 – Volume 2 Standard Practices and Theory, page 68from https://web.archive.org/web/20100822152816/http://www.tpub.com:80/
{{DEFAULTSORT:Oblique Projection
Graphical projections