Weyl equation

2014-10-01T10:46:41Z

82.45.20.45:

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Nicolas Bourbaki

2014-01-12T11:41:07Z

82.45.189.199: Citation needed for influence of lack of category theory.

{{for|isometric projection in video games|Video games with isometric graphics}}
{{Views}}
[[File:3D shapes in isometric projection.svg|thumb|Some 3D shapes in isometric projection. '''Black labels''' denote dimensions of the 3D object, while '''red labels''' denote dimensions of the 2D projection (drawing).]]
'''Isometric projection''' is a method for visually representing three-dimensional objects in two dimensions in [[technical drawing|technical]] and [[engineering drawing]]s. It is an [[axonometric projection]] in which the three [[Cartesian coordinate system|coordinate axes]] appear equally foreshortened and the angles between any two of them are 120 degrees.

== Overview ==
{{double image|left|Perspective isometrique cube gris.svg|100|Wikipedia_isometric_cube_3.jpg|100|Isometric drawing of a cube.|Camera rotations needed to achieve this perspective.||}}
The term "isometric" comes from the [[Greek language|Greek]] for "equal measure", reflecting that the [[scale (measurement)|scale]] along each axis of the projection is the same (unlike some other forms of [[graphical projection]]).

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'' [[coordinate axis|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°) or arctan(sin 45°) ) 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. There is isometric paper that can be placed under your normal piece of drawing paper as an aid.

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 axis by about 35.264° as above, and then rotated ±45° around the vertical axis.

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.

The term "isometric" is often mistakenly used to refer to [[axonometric]] projections in general. (There are three types of axonometric projections: isometric, dimetric and trimetric.)

==Rotation angles==
{{double image|right|CubeIsometricRotation1.png|200|CubeIsometricRotation2.png|200|Rotate on x for 45°|Rotate on y for <math>\scriptstyle arctan(1/\sqrt{2})</math>||}}

From the two angles needed for an isometric projection, the value of the second may seem counter intuitive and deserves some further explanation. Let’s first imagine a cube with sides of length 2, and its center positioned at the axis origin. We can calculate the length of the line from its center to the middle of any edge as <math>\scriptstyle \sqrt{2}</math> using [[Pythagorean theorem|Pythagoras' theorem]] . By rotating the cube by 45° on the x axis, the point (1, 1, 1) will therefore become (1, 0, <math>\scriptstyle \sqrt{2}</math>) as depicted in the diagram. The second rotation aims to bring the same point on the positive z axis and so needs to perform a rotation of value equal to the [[arctangent]] of <math>\scriptstyle 1/\sqrt{2}</math> which is approximately 35.264°.

==Mathematics==
There are eight different orientations to obtain an isometric view, depending into which [[octant (solid geometry)|octant]] the viewer looks. The isometric transform from a point <math>a_{x,y,z}</math> in 3D space to a point <math>b_{x,y}</math> in 2D space looking into the first octant can be written mathematically with [[rotation matrix|rotation matrices]] as:
:<math>
\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}
</math>

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

The other 7 possibilities are obtained by either rotating to the opposite sides or not, and then inverting the view direction or not.<ref>{{cite journal
|author=Ingrid Carlbom, Joseph Paciorek , Dan Lim
|title=Planar Geometric Projections and Viewing Transformations
|journal=[[ACM Computing Surveys]]
|volume=10|issue=4
|pages=465–502
|date=December 1978
|publisher=[[Association for Computing Machinery|ACM]]
|doi=10.1145/356744.356750
}}</ref>

== History and limitations ==
{{multiple image
| width = 150
| image1 = Optimal-grinding engine model.jpg
| caption1 = Optimal-grinding engine model (1822), drawn in 30° isometric.<ref>William Farish (1822) "On Isometrical Perspective". In: ''Cambridge Philosophical Transactions''. 1 (1822).</ref>
| image2 = Sanguo2.PNG
| caption2 = Example of Chinese art in an illustrated edition of the ''[[Romance of the Three Kingdoms]]'', China, c. 15th century.
}}

{{main|Axonometric projection}}
First formalized by Professor [[William Farish (professor)|William Farish]] (1759–1837), the concept of [[isometry]] had existed in a rough empirical form for centuries.<ref>Barclay G. Jones (1986). ''Protecting historic architecture and museum collections from natural disasters''. University of Michigan. ISBN 0-409-90035-4. p.243.</ref><ref>Charles Edmund Moorhouse (1974). ''Visual messages: graphic communication for senior students''.</ref> From the middle of the 19th century 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."<ref name="Kri96">J. Krikke (1996). "[http://www.iias.nl/iiasn/iiasn9/eastasia/krikke.html A Chinese perspective for cyberspace?]". In: ''International Institute for Asian Studies Newsletter'', 9, Summer 1996.</ref> According to Jan Krikke (2000)<ref name="Kri00">Jan Krikke (2000). "Axonometry: a matter of perspective". In: ''Computer Graphics and Applications, IEEE'' Jul/Aug 2000. Vol 20 (4), pp. 7–11.</ref> 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".<ref name="Kri00"/>

{{multiple image
| width = 150
| image1 = IsometricFlaw 2.svg
| caption1 = An example of the limitations of isometric projection. The height difference between the red and blue balls cannot be determined locally.
| image2 = Impossible staircase.svg
| caption2 = 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 isometric projection do not appear larger or smaller as they extend closer to or away from the viewer. While advantageous for [[architectural drawing]]s where measurements need to be taken directly, the result is 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.
{{clear}}

==See also==
* [[Graphical projection]]
* [[Isometric graphics in video games and pixel art]]

== References ==
{{Reflist}}

==External links==
{{Commons category|Isometric projection}}
* [http://www.gandraxa.com/isometric_projection.xml Isometric Projection]
* [http://gersic.com/blog.php?id=52 Rendering Isometric Tiles in Blender 3D]
* [http://www.flashgameblog.at/blog-post/2010/04/24/rendering-iso-tiles-with-blender/ A prototype project for Blender to render isometric Tiles]

{{visualization}}

[[Category:Graphical projections]]

[[de:Perspektive#Isometrische Axonometrie, nach DIN 5]]
[[it:Assonometria#Assonometria isometrica]]

Function and Concept

2012-05-31T12:07:19Z

82.45.193.165: /* External links */

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Weyl equation

Nicolas Bourbaki

Function and Concept