A 3D projection (or graphical projection) is a design technique used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These projections rely on visual perspective and aspect analysis to project a complex object for viewing capability on a simpler plane. This concept of extending 2D geometry to 3D was mastered by Heron of Alexandria in the first century. Heron could be called the father of 3D. 3D Projection is the basis of the concept for Computer Graphics simulating fluid flows to imitate realistic effects. Lucas Films 'ILM group is credited with introducing the concept (and even the term "Particle effect"). In 1982 the first all digital computer generated sequence for a motion picture file was in: Star Trek II: Wrath of Khan. A 1984 patent related to this concept was written by William E Masters, "Computer automated manufacturing process and system" US4665492A using mass particles to fabricate a cup. The process of particle deposition is one technology of 3D Printing.
3D projections use the primary qualities of an object's basic shape to create a map of points, that are then connected to one another to create a visual element. The result is a graphic that contains conceptual properties to interpret that the figure or image is not actually flat (2D), but rather, is a solid object (3D) being viewed on a 2D display.
3D objects are largely displayed on two-dimensional mediums (i.e. paper and computer monitors). As such, graphical projections are a commonly used design element; notably, in engineering drawing, drafting, and computer graphics. Projections can be calculated through employment of mathematical analysis and formulae, or by using various geometric and optical techniques.

Overview

Projection is achieved by the use of imaginary "projectors"; the projected, mental image becomes the technician's vision of the desired, finished picture. Methods provide a uniform imaging procedure among people trained in technical graphics (mechanical drawing, computer aided design, etc.). By following a method, the technician may produce the envisioned picture on a planar surface such as drawing paper. There are two graphical projection categories, each with its own method: *parallel projection *perspective projection
File:Orthographic perspective arch.svg|Multiview projection (elevation)
File:Isometrie.png|Axonometric projection (isometric)
File:Militärperspektive.PNG|Oblique projection (military)
File:Cabinet perspective 45.svg|Oblique projection (cabinet)
File:1ptPerspective.svg|One-point perspective
File:2-punktperspektive.svg|Two-point perspective
File:3-punktperspektive 1.svg|Three-point perspective

Parallel projection

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 lens and focal point), or "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

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 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 $a\_x$, $a\_y$, $a\_z$ onto the 2D point $b\_x$, $b\_y$ using an orthographic projection parallel to the y axis (where positive ''y'' represents forward direction - profile view), the following equations can be used: :$b\_x\; =\; s\_x\; a\_x\; +\; c\_x$ :$b\_y\; =\; s\_z\; a\_z\; +\; c\_z$ 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: :$\backslash begin\; b\_x\; \backslash \backslash \; b\_y\; \backslash end\; =\; \backslash begin\; s\_x\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; s\_z\; \backslash end\backslash begin\; a\_x\; \backslash \backslash \; a\_y\; \backslash \backslash \; a\_z\; \backslash end\; +\; \backslash begin\; c\_x\; \backslash \backslash \; c\_z\; \backslash end.$ 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

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.

Axonometric projection

Within orthographic projection there is an ancillary category known as ''orthographic pictorial'' or ''axonometric projection''. 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. 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. ''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. A typical characteristic of orthographic pictorials is that one axis of space is usually displayed as vertical. ''Axonometric projections'' are also sometimes known as ''auxiliary views'', as opposed to the ''primary views'' of ''multiview projections''.

=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.

Oblique projection

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 projections, 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. 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) = 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''.

Limitations of parallel projection

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 drawings, 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'' (1961), while not strictly using 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 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.

