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# Isometric projection

[ahy-suh-me-trik]
Isometric projection is a form of graphical projection—more specifically, an axonometric projection. It is a method of visually representing three-dimensional objects in two dimensions, in which the three coordinate axes appear equally foreshortened and the angles between any two of them are 120°. Isometric projection is one of the projections used in drafting engineering drawings.

The term isometric comes from the Greek for "equal measure", reflecting that the scale along each axis of the projection is the same (this is not true of some other forms of graphical projection).

One of the advantages of isometric perspective in engineering drawings is that 60° angles are easy to construct using only a compass and straightedge.

## Visualization

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°) ) about the horizontal axis.

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 axes.

Another way in which isometric projection can be visualized is by considering the view of a cubical room from an upper corner, looking towards the opposite lower corner. The x-axis is diagonally down and right, the y-axis is 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.

## Mathematical

There are 8 different orientations to obtain an isometric view, depending into which octant the viewer looks. The isometric transform from a point $a_\left\{x,y,z\right\}$ in 3D space to a point $b_\left\{x,y\right\}$ 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 & {cosalpha} & {sinalpha} 0 & { - sinalpha} & {cosalpha} end{bmatrix}begin{bmatrix} {cosbeta } & 0 & { - sinbeta }
0 & 1 & 0
{sinbeta } & 0 & {cosbeta } 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 $alpha = arcsin\left(tan30^circ\right)approx35.264^circ$ 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 seven possibilities are obtained by either rotating to the opposite sides or not, and then inverting the view direction or not.

## Limits of axonometric projection

As with all types of parallel projection, objects drawn with axonometric projection do not appear larger or smaller as they extend closer to or away from the viewer. While advantageous for architectural drawings and sprite-based video games, this results in a perceived distortion, as unlike perspective projection, it is not how our eyes or photography usually work. It also can easily result in situations where depth and altitude are impossible to gauge, as is shown in the illustration to the right. An additional problem particular to isometric projection is when it becomes difficult to determine which face of the object is being observed. In the absence of proper shading—and for objects that are relatively perpendicular and similarly proportioned—it can become difficult to determine which is the top, bottom or side face of the object. This is because, in isometric projection, the projection of each face onto a two-dimensional plane has similar dimensions and area.

Most contemporary video games have avoided these situations by dropping axonometric projection in favor of perspective 3D rendering utilizing vanishing points. Some of the famous "impossible architecture" works of M. C. Escher, however, exploit them. Waterfall (1961) is a good example, in which the building is (roughly) isometric, but the faded background utilizes perspective projection.

## "Isometric" projection in video games and pixel art

In the fields of computer and video games and pixel art, axonometric projection has been popular because of the ease with which 2D sprites and tile-based graphics can be made to represent a 3D gaming environment. Because objects do not change size as they move about the game field, there is no need for the computer to scale sprites or do the calculations necessary to simulate visual perspective. This allowed older 8-bit and 16-bit game systems (and, more recently, handheld systems) to portray large 3D areas easily. While the depth confusion problems illustrated above can sometimes be a problem, good game design can alleviate this. With the advent of more powerful graphics systems, axonometric projection is becoming less common.

The projection used in videogames usually deviates slightly from "true" isometric due to the limitations of raster graphics. Lines in the x and y axes would not follow a neat pixel pattern if drawn in the required 30° to the horizontal. While modern computers can eliminate this problem using anti-aliasing, earlier computer graphics did not support enough colors or possess enough CPU power to accomplish this. So instead, a 2:1 pixel pattern ratio would be used to draw the x and y axes lines, resulting in these axes following a 26.565° (arctan 0.5) angle to the horizontal. (Game systems that do not use square pixels could, however, yield different angles, including true isometric.) Therefore, this form of projection is more accurately described as a variation of dimetric projection, since only two of the three angles between the axes are equal (116.565°, 116.565°, 126.87°). Many in video game and pixel art communities, however, continue to colloquially refer to this projection as "isometric perspective"; the terms "3/4 perspective" and "2.5D" are also commonly used.

The term has also been applied to games that do not use the 2:1 pixel pattern ratio common among video games. Fallout and SimCity 4, which use trimetric projection, have been referred to as "isometric". Games that use oblique projection, such as The Legend of Zelda: A Link to the Past and Ultima Online—as well as games that use perspective projection with a bird's eye view, such as The Age of Decadence and Silent Storm—are also sometimes referred to as being isometric, or "pseudo-isometric".

## History of isometric video games

While the history of computer games saw some true 3D games as soon as the early 1970s, the first video games to use the distinct visual style of isometric projection in the meaning described above were arcade games in the early 1980s.

Q*bert and Zaxxon were both released in 1982. Q*bert showed a static pyramid drawn in an isometric perspective, with the player controlling a person which could jump around on the pyramid. Zaxxon employed scrolling isometric levels where the player controlled a plane to fly through the levels. A year later in 1983 the arcade game Congo Bongo was released, running on the same hardware as Zaxxon . It allowed the player character to move around in bigger isometric levels, including true three-dimensional climbing and falling. The same was possible in the 1984 arcade title Marble Madness.

At this time, isometric games were no longer exclusive to the arcade market and also entered home computers with the release of Ant Attack for the ZX Spectrum in 1983. The ZX Crash magazine consequently awarded it 100% in the graphics category for this new "3D" technique . A year later the ZX saw the release of Knight Lore, which is generally regarded as a revolutionary title which defined the subsequent genre of isometric adventure games .

Following Knight Lore, many isometric titles were seen on home computers - to an extent that it was regarded as being the second most cloned piece of software after WordStar. One big success out of those was the 1987 game Head Over Heels . Isometric perspective was not limited to arcade/adventure games, though; for example, the 1989 strategy game Populous used isometric perspective.

Throughout the 1990s some very successful games like Civilization II and Diablo used a fixed isometric perspective. But with the advent of 3D acceleration on personal computers and gaming consoles, games using a 3D perspective generally started using true 3D instead of isometric perspective. This can be seen by successors of the above games, starting with Civilization IV the civilization series uses full 3D. Diablo II used a fixed perspective like its predecessor, but optionally allowed for perspective scaling of the sprites in the distance to lend a pseudo-3D perspective.

For a comprehensive list of isometric video games, see Isometric video games.