Isometric projection is a method for visually representing three-dimensional objects in two dimensions intechnical andengineering drawings. It is anaxonometric projection in which the threecoordinate axes appear equally foreshortened and the angle between any two of them is 120 degrees.
The term "isometric" comes from theGreek for "equal measure", reflecting that thescale along each axis of the projection is the same (unlike some other forms ofgraphical projection).
An isometric view of an object can be obtained by choosing the viewing direction such that the angles between the projections of thex,y, andzaxes are all the same, or 120°. For example, with 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 arcsin1⁄√3 or arctan1⁄√2, which is related to theMagic angle) about the horizontal axis. Note that with the cube (see image) the perimeter of the resulting 2D drawing is a perfect regular hexagon: all the black lines have equal length and all the cube's faces are the same area. Isometricgraph paper can be placed under a normal piece of drawing paper to help achieve the effect without calculation.
In a similar way, anisometric view can be obtained in a 3D scene. Starting with the camera aligned parallel to the floor and aligned to the coordinate axes, it is first rotated horizontally (around the vertical axis) by ±45°, then 35.264° around the horizontal axis.
Another way 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. Thex-axis extends diagonally down and right, they-axis extends diagonally down and left, and thez-axis is straight up. Depth is also shown by height on the image. Lines drawn along the axes are at 120° to one another.
In all these cases, as with allaxonometric andorthographic projections, such a camera would need anobject-space telecentric lens, in order that projected lengths not change with distance from the camera.
The term "isometric" is often mistakenly used to refer to axonometric projections, generally. There are, however, actually three types of axonometric projections:isometric,dimetric andoblique.
From the two angles needed for an isometric projection, the value of the second may seem counterintuitive and deserves some further explanation. Let's first imagine a cube with sides of length 2, and its center at the axis origin, which means all its faces intersect the axes at a distance of 1 from the origin. We can calculate the length of the line from its center to the middle of any edge as√2 usingPythagoras' theorem . By rotating the cube by 45° on thex-axis, the point (1, 1, 1) will therefore become (1, 0,√2) as depicted in the diagram. The second rotation aims to bring the same point on the positivez-axis and so needs to perform a rotation of value equal to thearctangent of1⁄√2 which is approximately 35.264°.
There are eight different orientations to obtain an isometric view, depending into whichoctant the viewer looks. The isometric transform from a pointax,y,z in 3D space to a pointbx,y in 2D space looking into the first octant can be written mathematically withrotation matrices as:
whereα = arcsin(tan 30°) ≈ 35.264° andβ = 45°. As explained above, this is a rotation around the vertical (herey) axis byβ, followed by a rotation around the horizontal (herex) axis byα. This is then followed by an orthographic projection to thexy-plane:
The other 7 possibilities are obtained by either rotating to the opposite sides or not, and then inverting the view direction or not.[1]
First formalized by ProfessorWilliam Farish (1759–1837), the concept ofisometry 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 tolinear 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]
As with all types ofparallel projection, objects drawn with isometric projection do not appear larger or smaller as they extend closer to or away from the viewer. While advantageous forarchitectural drawings where measurements need to be taken directly, the result is a perceived distortion, as unlikeperspective projection, it is not howhuman vision 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 or above. This can appear to create paradoxical orimpossible shapes, such as thePenrose stairs.
Isometric video game graphics are graphics employed invideo games andpixel art that utilize aparallel projection, but which angle theviewpoint to reveal facets of the environment that would otherwise not be visible from atop-down perspective orside view, thereby producing athree-dimensional effect. Despite the name, isometric computer graphics are not necessarily truly isometric—i.e., thex,y, andz axes are not necessarily oriented 120° to each other. Instead, a variety of angles are used, withdimetric projection and a 2:1 pixel ratio being the most common. The terms "3⁄4 perspective", "3⁄4 view", "2.5D", and "pseudo 3D" are also sometimes used, although these terms can bear slightly different meanings in other contexts.
Once common, isometric projection became less so with the advent of more powerful3D graphics systems, and as video games began to focus more on action and individual characters.[7] However, video games utilizing isometric projection—especiallycomputer role-playing games—have seen a resurgence in recent years within theindie gaming scene.[7][8]