** Perspective projection **

300px|Axonometric projection of a scheme displaying the relevant elements of a vertical [[picture plane perspective. The standing point (P.S.) is located on the ground plane ''π'', and the point of view (P.V.) is right above it. P.P. is its projection on the picture plane ''α''. L.O. and L.T. are the horizon and the ground lines (''linea d'orizzonte'' and ''linea di terra''). The bold lines s and q lie on ''π'', and intercept ''α'' at ''Ts'' and ''Tq'' respectively. The parallel lines through P.V. (in red) intercept L.O. in the vanishing points ''Fs'' and ''Fq'': thus one can draw the projections s′ and q′, and hence also their intersection R′ on R.
Perspective projection or perspective transformation is a linear projection where three dimensional objects are projected on a ''picture plane''. This has the effect that distant objects appear smaller than nearer objects.
It also means that lines which are parallel in nature (that is, meet at the point at infinity) appear to intersect in the projected image, for example if railways are pictured with perspective projection, they appear to converge towards a single point, called the vanishing point. Photographic lenses and the human eye work in the same way, therefore perspective projection looks most realistic. Perspective projection is usually categorized into ''one-point'', ''two-point'' and ''three-point perspective'', depending on the orientation of the projection plane towards the axes of the depicted object.
Graphical projection methods rely on the duality between lines and points, whereby two straight lines determine a point while two points determine a straight line. The orthogonal projection of the eye point onto the picture plane is called the ''principal vanishing point'' (P.P. in the scheme on the left, from the Italian term ''punto principale'', coined during the renaissance).
Two relevant points of a line are:
*its intersection with the picture plane, and
*its vanishing point, found at the intersection between the parallel line from the eye point and the picture plane.
The principal vanishing point is the vanishing point of all horizontal lines perpendicular to the picture plane. The vanishing points of all horizontal lines lie on the horizon line. If, as is often the case, the picture plane is vertical, all vertical lines are drawn vertically, and have no finite vanishing point on the picture plane. Various graphical methods can be easily envisaged for projecting geometrical scenes. For example, lines traced from the eye point at 45° to the picture plane intersect the latter along a circle whose radius is the distance of the eye point from the plane, thus tracing that circle aids the construction of all the vanishing points of 45° lines; in particular, the intersection of that circle with the horizon line consists of two ''distance points''. They are useful for drawing chessboard floors which, in turn, serve for locating the base of objects on the scene. In the perspective of a geometric solid on the right, after choosing the principal vanishing point —which determines the horizon line— the 45° vanishing point on the left side of the drawing completes the characterization of the (equally distant) point of view. Two lines are drawn from the orthogonal projection of each vertex, one at 45° and one at 90° to the picture plane. After intersecting the ground line, those lines go toward the distance point (for 45°) or the principal point (for 90°). Their new intersection locates the projection of the map. Natural heights are measured above the ground line and then projected in the same way until they meet the vertical from the map.
While orthographic projection ignores perspective to allow accurate measurements, perspective projection shows distant objects as smaller to provide additional realism.

