Axonometric projection

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Axonometric projection is a type of orthographic projection used for creating a pictorial drawing of an object, where the lines of sight are perpendicular to the plane of projection, and the object is rotated around one or more of its axes to reveal multiple sides.[1]

Overview[edit]

"Axonometry" 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 of the object in the same picture. Whereas multiview projections involve depictions of objects from the six primary views (e.g. front, right, left, top, bottom, or back) in which the principal axes or planes of the object are parallel with the projection plane (thus revealing only one side of an object at a time), in axonometric projection the principal planes or axes of the object are always drawn not parallel to the projection plane.

With axonometric projections the scale of distant features is the same as for near features, so such pictures will look distorted, as it is not how human vision or photography works. This distortion, the direct result of a presence or absence of foreshortening, 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.

In German literature, axonometry is more generally defined based on Pohlke's theorem, the fundamental theorem of axonometry. Thus, in German literature, the scope of axonometric projection is broadened to encompass all types of parallel projection.

Like oblique projection, axonometric projection is typically used to create pictorials. In the context of multiviews, axonometric pictorials are sometimes known as auxiliary views.

Three types[edit]

Comparison of several types of graphical projection
Various projections and how they are produced
The three axonometric views. The percentages show the amount of foreshortening.

The three types of axonometric projection are isometric projection, dimetric projection, and trimetric projection, depending on the exact angle at which the view deviates from the orthogonal.[2][3] Typically in axonometric drawing, as in other types of pictorials, one axis of space is shown as the vertical.

In isometric projection, the most commonly used form of axonometric projection in engineering drawing,[4] 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 of all sides and lengths are preserved, and the axes share a common scale. This enables measurements to be read or taken directly from the drawing. Another advantage is that 120° 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. Approximations are common in dimetric drawings.[clarification needed]

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. Approximations in trimetric drawings are common,[clarification needed] and trimetric perspective is seldom used.[3]

History[edit]

The concept of isometry had existed in a rough empirical form for centuries, well before Professor William Farish (1759–1837) of Cambridge University was the first to provide detailed rules for isometric drawing.[5][6]

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".[7]

From the middle of the 19th century, according to Jan Krikke (2006)[7] 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 and De Stijl embraced it".[7] De Stijl architects like Theo van Doesburg used axonometry for their architectural designs, which caused a sensation when exhibited in Paris in 1923".[7]

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 systems and other visual computing tools.[8]

According to Jan Krikke (2000), "axonometry originated in China. 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".[8][clarification needed]

Limitations[edit]

In this 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.
The Penrose stairs depicts a staircase which seems to ascend (anticlockwise) or descend (clockwise) yet forms a continuous loop.

As with all 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, 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 op art, as well as "impossible object" drawings. Though not strictly axonometric, M. C. Escher's Waterfall (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.

References[edit]

  1. ^ Gary R. Bertoline et al. (2002) Technical Graphics Communication. McGraw–Hill Professional, 2002. ISBN 0-07-365598-8, p. 330.
  2. ^ Maynard, Patric (2005). Drawing distinctions: the varieties of graphic expression. Cornell University Press. p. 22. ISBN 0-8014-7280-6. 
  3. ^ a b McReynolds, Tom; David Blythe (2005). Advanced graphics programming using openGL. Elsevier. p. 502. ISBN 1-55860-659-9. 
  4. ^ Godse, A. P. (1984). Computer graphics. Technical Publications. p. 29. ISBN 81-8431-558-9. 
  5. ^ Barclay G. Jones (1986). Protecting historic architecture and museum collections from natural disasters. University of Michigan. ISBN 0-409-90035-4. p. 243.
  6. ^ Charles Edmund Moorhouse (1974). Visual messages: graphic communication for senior students.
  7. ^ a b c d J. Krikke (1996). "A Chinese perspective for cyberspace?". In: International Institute for Asian Studies Newsletter, 9, Summer 1996.
  8. ^ a b Jan Krikke (2000). "Axonometry: a matter of perspective". In: Computer Graphics and Applications, IEEE Jul/Aug 2000. Vol 20 (4), pp. 7–11.
  9. ^ William Farish (1822) "On Isometrical Perspective". In: Cambridge Philosophical Transactions. 1 (1822).

Further reading[edit]

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