Axonometric projection
Encyclopedia
Axonometric projection is a type of parallel projection
Parallel projection
Parallel projections have lines of projection that are parallel both in reality and in the projection plane.Parallel projection corresponds to a perspective projection with an infinite focal length , or "zoom".Within parallel projection there is an ancillary category known as "pictorials"...

, more specifically a type of orthographic projection
Orthographic projection
Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...

, used to create a pictorial drawing of an object, where the object is rotated along one or more of its axes relative to the plane of projection.

There are three main types of axonometric projection: isometric, dimetric, and trimetric projection.

"Axonometric" means "to measure along axes". Axonometric projection shows an image of an object as viewed from a skew direction in order to reveal more than one side in the same picture.Whereas the term orthographic is sometimes reserved specifically for depictions of objects where the axis or plane of the object is parallel with the projection plane, in axonometric projection the plane or axis of the object is always drawn not parallel to the projection plane.

With axonometric projections the scale of distant features is the same as for near features, such pictures will look distorted, as it is not how our eyes or photography work. This distortion is especially evident if the object to view is mostly composed of rectangular features. Despite this limitation, axonometric projection can be useful for purposes of illustration.

History

The concept of an isometric
Isometric
The term isometric comes from the Greek for "having equal measurement".isometric may mean:* Isometric projection , a method for the visual representation of three-dimensional objects in two dimensions; a form of orthographic projection, or more specifically, an axonometric projection.* Isometry and...

 had existed in a rough empirical form for centuries, well before Professor William Farish
William Farish (professor)
William Farish was a British scientist who was a professor of Chemistry and Natural Philosophy at the University of Cambridge, known for the development of the method of isometric projection and development of the first written university examination.- Biography :Farish's father was the Reverend...

 (1759–1837) of Cambridge University was the first to provide detailed rules for isometric drawing.

Farish published his ideas in the 1822 paper "On Isometrical 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".

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 and the U.S. The popular acceptance of axonometry came in the 1920s, when modernist architects from the Bauhaus
Bauhaus
', commonly known simply as Bauhaus, was a school in Germany that combined crafts and the fine arts, and was famous for the approach to design that it publicized and taught. It operated from 1919 to 1933. At that time the German term stood for "School of Building".The Bauhaus school was founded by...

 and De Stijl
De Stijl
De Stijl , propagating the group's theories. Next to van Doesburg, the group's principal members were the painters Piet Mondrian , Vilmos Huszár , and Bart van der Leck , and the architects Gerrit Rietveld , Robert van 't Hoff , and J.J.P. Oud...

 embraced it". De Stijl architects like Theo van Doesburg
Theo van Doesburg
Theo van Doesburg was a Dutch artist, practicing in painting, writing, poetry and architecture. He is best known as the founder and leader of De Stijl.-Biography:-Early life:...

 used axonometry for their architectural design
Architectural Design
Architectural Design, also known as AD, is a UK-based architectural journal first launched in 1930.In its early days it was more concerned with the British scene, but gradually became more international. It also moved away from presenting mostly news towards theme-based issues...

s, which caused a sensation when exhibited in Paris 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 3D space on the 2D picture plane. It usually comes as a standard feature of CAD systems and other visual computing tools.

According to Jan Krikke (2000) however, "axonometry originated in China
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...

. Its function in Chinese art was similar to linear perspective in European art. Axonometry, and the pictorial grammar that goes with it, has taken on a new significance with the advent of visual computing".

Three types of axonometric projections

The three types of axonometric projections are isometric projection
Isometric projection
Isometric projection is a method for visually representing three-dimensional objects in two dimensions in technical and engineering drawings...

, dimetric projection, and trimetric projection, depending on the exact angle at which the view deviates from the orthogonal. Typically in axonometric drawing, one axis of space is shown as the vertical.
  • In isometric projection
    Isometric projection
    Isometric projection is a method for visually representing three-dimensional objects in two dimensions in technical and engineering drawings...

    , 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, of which the displayed angles among them and also the scale of foreshortening are universally known. However in creating a final, isometric instrument drawing, in most cases a full-size scale, i.e., without using a foreshortening factor, is employed to good effect because the resultant distortion is difficult to perceive. Another advantage is that, in engineering drawings, 60° angles are more easily constructed using only a 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 (vertical) 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, and is found in only a few video games
    Video games with isometric graphics
    In video games, "isometric" refers to some form of parallel projection where the viewpoint is rotated slightly to reveal other facets of the game environment than are visible from a top-down perspective or side view, thereby producing a three-dimensional effect...

     (Fallout
    Fallout (computer game)
    Fallout is a computer role-playing game produced by Tim Cain, developed and published by Interplay in 1997. The game has a post-apocalyptic and retro-futuristic setting in the mid-22nd century, featuring an alternate history which deviates some time after World War II, where technology, politics...

    , SimCity 4
    SimCity 4
    SimCity 4 is a city-building/urban planning simulation computer game developed by Maxis, a subsidiary of Electronic Arts. It was released on January 14, 2003. It is the fourth installment in the SimCity series. SimCity 4 has a single expansion pack called Rush Hour which adds features to the game...

    ).


Approximations are common in dimetric and trimetric drawings.

Limitations of axonometric projection


As with all types of parallel projection
Parallel projection
Parallel projections have lines of projection that are parallel both in reality and in the projection plane.Parallel projection corresponds to a perspective projection with an infinite focal length , or "zoom".Within parallel projection there is an ancillary category known as "pictorials"...

, objects drawn with axonometric projection do not appear larger or smaller as they extend closer to or away from the viewer. While advantageous for architectural drawing
Architectural drawing
An architectural drawing or architect's drawing is a technical drawing of a building that falls within the definition of architecture...

s, where measurements must be taken directly from the image, the result is a perceived distortion, since unlike perspective projection, this is not how our eyes or photography normally work. It also can easily result in situations where depth and altitude are difficult to gauge, as is shown in the illustration to the right.

In this isometric drawing, the blue sphere is two units higher than the red one. However, this difference in elevation is not apparent if one covers the right half of the picture, as the boxes (which serve as clues suggesting height) are then obscured.

This visual ambiguity has been exploited in op art
Op art
Op art, also known as optical art, is a style of visual art that makes use of optical illusions."Optical art is a method of painting concerning the interaction between illusion and picture plane, between understanding and seeing." Op art works are abstract, with many of the better known pieces made...

, including "impossible object" drawings. M. C. Escher
M. C. Escher
Maurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...

's Waterfall
Waterfall (M. C. Escher)
Waterfall is a lithography print by the Dutch artist M. C. Escher which was first printed in October, 1961. It shows an apparent paradox where water from the base of a waterfall appears to run downhill before reaching the top of the waterfall....

(1961) is a well-known example, 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
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...

.

Further reading

  • Yve-Alain Bois, "Metamorphosis of Axonometry," Daidalos, no. 1 (1981), pp. 41–58
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