Oblique projection


 * This article discusses imaging of three-dimensional objects. For an abstract mathematical discussion, see Projection (linear algebra).

Oblique projection is a simple type of graphical projection used for producing pictorial, two-dimensional images of three-dimensional objects. If neither Equator nor the central meridian are aligned with and centered on the map axes, the result is commonly called an oblique projection.

Overview
Oblique projection is a type of parallel projection: In both oblique projection and orthographic projection, parallel lines of the source object produce parallel lines in the projected image. The projectors in oblique projection intersect the projection plane at an oblique angle to produce the projected image, as opposed to the perpendicular angle used in orthographic projection.
 * it projects an image by intersecting parallel rays (projectors)
 * from the three-dimensional source object with the drawing surface (projection plan).

On a map, this type of projection is very useful when an area of interest would be in a location of high distortion using another projection style. By using an oblique projection the larger, more important areas can be placed in position with less distortion. This can clarify the purpose of a map and provide the users with a greater understanding.

Mathematically, the parallel projection of the point $$(x, y, z)$$ on the $$xy$$-plane gives $$(x+az, y+bz, 0)$$. The constants $$a$$ and $$b$$ uniquely specify a parallel projection. When $$a = b = 0$$, the projection is said to be "orthographic" or "orthogonal". Otherwise, it is "oblique". The constants $$a$$ and $$b$$ are not necessarily less than 1, and as a consequence lengths measured on an oblique projection may be either larger or shorter than they were in space. In a general oblique projection, spheres of the space are projected as ellipses on the drawing plane, and not as circles as you would expect them from an orthogonal projection.



Oblique drawing is also the crudest "3D" drawing method but the easiest to master. Oblique is not really a 3D system but a 2 dimensional view of an object with 'forced depth'. One way to draw using an oblique view is to draw the side of the object you are looking at in two dimensions, i.e. flat, and then draw the other sides at an angle of 45 degrees, but instead of drawing the sides full size they are only drawn with half the depth creating 'forced depth' - adding an element of realism to the object. Even with this 'forced depth', oblique drawings look very unconvincing to the eye. For this reason oblique is rarely used by professional designer and engineers.

Oblique pictorial
In an oblique pictorial drawing, the angles displayed among the axes, as well as the foreshortening factors (scale) are arbitrary. More precisely, any given set of three coplanar segments originating from the same point may be construed as forming some oblique perspective of three sides of a cube. This result is known as Pohlke's theorem, from the German mathematician Pohlke, who published it in the early 19th century.

The resulting distortions make the technique unsuitable for formal, working drawings. Nevertheless, the distortions are partially overcome by aligning one plane of the image parallel to the plane of projection. Doing so creates a true shape image of the chosen plane. This specific category of oblique projections, whereby lengths along the directions $$x$$ and $$y$$ are preserved, but lengths along direction $$z$$ are drawn at angle using a reduction factor is very much in use for industrial drawings.
 * Cavalier projection is the name of such a projection, where the length along the $$z$$ axis remains unscaled.


 * Cabinet projection, popular in furniture illustrations, is an example of such a technique, wherein the receding axis is scaled to half-size (sometimes also two thirds the original ).

Examples
Besides technical drawing and illustrations, computer games (especially from the era before the advent of 3D games) also often use a form of oblique projection. Examples of such games include Ultima VII, Tibia, EarthBound, and Paperboy.