Axonometric projection is a type of parallel projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical, more specifically a type of orthographic projection Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where the view direction is orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface. It is further divided into multiview orthographic, 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.[1]

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

Contents

Overview

"Axonometric" means "to measure along axes".[2] 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.[3] 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,[4] in axonometric projection the plane or axis of the object is always drawn not parallel to the projection plane.[5][6][7]

Because 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.[8]

History

Optical-grinding engine model (1822), drawn in 30° isometric perspective.[9]

The concept of an isometric had existed in a rough empirical form for centuries, well before Professor William Farish William Farish was a British professor in 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 (1759-1837) of Cambridge University The University of Cambridge is the second oldest university in England and the fourth oldest in the world. In post-nominals the university's name is abbreviated as Cantab, a shortened form of Cantabrigiensis (an adjective derived from Cantabrigia, the Latinised form of Cambridge) was the first to provide detailled rules for isometric drawing.[10][11]

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".[12]

Axonometry in medieval Asian art.

From the middle of the 19th century, according to Jan Krikke (2006)[12] 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 Staatliches 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. The term Bauhaus (help·info) is German for ("House of Building" or "Building School") and De Stijl De Stijl , Dutch for "The Style", also known as neoplasticism, was a Dutch artistic movement founded in 1917. In a narrower sense, the term De Stijl is used to refer to a body of work from 1917 to 1931 founded in the Netherlands. De Stijl is also the name of a journal that was published by the Dutch painter, designer, writer, and critic embraced it".[12] De Stijl architects like 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 used axonometry for their architectural designs Architectural design values make up an important part of what influences an architect and designer when they make their design decisions. However, architects and designers are not always influenced by the same values and intentions. Value and intentions differ between different architectural movements. It also differs between different schools of, which caused a sensation when exhibited in Paris in 1923".[12] 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.[13]

According to Jan Krikke (2000)[13] however, "axonometry originated in China China is seen variously as an ancient civilization extending over a large area in East Asia, a nation and/or a multinational entity. Its function in Chinese art was similar to linear perspective Perspective in the graphic arts, such as drawing, is an approximate representation, on a flat surface (such as paper), of an image as it is seen by the eye. The two most characteristic features of perspective are that objects are drawn: in European art. Axonometry, and the pictorial grammar that goes with it, has taken on a new significance with the advent of visual computing".[13]

Three types of axonometric projections

Comparison of several types of graphical projection Graphical projection is a protocol by which an image of an three-dimensional object is projected onto a planar surface without the aid of mathematical calculation, used in technical drawing.

The three types of axonometric projections are isometric projection Isometric projection is a form of graphical projection, more specifically, a form of axonometric projection. It is a method of visually representing three-dimensional objects in two dimensions, in which the three coordinate axes appear equally foreshortened and the angles between any two of them are 120 degrees, dimetric projection, and trimetric projection, depending on the exact angle at which the view deviates from the orthogonal.[4][7] Typically in axonometric drawing, one axis of space is shown as the vertical.

Example of a dimetric axonometric drawing from a US Patent (1874).

Approximations are common in dimetric and trimetric drawings.[clarification needed]

Limitations of axonometric projection

Example of limitations. See also: Impossible object

As with all types of parallel projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical, 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 drawings An architectural drawing or architect's drawing is a technical drawing of a building that falls within the definition of architecture. Architectural drawings are used by architects and others for a number of purposes: to develop a design idea into a coherent proposal, to communicate ideas and concepts, to convince clients of the merits of a design,, where measurements must be taken directly from the image, the result is a perceived distortion, since unlike perspective projection 3D projection is any method of mapping three-dimensional points to a two-dimensional plane. As most current methods for displaying graphical data are based on planar two-dimensional media, the use of this type of projection is widespread, especially in computer graphics, engineering and drafting, 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, also known as optical art, is a style of visual art that makes use of optical illusions, including "impossible object" drawings. M. C. Escher Maurits Cornelis Escher , usually referred to as M. C. Escher (English pronunciation: /ˈɛʃər/, Dutch: [ˈmʌurɪts kɔrˈneːlɪs ˈɛʃər] ( listen)), was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints. These feature impossible constructions, explorations of infinity,'s Waterfall Waterfall is a lithograph 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 uphill before reaching the top of the waterfall (1961) is a well-known example, in which a channel of water seems to travel unaided along an upward path, only to paradoxically fall once again as it returns to its point of origin.

