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Axonometric projection is a type of
orthographic projection Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogona ...
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, such that the scope of axonometric projection could encompass ''every'' type of parallel projection, including not only
orthographic projection Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogona ...
(and multiview projection), 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 refle ...
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 emplo ...
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, 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

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, 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 List of countries and dependencies by population, most populous country, with a Population of China, population exceeding 1.4 billion, slig ...
. Unlike the linear perspective 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'' ...
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 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., 20 ...
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 ...
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 Nell ...
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 complica ...
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. Si ...
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 c ...
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 ...
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 comp ...
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 o ...
. File:Optical-grinding engine model.jpg, Optical-grinding engine model (1822), drawn in 30° isometric perspectiveWilliam 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 List of cities in China, city and Special administrative regions of China, special ...
. 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 ...
'', 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'' 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, 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 Op art, short for optical art, is a style of visual art that uses optical illusions. Op artworks are abstract, with many better-known pieces created in black and white. Typically, they give the viewer the impression of movement, hidden image ...
, 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 ...
'' (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