Parallel projection
A parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane, known as the "projection plane" or "image plane", where the "rays", known as "lines of sight" or "projection lines", are parallel to each other. It is a basic tool in descriptive geometry. The projection is called "orthographic" if the rays are perpendicular (orthogonal) to the image plane, and "oblique" or "skew" if they are not. Overview. A parallel projection is a particular case of "projection" in mathematics and "graphical projection" in technical drawing. Parallel projections can be seen  as the limit of a central or perspective projection, in which the rays pass through a fixed point called the "center" or "viewpoint", as this point is moved towards infinity. Put differently, a parallel projection corresponds to a perspective projection with an infinite focal length (the distance between the lens and the focal point in photography) or "zoom". Further, in parallel projections, lines that are parallel in three-dimensional space remain parallel in the two-dimensionally projected image. A perspective projection of an object is often considered more realistic than a parallel projection, since it more closely resembles human vision  and photography. However, parallel projections are popular in technical applications, since the parallelism of an object's lines and faces is preserved, and direct measurements can be taken from the image. Among parallel projections, "orthographic projections" are seen as the most realistic, and are commonly used by engineers. On the other hand, certain types of "oblique projections" (for instance cavalier projection, military projection) are very simple to implement, and are used to create quick and informal pictorials of objects. The term "parallel projection" is used in the literature to describe both the "procedure itself" (a mathematical mapping function) as well  as the resulting image "produced by the procedure". Properties. Every parallel projection has the following properties: Types. Orthographic projection. Orthographic projection is derived from the principles of descriptive geometry, and is a type of parallel projection where the projection rays are perpendicular to the projection plane. It is the projection type of choice for working drawings. The term "orthographic" is sometimes reserved specifically for depictions of objects where the principal axes or planes of the object are also parallel with the projection plane (or the paper on which the orthographic or parallel projection is drawn). However, the term "primary view  is also used. In multiview projections", up to six pictures of an object are produced, with each projection plane perpendicular to one of the coordinate axes. However, when the principal planes or axes of an object are "not" parallel with the projection plane, but are rather tilted to some degree to reveal multiple sides of the object, they are called "auxiliary views" or "pictorials". Sometimes, the term "axonometric projection" is reserved solely for these views, and is juxtaposed with the term "orthographic projection". But "axonometric projection" might be more accurately described as being synonymous with "parallel projection", and "orthographic projection  a type of axonometric projection". The "primary views" include "plans", "elevations" and "sections"; and the "isometric", "dimetric" and "trimetric projections" could be considered "auxiliary views". A typical (but non-obligatory) characteristic of multiview orthographic projections is that one axis of space usually is displayed as vertical. When the viewing direction is perpendicular to the surface of the depicted object, regardless of the object's orientation, it is referred to as a "normal projection". Thus, in the case of a cube oriented with a space's coordinate system, the "primary views" of the cube would be considered "normal projections". Oblique projection . In an "oblique projection", the parallel projection rays are not perpendicular to the viewing plane, but strike the projection plane at an angle other than ninety degrees. In both orthographic and oblique projection, parallel lines in space appear parallel on the projected image. Because of its simplicity, oblique projection is used exclusively for pictorial purposes rather than for formal, working drawings. In an oblique pictorial drawing, the displayed angles separating the coordinate axes as well as the foreshortening factors (scaling) are arbitrary. The distortion created thereby is usually attenuated by aligning one plane of the imaged object to be parallel  with the plane of projection, creating a truly-formed, full-size image of the chosen plane. Special types of oblique projections include "military", "cavalier" and "cabinet projection". Analytic representation. If the image plane is given by equation formula_7 and the direction of projection by formula_1, then the projection line through the point formula_9 is parametrized by The image formula_12 of formula_13 is the intersection of line formula_14 with plane formula_15; it is given by the equation In several cases, these formulas can be simplified. (S1) If one can choose the vectors formula_17 and formula_18 such that formula_19, the formula for  the image simplifies to (S2) In an orthographic projection, the vectors formula_17 and formula_18 are parallel. In this case, one can choose formula_23 and one gets (S3) If one can choose the vectors formula_17 and formula_18 such that formula_19, and if the image plane contains the origin, one has formula_28 and the parallel projection is a linear mapping: From this analytic representation of a parallel projection one can deduce most of the properties stated in the previous sections. History. Axonometry originated in China. Its function in Chinese art was unlike the linear perspective in European art since its perspective was  not objective, or looking from the outside. Instead, its patterns used parallel projections within the painting that allowed the viewer to consider both the space and the ongoing progression of time in one scroll. According to science author and Medium journalist Jan Krikke, axonometry, and the pictorial grammar that goes with it, had taken on a new significance with the introduction of visual computing and engineering drawing.<ref>