Axonometric projection is a type of
orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Two-dimensional space, two dimensions. Orthographic projection is a form of parallel projection in ...
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 Graphics Communication''. McGraw–Hill Professional, 2002. , p. 330.]
Overview
"Axonometry" means "to measure along the axes". In German literature,
axonometry
Axonometry is a graphical procedure belonging to descriptive geometry that generates a planar image of a three-dimensional object. The term "axonometry" means "to measure along axes", and indicates that the dimensions and scaling of the coord ...
is based on
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 ...
, such that the scope of axonometric projection could encompass ''every'' type of
parallel projection
In three-dimensional geometry, a parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane, known as the ''projection plane'' or '' image plane'', where the ''rays'', known as '' li ...
, including not only
orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Two-dimensional space, two dimensions. Orthographic projection is a form of parallel projection in ...
(and
multiview projection
In technical drawing and computer graphics, a multiview projection is a technique of illustration by which a standardized series of orthographic two-dimensional pictures are constructed to represent the form of a three-dimensional object. Up to ...
), but also
oblique projection
Oblique projection is a simple type of technical drawing of graphical projection used for producing two-dimensional (2D) images of three-dimensional (3D) objects.
The objects are not in perspective and so do not correspond to any view of an ...
. However, outside of German literature, the term "axonometric" is sometimes used only to distinguish between orthographic views where the principal axes of an object are ''not'' orthogonal to the projection plane, and orthographic views in which the principal axes of the object ''are'' orthogonal to the projection plane. (In multiview projection these would be called ''auxiliary views'' and ''primary views'', respectively.) Confusingly, the term "orthographic projection" is also sometimes reserved only for the primary views.
Thus, in German literature, "axonometric projection" might be considered synonymous with "parallel projection", overall; but in English literature, an "axonometric projection" might be considered synonymous with an "auxiliary view" (versus a "primary view") in a "multiview orthographic projection".
With an axonometric projection, the scale of an object does not depend on its location (i.e., an object in the "foreground" has the same scale as an object in the "background"); consequently, such pictures look distorted, as
human vision
Visual perception is the ability to interpret the surrounding environment through photopic vision (daytime vision), color vision, scotopic vision (night vision), and mesopic vision (twilight vision), using light in the visible spectrum reflec ...
and
photography
Photography is the art, application, and practice of creating durable images by recording light, either electronically by means of an image sensor, or chemically by means of a light-sensitive material such as photographic film. It is employed ...
use
perspective projection
Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
, in which the perceived scale of an object depends on its distance and location from the viewer. This distortion, the direct result of a presence or absence of
foreshortening
Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
, is especially evident if the object is mostly composed of rectangular features. Despite this limitation, axonometric projection can be useful for purposes of illustration, especially because it allows for simultaneously relaying precise measurements.
Three types
![Various projections of cube above plane](https://upload.wikimedia.org/wikipedia/commons/4/4b/Various_projections_of_cube_above_plane.svg)
The three types of axonometric projection are ''
isometric projection
Isometric projection is a method for visually representing three-dimensional objects in two dimensions in technical and engineering drawings. It is an axonometric projection in which the three coordinate axes appear equally foreshortened and ...
'', ''dimetric projection'', and ''trimetric projection'', depending on the exact angle by which the view deviates from the
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
.
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,
the direction of viewing is such that the three axes of space appear equally
foreshortened
Linear or point-projection perspective (from la, perspicere 'to see through') 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 r ...
, 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
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
.
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. Dimensional approximations are common in dimetric drawings.
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. Dimensional approximations in trimetric drawings are common, and trimetric perspective is seldom used in technical drawings.
History
Axonometry originated in
China
China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's most populous country, with a population exceeding 1.4 billion, slightly ahead of India. China spans the equivalent of five time zones and ...
.
[ Unlike the ]linear perspective
Linear or point-projection perspective (from la, perspicere 'to see through') 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 r ...
in European art whose perspective was objective, or looking from the outside, Chinese art used parallel projections within the painting that allowed the viewer to consider both the space and the ongoing progression of time in one scroll. The concept of isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
had existed in a rough empirical form for centuries, well before Professor William Farish William Farish may refer to:
Senin varyoxunu... Dalbayov Gicdıllaq
* William Farish (chemist) (1759–1837), tutor at the University of Cambridge
* William Stamps Farish I (1843–1899)
* William Stamps Farish II (1881–1942), Standard Oil preside ...
(1759–1837) of Cambridge University
, mottoeng = Literal: From here, light and sacred draughts.
Non literal: From this place, we gain enlightenment and precious knowledge.
, established =
, other_name = The Chancellor, Masters and Schola ...
was the first to provide detailed rules for isometric drawing.
Farish published his ideas in the 1822 paper "On Isometric Perspective", in which he recognized the "need for accurate technical working drawings free of optical distortion. This would lead him to formulate isometry. Isometry means "equal measures" because the same scale is used for height, width, and depth".[J. Krikke (1996).]
A Chinese perspective for cyberspace?
". In: ''International Institute for Asian Studies Newsletter'', 9, Summer 1996.
From the middle of the 19th century, according to Jan Krikke (2006) isometry became an "invaluable tool for engineers, and soon thereafter axonometry and isometry were incorporated in the curriculum of architectural training courses in Europe
Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a Continent#Subcontinents, subcontinent of Eurasia ...
and the U.S.
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territori ...