Mathematical formula

The perspective projection requires a more involved definition as compared to orthographic projections. A conceptual aid to understanding the mechanics of this projection is to imagine the 2D projection as though the object(s) are being viewed through a camera viewfinder. The camera's position, orientation, and field of view control the behavior of the projection transformation. The following variables are defined to describe this transformation: * $\backslash mathbf\_$ – the 3D position of a point ''A'' that is to be projected. * $\backslash mathbf\_$ – the 3D position of a point ''C'' representing the camera. * $\backslash mathbf\_$ – The orientation of the camera (represented by Tait–Bryan angles). * $\backslash mathbf\_$ - the display surface's position relative to the camera pinhole C. Most conventions use positive z values (the plane being in front of the pinhole), however negative z values are physically more correct, but the image will be inverted both horizontally and vertically. Which results in: * $\backslash mathbf\_$ - the 2D projection of $\backslash mathbf.$ When $\backslash mathbf\_=\backslash langle\; 0,0,0\backslash rangle,$ and $\backslash mathbf\_\; =\; \backslash langle\; 0,0,0\backslash rangle,$ the 3D vector $\backslash langle\; 1,2,0\; \backslash rangle$ is projected to the 2D vector $\backslash langle\; 1,2\; \backslash rangle$. Otherwise, to compute $\backslash mathbf\_$ we first define a vector $\backslash mathbf\_$ as the position of point ''A'' with respect to a coordinate system defined by the camera, with origin in ''C'' and rotated by $\backslash mathbf$ with respect to the initial coordinate system. This is achieved by subtracting $\backslash mathbf$ from $\backslash mathbf$ and then applying a rotation by $-\backslash mathbf$ to the result. This transformation is often called a , and can be expressed as follows, expressing the rotation in terms of rotations about the ''x,'' ''y,'' and ''z'' axes (these calculations assume that the axes are ordered as a left-handed system of axes): :$\backslash begin\; \backslash mathbf\_x\; \backslash \backslash \; \backslash mathbf\_y\; \backslash \backslash \; \backslash mathbf\_z\; \backslash end=\backslash begin\; 1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; \backslash cos\; (\; \backslash mathbf\_x\; )\; \&\; \backslash sin\; (\; \backslash mathbf\_x\; )\; \backslash \backslash \; 0\; \&\; -\; \backslash sin\; (\; \backslash mathbf\_x\; )\; \&\; \backslash cos\; (\; \backslash mathbf\_x\; )\; \backslash end\backslash begin\; \backslash cos\; (\; \backslash mathbf\_y\; )\; \&\; 0\; \&\; -\; \backslash sin\; (\; \backslash mathbf\_y\; )\; \backslash \backslash \; 0\; \&\; 1\; \&\; 0\; \backslash \backslash \; \backslash sin\; (\; \backslash mathbf\_y\; )\; \&\; 0\; \&\; \backslash cos\; (\; \backslash mathbf\_y\; )\; \backslash end\backslash begin\; \backslash cos\; (\; \backslash mathbf\_z\; )\; \&\; \backslash sin\; (\; \backslash mathbf\_z\; )\; \&\; 0\; \backslash \backslash \; -\backslash sin\; (\; \backslash mathbf\_z\; )\; \&\; \backslash cos\; (\; \backslash mathbf\_z\; )\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \backslash end\backslash left(\; \backslash right)$ This representation corresponds to rotating by three Euler angles (more properly, Tait–Bryan angles), using the ''xyz'' convention, which can be interpreted either as "rotate about the ''extrinsic'' axes (axes of the ''scene'') in the order ''z'', ''y'', ''x'' (reading right-to-left)" or "rotate about the ''intrinsic'' axes (axes of the ''camera'') in the order ''x, y, z'' (reading left-to-right)". Note that if the camera is not rotated ($\backslash mathbf\_\; =\; \backslash langle\; 0,0,0\backslash rangle$), then the matrices drop out (as identities), and this reduces to simply a shift: $\backslash mathbf\; =\; \backslash mathbf\; -\; \backslash mathbf.