Editing 3D projection (section)

==Parallel projection==
{{Main|Parallel projection}}
[[File:Camera focal length distance house animation.gif|thumb|Parallel projection corresponds to a perspective projection with a hypothetical viewpoint; i.e. one where the camera lies an infinite distance away from the object and has an infinite focal length, or "zoom".]]
In parallel projection, the lines of sight from the object to the [[projection plane]] are parallel to each other. Thus, lines that are parallel in three-dimensional space remain parallel in the two-dimensional projected image. Parallel projection also corresponds to a [[perspective projection]] with an infinite [[focal length]] (the distance from a camera's [[camera lens|lens]] and [[focus (optics)|focal point]]), or "[[zoom lens|zoom]]".

Images drawn in parallel projection rely upon the technique of [[axonometry]] ("to measure along axes"), as described in [[Pohlke's theorem]]. In general, the resulting image is ''oblique'' (the rays are not perpendicular to the image plane); but in special cases the result is ''orthographic'' (the rays are perpendicular to the image plane). ''Axonometry'' should not be confused with ''axonometric projection'', as in English literature the latter usually refers only to a specific class of pictorials (see below).

===Orthographic projection===
{{Main|Orthographic projection}}
{{See also|Geometric transformation}}
The orthographic projection is derived from the principles of [[descriptive geometry]] and is a two-dimensional representation of a three-dimensional object. It is a parallel projection (the lines of projection are parallel both in reality and in the projection plane). It is the projection type of choice for [[plan (drawing)|working drawings]].

If the normal of the viewing plane (the camera direction) is parallel to one of the primary axes (which is the ''x'', ''y'', or ''z'' axis), the mathematical transformation is as follows;
To project the 3D point <math>a_x</math>, <math>a_y</math>, <math>a_z</math> onto the 2D point <math>b_x</math>, <math>b_y</math> using an orthographic projection parallel to the y axis (where positive ''y'' represents forward direction - profile view), the following equations can be used:
:<math>
b_x = s_x a_x + c_x
</math>
:<math>
b_y = s_z a_z + c_z
</math>
where the vector '''s''' is an arbitrary scale factor, and '''c''' is an arbitrary offset. These constants are optional, and can be used to properly align the viewport. Using [[matrix multiplication]], the equations become:
:<math>
 \begin{bmatrix}
   b_x  \\
   b_y
\end{bmatrix} = \begin{bmatrix}
   s_x & 0 & 0 \\
   0 & 0 & s_z
\end{bmatrix}\begin{bmatrix}
   a_x  \\
   a_y  \\
   a_z
\end{bmatrix} + \begin{bmatrix}
   c_x  \\
   c_z
\end{bmatrix}.
</math>

While orthographically projected images represent the three dimensional nature of the object projected, they do not represent the object as it would be recorded photographically or perceived by a viewer observing it directly. In particular, parallel lengths at all points in an orthographically projected image are of the same scale regardless of whether they are far away or near to the virtual viewer. As a result, lengths are not foreshortened as they would be in a perspective projection.

====Multiview projection====
{{Main|Multiview projection}}
[[File:Convention placement vues dessin technique.svg|thumb|right|Symbols used to define whether a multiview projection is either First Angle (left) or Third Angle (right).]]
With ''multiview projections'', up to six pictures (called ''primary views'') of an object are produced, with each projection plane parallel to one of the coordinate axes of the object. The views are positioned relative to each other according to either of two schemes: ''first-angle'' or ''third-angle'' projection. In each, the appearances of views may be thought of as being ''projected'' onto planes that form a 6-sided box around the object. Although six different sides can be drawn, ''usually'' three views of a drawing give enough information to make a 3D object. These views are known as ''front view'', ''top view'', and ''end view''. The terms ''elevation'', ''plan'' and ''section'' are also used.

===Oblique projection===
{{Main|Oblique projection}}
{{multiple image
| width= 120
| image1= Potting-bench-cabinet-view.png
| caption1= [[Potting bench]] drawn in '''cabinet projection''' with an angle of 45° and a ratio of 2/3
| image2= Militärperspektive.PNG
| caption2= Stone arch drawn in '''military perspective'''
}}
In ''oblique projections'' the parallel projection rays are not perpendicular to the viewing plane as with orthographic projection, 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 among the axes as well as the foreshortening factors (scale) 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 thereby creating a true shape, full-size image of the chosen plane. Special types of oblique projections are:

====Cavalier projection (45°)====
In '''cavalier projection''' (sometimes '''cavalier perspective''' or '''high view point''') a point of the object is represented by three coordinates, ''x'', ''y'' and ''z''. On the drawing, it is represented by only two coordinates, ''x″'' and ''y″''. On the flat drawing, two axes, ''x'' and ''z'' on the figure, are [[perpendicular]] and the length on these axes are drawn with a 1:1 scale; it is thus similar to the [[dimetric projection]]s, although it is not an [[axonometric projection]], as the third axis, here ''y'', is drawn in diagonal, making an arbitrary angle with the ''x″'' axis, usually 30 or 45°. The length of the third axis is not scaled.

====Cabinet projection====
The term '''cabinet projection''' (sometimes '''cabinet perspective''') stems from its use in illustrations by the furniture industry.{{citation needed|date=August 2010}} Like cavalier perspective, one face of the projected object is parallel to the viewing plane, and the third axis is projected as going off in an angle (typically 30° or 45° or arctan(2)&nbsp;=&nbsp;63.4°). Unlike cavalier projection, where the third axis keeps its length, with cabinet projection the length of the receding lines is cut in half.