References

This article needs additional citations for verification. Please help improve this article by adding reliable references. Unsourced material may be and removed. (November 2009)
  1. ^ Gary R. Bertoline et al. (2002) Technical Graphics Communication‎. McGraw-Hill Professional, 2002. ISBN 0073655988, p.330.
  2. ^ Etymology from yahoo.com Yahoo! Inc. is an American public corporation headquartered in Sunnyvale, California, (in Silicon Valley), that provides Internet services worldwide. The company is perhaps best known for its web portal, search engine (Yahoo! Search), Yahoo! Directory, Yahoo! Mail, Yahoo! News, advertising, online mapping (Yahoo! Maps), video sharing (Yahoo! Video)
  3. ^ Mitchell, William; Malcolm McCullough (1994). Digital design media. John Wiley and Sons. pp. 169. ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 0471286664. http://books.google.com/books?id=JrZoGgQEhKkC&pg=PA169&dq=axonometric+orthographic#v=onepage&q=axonometric%20orthographic&f=false.
  4. ^ a b Maynard, Patric (2005). Drawing distinctions: the varieties of graphic expression. Cornell University Press. pp. 22. ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 0801472806. http://books.google.com/books?id=4Y_YqOlXoxMC&pg=PA22&lpg=PA22&dq=axonometric+orthographic&source=bl&ots=wqg9m-MaKI&sig=xZ9MEJ8JjFcg28YefGxBW4zcZ2Y&hl=en&ei=Tjj2SvO_CM-n8AaI8bTzCQ&sa=X&oi=book_result&ct=result&resnum=6&ved=0CBoQ6AEwBQ#v=twopage&q=axonometric%20orthographic&f=false.
  5. ^ Desai, Apurva A.. Computer Graphics. PHI Learning Pvt. Ltd.. pp. 232. ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 8120335244. http://books.google.com/books?id=WQiIj8ZS0IoC&pg=PA231&dq=axonometric+orthographic#v=onepage&q=axonometric%20orthographic&f=false.
  6. ^ a b Godse, A. P. (1980). Computer graphics. Technical Publications. pp. 29. ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 8184315589. http://books.google.com/books?id=YkVp-2ZrmyMC&pg=PT224&dq=axonometric+orthographic#v=onepage&q=axonometric%20orthographic&f=false.
  7. ^ a b c McReynolds, Tom; David Blythe (2005). Advanced graphics programming using openGL. Elsevier. pp. 502. ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 1558606599. http://books.google.com/books?id=H4eYq7-2YhYC&pg=PA502&dq=axonometric+orthographic#v=onepage&q=axonometric%20orthographic&f=false.
  8. ^ Ingrid Carlbom, Joseph Paciorek (December 1978), "Planar Geometric Projections and Viewing Transformations", ACM Computing Surveys 10 (4): 465–502, doi A digital object identifier is a character string used to uniquely identify an electronic document or other object. Metadata about the object is stored in association with the DOI name and this metadata may include a location, such as a URL, where the object can be found. The DOI for a document is permanent, whereas its location and other metadata:10.1145/356744.356750
  9. ^ William Farish (1822) "On Isometrical Perspective". In: Cambridge Philosophical Transactions. 1 (1822).
  10. ^ Barclay G. Jones (1986). Protecting historic architecture and museum collections from natural disasters. University of Michigan. ISBN 0409900354. p.243.
  11. ^ Charles Edmund Moorhouse (1974). Visual messages: graphic communication for senior students‎.
  12. ^ a b c d J. Krikke (1996). "A Chinese perspective for cyberspace?". In: International Institute for Asian Studies Newsletter, 9, Summer 1996.
  13. ^ a b c Jan Krikke (2000). "Axonometry: a matter of perspective". In: Computer Graphics and Applications, IEEE Jul/Aug 2000. Vol 20 (4), pp. 7-11.

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