$ Alternatively, without using matrices (let us replace $a\_x\; -\; c\_x$ with $\backslash mathbf$ and so on, and abbreviate $\backslash cos\backslash left(\backslash theta\_\backslash alpha\backslash right)$ to $c\_\backslash alpha$ and $\backslash sin\backslash left(\backslash theta\_\backslash alpha\backslash right)$ to $s\_\backslash alpha$): :$\backslash begin\; \backslash mathbf\_x\; \&\; =\; c\_y\; (s\_z\; \backslash mathbf+c\_z\; \backslash mathbf)-s\_y\; \backslash mathbf\; \backslash \backslash \; \backslash mathbf\_y\; \&\; =\; s\_x\; (c\_y\; \backslash mathbf+s\_y\; (s\_z\; \backslash mathbf+c\_z\; \backslash mathbf))+c\_x\; (c\_z\; \backslash mathbf-s\_z\; \backslash mathbf)\; \backslash \backslash \; \backslash mathbf\_z\; \&\; =\; c\_x\; (c\_y\; \backslash mathbf+s\_y\; (s\_z\; \backslash mathbf+c\_z\; \backslash mathbf))-s\_x\; (c\_z\; \backslash mathbf-s\_z\; \backslash mathbf)\; \backslash end$ This transformed point can then be projected onto the 2D plane using the formula (here, ''x''/''y'' is used as the projection plane; literature also may use ''x''/''z''): :$\backslash begin\; \backslash mathbf\_x\; \&=\; \backslash frac\; \backslash mathbf\_x\; +\; \backslash mathbf\_x,\; \backslash \backslash pt\backslash mathbf\_y\; \&=\; \backslash frac\; \backslash mathbf\_y\; +\; \backslash mathbf\_y.\; \backslash end$ Or, in matrix form using homogeneous coordinates, the system :$\backslash begin\; \backslash mathbf\_x\; \backslash \backslash \; \backslash mathbf\_y\; \backslash \backslash \; \backslash mathbf\_w\; \backslash end=\backslash begin\; 1\; \&\; 0\; \&\; \backslash frac\; \backslash \backslash \; 0\; \&\; 1\; \&\; \backslash frac\; \backslash \backslash \; 0\; \&\; 0\; \&\; \backslash frac\; \backslash end\backslash begin\; \backslash mathbf\_x\; \backslash \backslash \; \backslash mathbf\_y\; \backslash \backslash \; \backslash mathbf\_z\; \backslash end$ in conjunction with an argument using similar triangles, leads to division by the homogeneous coordinate, giving :$\backslash begin\; \backslash mathbf\_x\; \&=\; \backslash mathbf\_x\; /\; \backslash mathbf\_w\; \backslash \backslash \; \backslash mathbf\_y\; \&=\; \backslash mathbf\_y\; /\; \backslash mathbf\_w\; \backslash end$ The distance of the viewer from the display surface, $\backslash mathbf\_z$, directly relates to the field of view, where $\backslash alpha=2\; \backslash cdot\; \backslash arctan(1/\backslash mathbf\_z)$ is the viewed angle. (Note: This assumes that you map the points (-1,-1) and (1,1) to the corners of your viewing surface) The above equations can also be rewritten as: :$\backslash begin\; \backslash mathbf\_x\; \&\; =\; (\backslash mathbf\_x\; \backslash mathbf\_x\; )\; /\; (\backslash mathbf\_z\; \backslash mathbf\_x)\; \backslash mathbf\_z,\; \backslash \backslash \; \backslash mathbf\_y\; \&\; =\; (\backslash mathbf\_y\; \backslash mathbf\_y\; )\; /\; (\backslash mathbf\_z\; \backslash mathbf\_y)\; \backslash mathbf\_z.\; \backslash end$ In which $\backslash mathbf\_$ is the display size, $\backslash mathbf\_$ is the recording surface size (CCD or film), $\backslash mathbf\_z$ is the distance from the recording surface to the entrance pupil (camera center), and $\backslash mathbf\_z$ is the distance, from the 3D point being projected, to the entrance pupil. Subsequent clipping and scaling operations may be necessary to map the 2D plane onto any particular display media.

