Isometric projection

From Wikipedia, the free encyclopedia

Jump to: navigation, search
Part of a series on:
Graphical projection
v • d • e

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.

Isometric projection is one of the projections used in technical and engineering drawings.

Contents

[edit] Overview

The term "isometric" comes from the Greek for "equal measure", reflecting that the scale along each axis of the projection is the same (this is not true of some other forms of graphical projection).

Isometric drawing of a cube.

An isometric view of an object can be obtained by choosing the viewing direction in a way that the angles between the projection of the x, y, and z axes are all the same, or 120°. For example when taking a cube, this is done by first looking straight towards one face. Next the cube is rotated ±45° about the vertical axis, followed by a rotation of approximately ±35.264° (precisely arcsin(tan 30°) ) about the horizontal axis. Note that with the cube (see image) the perimeter of the 2D drawing is a perfect regular hexagon: all the black lines are of equal length and all the cube's faces are the same area.

In a similar way an isometric view can be obtained for example in a 3D scene editor. Starting with the camera aligned parallel to the floor and aligned to the coordinate axes, it is first rotated downwards around the horizontal axes by about 35.264° as above, and then rotated ±45° around the vertical axes.

Another way in which isometric projection can be visualized is by considering a view within a cubical room starting in an upper corner and looking towards the opposite, lower corner. The x-axis extends diagonally down and right, the y-axis extends diagonally down and left, and the z-axis is straight up. Depth is also shown by height on the image. Lines drawn along the axes are at 120° to one another.

[edit] History

Optical-grinding engine model (1822), drawn in 30° isometric.[1]

Although the concept of an isometric had existed in a rough way for centuries, Professor William Farish (1759-1837) of Cambridge University was the first to provide rules for isometric drawing.[2] Isometric projection seems to have been known, in an empirical form at least, well before his time.[3]

In the 1822 paper "On Isometrical Perspective" Farish 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".[4]

From the middle of the 19th century, according to Jan Krikke (2006)[4] 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".[4] De Stijl architects like Theo van Doesburg used axonometry for their architectural designs, which caused a sensation when exhibited in Paris in 1923".[4] 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.[5]

According to Jan Krikke (2000)[5] however, 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.[5]

[edit] Mathematics

There are 8 different orientations to obtain an isometric view, depending into which octant the viewer looks. The isometric transform from a point ax,y,z in 3D space to a point bx,y in 2D space looking into the first octant can be written mathematically with rotation matrices as:

\begin{bmatrix} \mathbf{c}_x \\ \mathbf{c}_y \\ \mathbf{c}_z \\ \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ 0 & {\cos\alpha} & {\sin\alpha} \\ 0 & { - \sin\alpha} & {\cos\alpha} \\ \end{bmatrix}\begin{bmatrix} {\cos\beta } & 0 & { - \sin\beta } \\ 0 & 1 & 0 \\ {\sin\beta } & 0 & {\cos\beta } \\ \end{bmatrix}\begin{bmatrix} \mathbf{a}_x \\ \mathbf{a}_y \\ \mathbf{a}_z \\ \end{bmatrix}=\frac{1}{\sqrt{6}}\begin{bmatrix} \sqrt{3} & 0 & -\sqrt{3} \\ 1 & 2 & 1 \\ \sqrt{2} & -\sqrt{2} & \sqrt{2} \\ \end{bmatrix}\begin{bmatrix} \mathbf{a}_x \\ \mathbf{a}_y \\ \mathbf{a}_z \\ \end{bmatrix}

where \alpha = \arcsin(\tan30^\circ)\approx35.264^\circ and \beta = 45^\circ. As explained above, this is a rotation around the vertical (here y) axis by β, followed by a rotation around the horizontal (here x) axis by α. This is then followed by an orthographic projection to the x-y plane:

\begin{bmatrix} \mathbf{b}_x \\ \mathbf{b}_y \\ 0 \\ \end{bmatrix}= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}\begin{bmatrix} \mathbf{c}_x \\ \mathbf{c}_y \\ \mathbf{c}_z \\ \end{bmatrix}

The other seven possibilities are obtained by either rotating to the opposite sides or not, and then inverting the view direction or not.[6]

[edit] Limits of axonometric projection

Example of limitations

As with all types of parallel projection, 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, this results in a perceived distortion, as unlike perspective projection, it 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.

Once popular in sprite-based video games, most contemporary video games have avoided these sorts of situations by switching over to perspective 3D instead.

Such illusions were also popular in Op art such as M. C. Escher's "impossible architecture" drawings. Waterfall (1961), in which the drawing of the building makes use of axonometric projection, but the faded background uses perspective projection, is a well-known example. Another advantage is that, in engineering drawings, 60° angles are easier for novices to construct using only a compass and straightedge.

In this isometric drawing for example, the blue sphere is two levels 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.

[edit] See also

[edit] References

  1. ^ William Farish (1822) "On Isometrical Perspective". In: Cambridge Philosophical Transactions. 1 (1822).
  2. ^ Barclay G. Jones (1986). Protecting historic architecture and museum collections from natural disasters. University of Michigan. ISBN 0409900354. p.243.
  3. ^ Charles Edmund Moorhouse (1974). Visual messages: graphic communication for senior students‎.
  4. ^ a b c d J. Krikke (1996). "A Chinese perspective for cyberspace?". In: International Institute for Asian Studies Newsletter, 9, Summer 1996.
  5. ^ 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.
  6. ^ Ingrid Carlbom, Joseph Paciorek (December 1978). "Planar Geometric Projections and Viewing Transformations". ACM Computing Surveys (CSUR) (ACM) 10 (4): 465–502. doi:10.1145/356744.356750. 

[edit] External links

Search Wikimedia Commons Wikimedia Commons has media related to: Isometric projection
Retrieved from "http://en.wikipedia.org/wiki/Isometric_projection"
Personal tools