Isometric projection

Part of a series on
Graphical projection
Planar Parallel projection Orthographic projection Isometric projection Oblique projection Perspective projection Curvilinear perspective Reverse perspective
Views Bird's-eye view Cross section Cutaway drawing Exploded view drawing Fisheye lens Multiviews Panorama Worm's-eye view Zoom lens
Topics 3D projection Anamorphosis Axonometry Computer graphics Computer-aided design Descriptive geometry Engineering drawing Map projection Picture plane Plans (drawings) Projection (linear algebra) Projection plane Projective geometry Stereoscopy Technical drawing True length Vanishing point Video game graphics Viewing frustum
v t e

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 angle between any two of them is 120 degrees.

Isometric drawing of a cube

Camera rotations needed to achieve this perspective

The term "isometric" comes from the

An isometric view of an object can be obtained by choosing the viewing direction such that the angles between the projections of the x, y, and z

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

where α = arcsin(tan 30°) ≈ 35.264° and β = 45°. 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 xy-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 7 possibilities are obtained by either rotating to the opposite sides or not, and then inverting the view direction or not.^[1]

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

Example of axonometric art in an illustrated edition of the Romance of the Three Kingdoms, China, c. 15th century.

First formalized by Professor

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

Isometric video game graphics are graphics employed in

External links

Wikimedia Commons has media related to Isometric projection.

• Isometric Projection