** Weak perspective projection **

A "weak" perspective projection uses the same principles of an orthographic projection, but requires the scaling factor to be specified, thus ensuring that closer objects appear bigger in the projection, and vice versa. It can be seen as a hybrid between an orthographic and a perspective projection, and described either as a perspective projection with individual point depths $Z\_i$ replaced by an average constant depth $Z\_\backslash text$, or simply as an orthographic projection plus a scaling.
The weak-perspective model thus approximates perspective projection while using a simpler model, similar to the pure (unscaled) orthographic perspective.
It is a reasonable approximation when the depth of the object along the line of sight is small compared to the distance from the camera, and the field of view is small. With these conditions, it can be assumed that all points on a 3D object are at the same distance $Z\_\backslash text$ from the camera without significant errors in the projection (compared to the full perspective model).
''Equation''
:$\backslash begin\; \&\; P\_x\; =\; \backslash frac\; X\; \backslash \backslash pt\&\; P\_y\; =\; \backslash frac\; Y\; \backslash end$
assuming focal length

See also

*3D computer graphics *Camera matrix *Computer graphics *Cross section (geometry) *Cross-sectional view *Curvilinear perspective *Cutaway drawing *Descriptive geometry *Engineering drawing *Exploded-view drawing *Homogeneous coordinates *Homography *Map projection (including Cylindrical projection) *Multiview projection *Perspective (graphical) *Plan (drawing) *Technical drawing *Texture mapping *Transform, clipping, and lighting *Video card *Viewing frustum *Virtual globe

References

** Further reading **

*
*

External links

Creating 3D Environments from Digital Photographs

{{Visualization Category:3D computer graphics Category:3D imaging Category:Display devices Category:Euclidean solid geometry Category:Functions and mappings Category:Graphical projections Category:Linear algebra Category:Projective geometry

Overview

Projection is achieved by the use of imaginary "projectors"; the projected, mental image becomes the technician's vision of the desired, finished picture. Methods provide a uniform imaging procedure among people trained in technical graphics (mechanical drawing, computer aided design, etc.). By following a method, the technician may produce the envisioned picture on a planar surface such as drawing paper. There are two graphical projection categories, each with its own method: *parallel projection *perspective projection

Parallel projection

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 lens and focal point), or "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

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 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 $a\_x$, $a\_y$, $a\_z$ onto the 2D point $b\_x$, $b\_y$ using an orthographic projection parallel to the y axis (where positive ''y'' represents forward direction - profile view), the following equations can be used: :$b\_x\; =\; s\_x\; a\_x\; +\; c\_x$ :$b\_y\; =\; s\_z\; a\_z\; +\; c\_z$ 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: :$\backslash begin\; b\_x\; \backslash \backslash \; b\_y\; \backslash end\; =\; \backslash begin\; s\_x\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; s\_z\; \backslash end\backslash begin\; a\_x\; \backslash \backslash \; a\_y\; \backslash \backslash \; a\_z\; \backslash end\; +\; \backslash begin\; c\_x\; \backslash \backslash \; c\_z\; \backslash end.$ 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

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.

Axonometric projection

Within orthographic projection there is an ancillary category known as ''orthographic pictorial'' or ''axonometric projection''. 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. 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. ''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. A typical characteristic of orthographic pictorials is that one axis of space is usually displayed as vertical. ''Axonometric projections'' are also sometimes known as ''auxiliary views'', as opposed to the ''primary views'' of ''multiview projections''.

=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.

Oblique projection

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 projections, 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. 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) = 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''.

Limitations of parallel projection

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 drawings, 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'' (1961), while not strictly using 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 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.

