# Isometric projection

For isometric projection in video games, see Video games with isometric graphics.
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 and engineering drawings. It is an axonometric projection in which the three coordinate axes appear equally foreshortened and the angles between any two of them are 120 degrees.

## Overview

Isometric drawing of a cube.
Camera rotations needed to achieve this perspective.

The term "isometric" comes from the Greek for "equal measure", reflecting that the 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 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

Rotate on x for 45°
Rotate on y for $\scriptstyle arctan(1/\sqrt{2})$

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 $\scriptstyle \sqrt{2}$ using Pythagoras' theorem . By rotating the cube by 45° on the x axis, the point (1, 1, 1) will therefore become (1, 0, $\scriptstyle \sqrt{2}$) 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 $\scriptstyle 1/\sqrt{2}$ which is approximately 35.264°.

## Mathematics

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 $\alpha = \arcsin(\tan30^\circ)\approx35.264^\circ$ 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 7 possibilities are obtained by either rotating to the opposite sides or not, and then inverting the view direction or not.[1]

## History and limitations

Optimal-grinding engine model (1822), drawn in 30° isometric.[2]
Example of Chinese art in an illustrated edition of the Romance of the Three Kingdoms, China, c. 15th century.

First formalized by Professor William Farish (1759–1837), the concept of isometry had existed in a rough empirical form for centuries.[3][4] 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."[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]

An example of the limitations of isometric projection. The height difference between the red and blue balls cannot be determined locally.
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 drawings 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. This can appear to create paradoxical or impossible shapes, such as the Penrose stairs.