====Military projection====
A variant of [[oblique projection]] is called ''military projection''. In this case, the horizontal sections are isometrically drawn so that the floor plans are not distorted and the verticals are drawn at an angle. The military projection is given by rotation in the ''xy''-plane and a vertical translation an amount ''z''.<ref>{{cite web |url=http://www.math.utah.edu/~treiberg/Perspect/Perspect.htm |title= The Geometry of Perspective Drawing on the Computer |first1=Andrejs |last1=Treibergs |publisher= University of Utah § Department of Mathematics | access-date=24 April 2015 |url-status=live |archive-url=https://web.archive.org/web/20150430055524/http://www.math.utah.edu/~treiberg/Perspect/Perspect.htm |archive-date= Apr 30, 2015 }}</ref>

===Axonometric projection===
{{Main|Axonometric projection}}
[[File:Axonometric projections.png|thumb|The three [[axonometric projection|axonometric views]], here of [[cabinetry]]]]
''Axonometric projections'' show an image of an object as viewed from a skew direction in order to reveal all three directions (axes) of space in one picture.<ref>{{cite book |last= Mitchell |first= William |author2=Malcolm McCullough |title= Digital design media |publisher= John Wiley and Sons |date= 1994 |page= 169 |url= https://books.google.com/books?id=JrZoGgQEhKkC&q=axonometric+orthographic&pg=PA169 |isbn= 978-0-471-28666-0}}</ref> Axonometric projections may be either ''orthographic'' or ''oblique''. Axonometric instrument drawings are often used to approximate graphical perspective projections, but there is attendant distortion in the approximation. Because pictorial projections innately contain this distortion, in instrument drawings of pictorials great liberties may then be taken for economy of effort and best effect.{{clarify|date=May 2017}}

''Axonometric projection'' is further subdivided into three categories: ''isometric projection'', ''dimetric projection'', and ''trimetric projection'', depending on the exact angle at which the view deviates from the orthogonal.<ref name="maynard">{{cite book|url=https://books.google.com/books?id=4Y_YqOlXoxMC&q=axonometric+orthographic&pg=PA22|title=Drawing distinctions: the varieties of graphic expression|last=Maynard|first=Patric|date=2005|publisher=Cornell University Press|isbn=978-0-8014-7280-0|page=22}}</ref><ref name="mcreynolds">{{cite book|url=https://books.google.com/books?id=H4eYq7-2YhYC&q=axonometric+orthographic&pg=PA502|title=Advanced graphics programming using openGL|last=McReynolds|first=Tom|date=2005|publisher=Elsevier|isbn=978-1-55860-659-3|page=502|author2=David Blythe}}</ref> A typical characteristic of orthographic pictorials is that one axis of space is usually displayed as vertical.

====Isometric projection====
In '''isometric pictorials''' (for methods, see [[Isometric projection]]), the direction of viewing is such that the three axes of space appear equally foreshortened, and there is a common angle of 120° between them. The distortion caused by [[foreshortening]] is uniform, therefore the proportionality of all sides and lengths are preserved, and the axes share a common scale. This enables measurements to be read or taken directly from the drawing.

====Dimetric projection====
In '''dimetric pictorials''' (for methods, see [[Dimetric projection]]), the direction of viewing is such that two of the three axes of space appear equally foreshortened, of which the attendant scale and angles of presentation are determined according to the angle of viewing; the scale of the third direction (vertical) is determined separately. Approximations are common in dimetric drawings.

====Trimetric projection====
In '''trimetric pictorials''' (for methods, see [[Trimetric projection]]), the direction of viewing is such that all of the three axes of space appear unequally foreshortened. The scale along each of the three axes and the angles among them are determined separately as dictated by the angle of viewing. Approximations in Trimetric drawings are common.

===Limitations of parallel projection===
{{See also|Impossible object}}
{{Multiple image
| width= 200
| image1= IsometricFlaw 2.svg
| caption1= An example of the limitations of isometric projection. The height difference between the red and blue balls cannot be determined locally.
| image2= Impossible staircase.svg
| caption2= The [[Penrose stairs]] depicts a staircase which seems to ascend (anticlockwise) or descend (clockwise) yet forms a continuous loop.
}}

Objects drawn with parallel projection do not appear larger or smaller as they extend closer to or away from the viewer. While advantageous for [[architectural drawing]]s, where measurements must be taken directly from the image, the result is a perceived distortion, since unlike [[perspective projection]], this is not how our eyes 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.

In this isometric drawing, the blue sphere is two units higher than the red one. However, this difference in elevation is not apparent if one covers the right half of the picture, as the boxes (which serve as clues suggesting height) are then obscured.

This visual ambiguity has been exploited in [[op art]], as well as "impossible object" drawings. [[M. C. Escher]]'s ''[[Waterfall (M. C. Escher)|Waterfall]]'' (1961), while not strictly utilizing parallel projection, is a well-known example, in which a channel of water seems to travel unaided along a downward path, only to then paradoxically fall once again as it returns to its source. The water thus appears to disobey the [[conservation of energy|law of conservation of energy]]. An extreme example is depicted in the film ''[[Inception]]'', where by a [[forced perspective]] trick an immobile stairway changes its connectivity. The video game ''[[Fez (video game)|Fez]]'' uses tricks of perspective to determine where a player can and cannot move in a puzzle-like fashion.