Mathematical formula

The perspective projection requires a more involved definition as compared to orthographic projections. A conceptual aid to understanding the mechanics of this projection is to imagine the 2D projection as though the object(s) are being viewed through a camera viewfinder. The camera's position, orientation, and field of view control the behavior of the projection transformation. The following variables are defined to describe this transformation: * $\backslash mathbf\_$ – the 3D position of a point ''A'' that is to be projected. * $\backslash mathbf\_$ – the 3D position of a point ''C'' representing the camera. * $\backslash mathbf\_$ – The orientation of the camera (represented by Tait–Bryan angles). * $\backslash mathbf\_$ - the display surface's position relative to the camera pinhole C. Most conventions use positive z values (the plane being in front of the pinhole), however negative z values are physically more correct, but the image will be inverted both horizontally and vertically. Which results in: * $\backslash mathbf\_$ - the 2D projection of $\backslash mathbf.$ When $\backslash mathbf\_=\backslash langle\; 0,0,0\backslash rangle,$ and $\backslash mathbf\_\; =\; \backslash langle\; 0,0,0\backslash rangle,$ the 3D vector $\backslash langle\; 1,2,0\; \backslash rangle$ is projected to the 2D vector $\backslash langle\; 1,2\; \backslash rangle$. Otherwise, to compute $\backslash mathbf\_$ we first define a vector $\backslash mathbf\_$ as the position of point ''A'' with respect to a coordinate system defined by the camera, with origin in ''C'' and rotated by $\backslash mathbf$ with respect to the initial coordinate system. This is achieved by subtracting $\backslash mathbf$ from $\backslash mathbf$ and then applying a rotation by $-\backslash mathbf$ to the result. This transformation is often called a , and can be expressed as follows, expressing the rotation in terms of rotations about the ''x,'' ''y,'' and ''z'' axes (these calculations assume that the axes are ordered as a left-handed system of axes): :$\backslash begin\; \backslash mathbf\_x\; \backslash \backslash \; \backslash mathbf\_y\; \backslash \backslash \; \backslash mathbf\_z\; \backslash end=\backslash begin\; 1\; \&\; 0\; \&\; 0\; \backslash \backslash \; 0\; \&\; \backslash cos\; (\; \backslash mathbf\_x\; )\; \&\; \backslash sin\; (\; \backslash mathbf\_x\; )\; \backslash \backslash \; 0\; \&\; -\; \backslash sin\; (\; \backslash mathbf\_x\; )\; \&\; \backslash cos\; (\; \backslash mathbf\_x\; )\; \backslash end\backslash begin\; \backslash cos\; (\; \backslash mathbf\_y\; )\; \&\; 0\; \&\; -\; \backslash sin\; (\; \backslash mathbf\_y\; )\; \backslash \backslash \; 0\; \&\; 1\; \&\; 0\; \backslash \backslash \; \backslash sin\; (\; \backslash mathbf\_y\; )\; \&\; 0\; \&\; \backslash cos\; (\; \backslash mathbf\_y\; )\; \backslash end\backslash begin\; \backslash cos\; (\; \backslash mathbf\_z\; )\; \&\; \backslash sin\; (\; \backslash mathbf\_z\; )\; \&\; 0\; \backslash \backslash \; -\backslash sin\; (\; \backslash mathbf\_z\; )\; \&\; \backslash cos\; (\; \backslash mathbf\_z\; )\; \&\; 0\; \backslash \backslash \; 0\; \&\; 0\; \&\; 1\; \backslash end\backslash left(\; \backslash right)$ This representation corresponds to rotating by three Euler angles (more properly, Tait–Bryan angles), using the ''xyz'' convention, which can be interpreted either as "rotate about the ''extrinsic'' axes (axes of the ''scene'') in the order ''z'', ''y'', ''x'' (reading right-to-left)" or "rotate about the ''intrinsic'' axes (axes of the ''camera'') in the order ''x, y, z'' (reading left-to-right)". Note that if the camera is not rotated ($\backslash mathbf\_\; =\; \backslash langle\; 0,0,0\backslash rangle$), then the matrices drop out (as identities), and this reduces to simply a shift: $\backslash mathbf\; =\; \backslash mathbf\; -\; \backslash mathbf.