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概要

Note that with the cube (see image) the perimeter of the 2D drawing is斜体文字 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 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.)

數學關係

There are eight different orientations to obtain an isometric view, depending into which octant the viewer looks. The isometric transform from a point ${\displaystyle a_{x,y,z}}$ in 3D space to a point ${\displaystyle b_{x,y}}$ in 2D space looking into the first octant can be written mathematically with rotation matrices as:

${\displaystyle {\begin{bmatrix}\mathbf {c} _{x}\\\mathbf {c} _{y}\\\mathbf {c} _{z}\\\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&{\cos \alpha }&{\sin \alpha }\\0&{-\sin \alpha }&{\cos \alpha }\\\end{bmatrix}}{\begin{bmatrix}{\cos \beta }&0&{-\sin \beta }\\0&1&0\\{\sin \beta }&0&{\cos \beta }\\\end{bmatrix}}{\begin{bmatrix}\mathbf {a} _{x}\\\mathbf {a} _{y}\\\mathbf {a} _{z}\\\end{bmatrix}}={\frac {1}{\sqrt {6}}}{\begin{bmatrix}{\sqrt {3}}&0&-{\sqrt {3}}\\1&2&1\\{\sqrt {2}}&-{\sqrt {2}}&{\sqrt {2}}\\\end{bmatrix}}{\begin{bmatrix}\mathbf {a} _{x}\\\mathbf {a} _{y}\\\mathbf {a} _{z}\\\end{bmatrix}}}$

where ${\displaystyle \alpha =\arcsin(\tan 30^{\circ })\approx 35.264^{\circ }}$ and ${\displaystyle \beta =45^{\circ }}$. As explained above, this is a rotation around the vertical (here y) axis by ${\displaystyle \beta }$, followed by a rotation around the horizontal (here x) axis by ${\displaystyle \alpha }$. This is then followed by an orthographic projection to the x-y plane:

${\displaystyle {\begin{bmatrix}\mathbf {b} _{x}\\\mathbf {b} _{y}\\0\\\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\\\end{bmatrix}}{\begin{bmatrix}\mathbf {c} _{x}\\\mathbf {c} _{y}\\\mathbf {c} _{z}\\\end{bmatrix}}}$

The other seven possibilities are obtained by either rotating to the opposite sides or not, and then inverting the view direction or not.[1]

歷史與使用的限制

Optical-grinding engine model (1822), drawn in 30° isometric.[2]

First formalized by Professor William Farish (1759–1837), the concept of an isometric had existed in a rough empirical form for centuries.[3][4] From the middle of the 19th century isometry became an "inv斜体文字aluable tool for engineers, and soon thereafter axonometry and isometry were incorporated in the curriculum of architectural training courses in Europe and the U.S."[5] According to Jan Krikke (2000)[6] 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".[6]

註解

1. ^ 這裡的旋轉軸是一條同時與z軸與前一步驟後得到的視角垂直的直線
2. ^ 準確值是${\displaystyle \arcsin({\frac {\sqrt {3}}{3}})}$${\displaystyle \arctan({\frac {\sqrt {2}}{2}})}$

參考資料

1. ^ Ingrid Carlbom, Joseph Paciorek. Planar Geometric Projections and Viewing Transformations. ACM Computing Surveys (ACM). 1978, 10 (4): 465–502. doi:10.1145/356744.356750. 已忽略未知参数|month=（建议使用|date=） (帮助)
2. ^ William Farish (1822) "On Isometrical Perspective". In: Cambridge Philosophical Transactions. 1 (1822).
3. ^ Barclay G. Jones (1986). Protecting historic architecture and museum collections from natural disasters. University of Michigan. ISBN 0409900354. p.243.
4. ^ Charles Edmund Moorhouse (1974). Visual messages: graphic communication for senior students‎.
5. ^ J. Krikke (1996). "A Chinese perspective for cyberspace?". In: International Institute for Asian Studies Newsletter, 9, Summer 1996.
6. Jan Krikke (2000). "Axonometry: a matter of perspective". In: Computer Graphics and Applications, IEEE Jul/Aug 2000. Vol 20 (4), pp. 7–11.