The popular acceptance of axonometry came in the 1920s, when modernist architects
Modern architecture, or modernist architecture, was an architectural movement or architectural style based upon new and innovative technologies of construction, particularly the use of glass, steel, and reinforced concrete; the idea that form ...
from the Bauhaus
The Staatliches Bauhaus (), commonly known as the Bauhaus (), was a German art school operational from 1919 to 1933 that combined crafts and the fine arts.Oxford Dictionary of Art and Artists (Oxford: Oxford University Press, 4th edn., 200 ...
and De Stijl
''De Stijl'' (; ), Dutch for "The Style", also known as Neoplasticism, was a Dutch art movement founded in 1917 in Leiden. De Stijl consisted of artists and architects. In a more narrow sense, the term ''De Stijl'' is used to refer to a body o ...
embraced it". De Stijl architects like Theo van Doesburg
Theo van Doesburg (, 30 August 1883 – 7 March 1931) was a Dutch artist, who practiced painting, writing, poetry and architecture. He is best known as the founder and leader of De Stijl. He was married to artist, pianist and choreographer Nelly ...
used axonometry for their architectural design
Building design refers to the broadly based architectural, engineering and technical applications to the design of buildings. All building projects require the services of a building designer, typically a licensed architect. Smaller, less complic ...
s, which caused a sensation when exhibited in Paris
Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. S ...
in 1923".
Since the 1920s axonometry, or parallel perspective, has provided an important graphic technique for artists, architects, and engineers. Like linear perspective, axonometry helps depict three-dimensional space on a two-dimensional picture plane. It usually comes as a standard feature of CAD
Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve co ...
systems and other visual computing tools.[Jan Krikke (2000). "Axonometry: a matter of perspective". In: ''Computer Graphics and Applications, IEEE'' Jul/Aug 2000. Vol 20 (4), pp. 7–11.] According to science author and Medium
Medium may refer to:
Science and technology
Aviation
*Medium bomber, a class of war plane
*Tecma Medium, a French hang glider design
Communication
* Media (communication), tools used to store and deliver information or data
* Medium of ...
journalist Jan Krikke, axonometry, and the pictorial grammar that goes with it, has taken on a new significance with the introduction of visual computing
Visual computing is a generic term for all computer science disciplines dealing with images and 3D models, such as computer graphics, image processing, visualization, computer vision, virtual and augmented reality and video processing. Visual compu ...
and engineering drawing
An engineering drawing is a type of technical drawing that is used to convey information about an object. A common use is to specify the geometry necessary for the construction of a component and is called a detail drawing. Usually, a number of ...
.
File:Optical-grinding engine model.jpg, Optical-grinding engine model (1822), drawn in 30° isometric perspective[William Farish (1822) "On Isometrical Perspective". In: ''Cambridge Philosophical Transactions''. 1 (1822).]
File:Axonometric example.gif, Example of a dimetric perspective drawing from a US Patent (1874)
File:Bank_of_China_Tower_massing_model.svg, Example of a trimetric projection showing the shape of the Bank of China Tower in Hong Kong
Hong Kong ( (US) or (UK); , ), officially the Hong Kong Special Administrative Region of the People's Republic of China ( abbr. Hong Kong SAR or HKSAR), is a city and special administrative region of China on the eastern Pearl River Delt ...
.
File:Sanguo2.PNG, Example of isometric projection in Chinese art in an illustrated edition of the ''Romance of the Three Kingdoms
''Romance of the Three Kingdoms'' () is a 14th-century historical novel attributed to Luo Guanzhong. It is set in the turbulent years towards the end of the Han dynasty and the Three Kingdoms period in Chinese history, starting in 184 AD and ...
'', China, c. 15th century CE.
File:Along the River During the Qingming Festival (detail of original).jpg, Detail of the original version of ''Along the River During the Qingming Festival
''Along the River During the Qingming Festival'' (''Qingming Shanghe Tu'') is a handscroll painting by the Song dynasty painter Zhang Zeduan (1085–1145) and copied many times in the following centuries. It captures the daily life of people and ...
'' attributed to Zhang Zeduan (1085–1145). Note that the picture switches back and forth between axonometric and perspective projection in different parts of the image, and is thus inconsistent.
Limitations
As with other types of parallel projection
In three-dimensional geometry, a parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane, known as the ''projection plane'' or '' image plane'', where the ''rays'', known as '' li ...
, objects drawn with axonometric projection do not appear larger or smaller as they lie closer to or farther away from the viewer. While advantageous for architectural drawings, where measurements must be taken directly from the image, the result is a perceived distortion, since unlike perspective projection
Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
, this is not how human vision or photography normally works. It also can easily result in situations where depth and altitude are difficult to gauge, as is shown in the illustration to the right.
This visual ambiguity has been exploited in optical art, as well as "impossible object" drawings. Though not strictly axonometric, M. C. Escher
Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, and mezzotints.
Despite wide popular interest, Escher was for most of his life neglected in t ...
's ''Waterfall
A waterfall is a point in a river or stream where water flows over a vertical drop or a series of steep drops. Waterfalls also occur where meltwater drops over the edge of a tabular iceberg or ice shelf.
Waterfalls can be formed in several wa ...
'' (1961) is a well-known image, in which a channel of water seems to travel unaided along a downward path, only to then paradoxically fall once again as it returns to its source. The water thus appears to disobey the law of conservation of energy
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means that ...
.
References
*
Further reading
* Yve-Alain Bois, "Metamorphosis of Axonometry," ''Daidalos'', no. 1 (1981), pp. 41–58
{{visualization
Graphical projections