$ Alternatively, without using matrices (let us replace $a\_x\; -\; c\_x$ with $\backslash mathbf$ and so on, and abbreviate $\backslash cos\backslash left(\backslash theta\_\backslash alpha\backslash right)$ to $c\_\backslash alpha$ and $\backslash sin\backslash left(\backslash theta\_\backslash alpha\backslash right)$ to $s\_\backslash alpha$): :$\backslash begin\; \backslash mathbf\_x\; \&\; =\; c\_y\; (s\_z\; \backslash mathbf+c\_z\; \backslash mathbf)-s\_y\; \backslash mathbf\; \backslash \backslash \; \backslash mathbf\_y\; \&\; =\; s\_x\; (c\_y\; \backslash mathbf+s\_y\; (s\_z\; \backslash mathbf+c\_z\; \backslash mathbf))+c\_x\; (c\_z\; \backslash mathbf-s\_z\; \backslash mathbf)\; \backslash \backslash \; \backslash mathbf\_z\; \&\; =\; c\_x\; (c\_y\; \backslash mathbf+s\_y\; (s\_z\; \backslash mathbf+c\_z\; \backslash mathbf))-s\_x\; (c\_z\; \backslash mathbf-s\_z\; \backslash mathbf)\; \backslash end$ This transformed point can then be projected onto the 2D plane using the formula (here, ''x''/''y'' is used as the projection plane; literature also may use ''x''/''z''): :$\backslash begin\; \backslash mathbf\_x\; \&=\; \backslash frac\; \backslash mathbf\_x\; +\; \backslash mathbf\_x,\; \backslash \backslash pt\backslash mathbf\_y\; \&=\; \backslash frac\; \backslash mathbf\_y\; +\; \backslash mathbf\_y.\; \backslash end$ Or, in matrix form using homogeneous coordinates, the system :$\backslash begin\; \backslash mathbf\_x\; \backslash \backslash \; \backslash mathbf\_y\; \backslash \backslash \; \backslash mathbf\_w\; \backslash end=\backslash begin\; 1\; \&\; 0\; \&\; \backslash frac\; \backslash \backslash \; 0\; \&\; 1\; \&\; \backslash frac\; \backslash \backslash \; 0\; \&\; 0\; \&\; \backslash frac\; \backslash end\backslash begin\; \backslash mathbf\_x\; \backslash \backslash \; \backslash mathbf\_y\; \backslash \backslash \; \backslash mathbf\_z\; \backslash end$ in conjunction with an argument using similar triangles, leads to division by the homogeneous coordinate, giving :$\backslash begin\; \backslash mathbf\_x\; \&=\; \backslash mathbf\_x\; /\; \backslash mathbf\_w\; \backslash \backslash \; \backslash mathbf\_y\; \&=\; \backslash mathbf\_y\; /\; \backslash mathbf\_w\; \backslash end$ The distance of the viewer from the display surface, $\backslash mathbf\_z$, directly relates to the field of view, where $\backslash alpha=2\; \backslash cdot\; \backslash arctan(1/\backslash mathbf\_z)$ is the viewed angle. (Note: This assumes that you map the points (-1,-1) and (1,1) to the corners of your viewing surface) The above equations can also be rewritten as: :$\backslash begin\; \backslash mathbf\_x\; \&\; =\; (\backslash mathbf\_x\; \backslash mathbf\_x\; )\; /\; (\backslash mathbf\_z\; \backslash mathbf\_x)\; \backslash mathbf\_z,\; \backslash \backslash \; \backslash mathbf\_y\; \&\; =\; (\backslash mathbf\_y\; \backslash mathbf\_y\; )\; /\; (\backslash mathbf\_z\; \backslash mathbf\_y)\; \backslash mathbf\_z.\; \backslash end$ In which $\backslash mathbf\_$ is the display size, $\backslash mathbf\_$ is the recording surface size (CCD or film), $\backslash mathbf\_z$ is the distance from the recording surface to the entrance pupil (camera center), and $\backslash mathbf\_z$ is the distance, from the 3D point being projected, to the entrance pupil. Subsequent clipping and scaling operations may be necessary to map the 2D plane onto any particular display media.

See also

*3D computer graphics *Camera matrix *Computer graphics *Cross section (geometry) *Cross-sectional view *Curvilinear perspective *Cutaway drawing *Descriptive geometry *Engineering drawing *Exploded-view drawing *Homogeneous coordinates *Homography *Map projection (including Cylindrical projection) *Multiview projection *Perspective (graphical) *Plan (drawing) *Technical drawing *Texture mapping *Transform, clipping, and lighting *Video card *Viewing frustum *Virtual globe

References

External links

Creating 3D Environments from Digital Photographs

{{Visualization Category:3D computer graphics Category:3D imaging Category:Display devices Category:Euclidean solid geometry Category:Functions and mappings Category:Graphical projections Category:Linear algebra Category:Projective geometry