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等角图是在工程制图中，一种在平面上表达立体图形的方式。在等角图中，直角座标系的三条轴被等比例地缩小(意即在图上的长度相同)，且在图上两两夹120度角。

概要

等角图下的正方体

等角图所使用的投影角度

为了达到前面所说等角、等长的特点，等角图的本质是特定角度的正投影。而这个特定角度就是(±1,±1,±1)的向量(扣除平行而反向的向量有4种可能)，或者可以说是：原本是正对着某一面的正投影，先绕z轴旋转45°(方向不拘)，再向上或向下^{[注 1]}旋转约35.264°^{[注 2]}

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.)

数学关系

There are eight different orientations to obtain an isometric view, depending into which octant the viewer looks. The isometric transform from a point $a_{x,y,z}$ in 3D space to a point $b_{x,y}$ 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 $\alpha =\arcsin(\tan 30^{\circ })\approx 35.264^{\circ }$ and $\beta =45^{\circ }$ . As explained above, this is a rotation around the vertical (here y) axis by $\beta$ , followed by a rotation around the horizontal (here x) axis by $\alpha$ . 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.^[1]

历史与使用的限制

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

运用了等角图的三国演义插图

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]

一个不适合画成等角图的例子。在图中难以分辨红蓝两球的高度关系

在潘洛斯阶梯中的无限循环阶梯

一如所有形式的正投影图，图中的物体不能透过大小表现出远近关系。等角图满足了工程制图所要求，能够准确地表现物体的尺寸，却造成失真。相较于正投影图，透视图要更接近肉眼、照相机。右边的两张图片则是利用距离难以分辨画出的图片。

注解

^ 这里的旋转轴是一条同时与z轴与前一步骤后得到的视角垂直的直线
^ 准确值是 $\arcsin({\frac {\sqrt {3}}{3}})$ 或 $\arctan({\frac {\sqrt {2}}{2}})$

参考资料

^ 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)
^ William Farish (1822) "On Isometrical Perspective". In: Cambridge Philosophical Transactions. 1 (1822).
^ Barclay G. Jones (1986). Protecting historic architecture and museum collections from natural disasters. University of Michigan. ISBN 0409900354. p.243.
^ Charles Edmund Moorhouse (1974). Visual messages: graphic communication for senior students‎.
^ J. Krikke (1996). "A Chinese perspective for cyberspace?". In: International Institute for Asian Studies Newsletter, 9, Summer 1996.
^ ^6.0 ^6.1 Jan Krikke (2000). "Axonometry: a matter of perspective". In: Computer Graphics and Applications, IEEE Jul/Aug 2000. Vol 20 (4), pp. 7–11.

外部链接

[1] 这里的旋转轴是一条同时与z轴与前一步骤后得到的视角垂直的直线

[2] 准确值是 $\arcsin({\frac {\sqrt {3}}{3}})$ 或 $\arctan({\frac {\sqrt {2}}{2}})$

[3] 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)

[4] William Farish (1822) "On Isometrical Perspective". In: Cambridge Philosophical Transactions. 1 (1822).

[5] Barclay G. Jones (1986). Protecting historic architecture and museum collections from natural disasters. University of Michigan. ISBN 0409900354. p.243.

[6] Charles Edmund Moorhouse (1974). Visual messages: graphic communication for senior students‎.

[Kri96-7] J. Krikke (1996). "A Chinese perspective for cyberspace?". In: International Institute for Asian Studies Newsletter, 9, Summer 1996.

[Kri00-8] 6.0 ^6.1 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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