3d drawing projections are aligned

Design technique

Classification of some 3D projections

A 3D projection (or graphical projection) is a pattern technique used to display a three-dimensional (3D) object on a 2-dimensional (2D) surface. These projections rely on visual perspective and aspect analysis to project a circuitous 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.^[one] Heron could be called the begetter of 3D. 3D Projection is the basis of the concept for Computer Graphics simulating fluid flows to imitate realistic effects.^[2] Lucas Films 'ILM grouping is credited with introducing the concept (and even the term "Particle effect").

In 1982, the first all-digital computer generated sequence for a movement picture file was in: Star Trek II: The Wrath of Khan. A 1984 patent related to this concept was written by William East Masters, "Figurer automated manufacturing procedure and organisation" US4665492A using mass particles to fabricate a cup.^[3] The process of particle deposition is one engineering science of 3D press.

3D projections use the principal qualities of an object'south basic shape to create a map of points, that are so connected to one another to create a visual element. The result is a graphic that contains conceptual properties to interpret that the figure or paradigm as non actually apartment (2nd), but rather, as a solid object (3D) existence viewed on a 2d brandish.

3D objects are largely displayed on two-dimensional mediums (i.e. paper and computer monitors). Every bit such, graphical projections are a commonly used blueprint element; notably, in technology drawing, drafting, and reckoner graphics. Projections can exist calculated through employment of mathematical assay and formulae, or by using diverse geometric and optical techniques.

Overview [edit]

Several types of graphical projection compared

Various projections and how they are produced

Projection is accomplished past the apply of imaginary "projectors"; the projected, mental prototype becomes the technician's vision of the desired, finished motion picture.^{[ farther explanation needed ]} Methods provide a uniform imaging procedure amid people trained in technical graphics (mechanical drawing, figurer aided design, etc.). By following a method, the technician may produce the envisioned moving picture on a planar surface such every bit cartoon paper.

There are two graphical projection categories, each with its own method:

• parallel projection
• perspective projection

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Parallel project [edit]

Parallel projection corresponds to a perspective projection with a hypothetical viewpoint; i.e. ane where the camera lies an infinite distance abroad from the object and has an space 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 photographic camera's lens and focal point), or "zoom".

Images fatigued in parallel project rely upon the technique of axonometry ("to mensurate forth axes"), as described in Pohlke'south theorem. In general, the resulting paradigm is oblique (the rays are not perpendicular to the image aeroplane); merely in special cases the result is orthographic (the rays are perpendicular to the image plane). Axonometry should not exist confused with axonometric projection, as in English language literature the latter normally refers only to a specific class of pictorials (see below).

Orthographic project [edit]

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 project (the lines of project are parallel both in reality and in the projection plane). It is the projection blazon of choice for working drawings.

If the normal of the viewing plane (the photographic camera direction) is parallel to one of the principal axes (which is the x, y, or z axis), the mathematical transformation is as follows; To project the 3D signal ${\displaystyle a_{x}}$ , ${\displaystyle a_{y}}$ , ${\displaystyle a_{z}}$ onto the 2D betoken ${\displaystyle b_{ten}}$ , ${\displaystyle b_{y}}$ using an orthographic projection parallel to the y axis (where positive y represents forwards direction - profile view), the following equations tin can be used:

${\displaystyle b_{x}=s_{x}a_{x}+c_{x}}$

${\displaystyle b_{y}=s_{z}a_{z}+c_{z}}$

where the vector s is an arbitrary scale factor, and c is an arbitrary commencement. These constants are optional, and can be used to properly align the viewport. Using matrix multiplication, the equations become:

${\displaystyle {\begin{bmatrix}b_{x}\\b_{y}\end{bmatrix}}={\begin{bmatrix}s_{x}&0&0\\0&0&s_{z}\end{bmatrix}}{\begin{bmatrix}a_{10}\\a_{y}\\a_{z}\end{bmatrix}}+{\begin{bmatrix}c_{x}\\c_{z}\end{bmatrix}}.}$

While orthographically projected images represent the three dimensional nature of the object projected, they do not represent the object as information technology would be recorded photographically or perceived past a viewer observing information technology directly. In item, 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 issue, lengths are not foreshortened every bit they would be in a perspective project.

Multiview projection [edit]

Symbols used to define whether a multiview projection is either 3rd Angle (right) or First Angle (left).

With multiview projections, upwards 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 exist idea of equally being projected onto planes that form a 6-sided box around the object. Although six unlike sides can be drawn, usually three views of a drawing give plenty information to make a 3D object. These views are known equally front view, top view, and end view. The terms elevation, plan and section are also used.

Oblique projection [edit]

Potting bench drawn in chiffonier projection with an angle of 45° and a ratio of ii/iii

Stone arch fatigued 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 (calibration) are arbitrary. The distortion created thereby is usually attenuated by aligning i plane of the imaged object to exist parallel with the airplane of projection thereby creating a truthful shape, full-size paradigm of the chosen plane. Special types of oblique projections are:

Cavalier projection (45°) [edit]

In cavalier project (sometimes condescending perspective or high view bespeak) a point of the object is represented by three coordinates, 10, y and z. On the drawing, it is represented by but 2 coordinates, x″ and y″. On the flat drawing, two axes, ten and z on the figure, are perpendicular and the length on these axes are drawn with a one:one calibration; it is thus similar to the dimetric projections, although it is not an axonometric project, as the 3rd axis, here y, is drawn in diagonal, making an arbitrary bending with the x″ axis, usually 30 or 45°. The length of the tertiary axis is not scaled.

Cabinet projection [edit]

The term cabinet projection (sometimes cabinet perspective) stems from its utilize in illustrations by the piece of furniture manufacture.^{[ commendation needed ]} Like condescending perspective, i face up of the projected object is parallel to the viewing plane, and the third axis is projected equally going off in an bending (typically xxx° or 45° or arctan(2) = 63.four°). Unlike cavalier projection, where the 3rd centrality keeps its length, with cabinet projection the length of the receding lines is cutting in half.

Military projection [edit]

A variant of oblique project is called armed services 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 project is given by rotation in the xy-plane and a vertical translation an amount z.^[4]

Axonometric projection [edit]

Axonometric projections testify an image of an object equally viewed from a skew direction in order to reveal all three directions (axes) of infinite in one pic.^[v] Axonometric projections may be either orthographic or oblique. Axonometric instrument drawings are often used to guess 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 and then be taken for economic system of effort and best effect.^{[ clarification needed ]}

Axonometric projection is further subdivided into 3 categories: isometric projection, dimetric project, and trimetric project, depending on the verbal angle at which the view deviates from the orthogonal.^{[half-dozen] [vii]} A typical characteristic of orthographic pictorials is that one axis of space is ordinarily displayed as vertical.

Axonometric projections are also sometimes known equally auxiliary views, as opposed to the principal views of multiview projections.

Isometric projection [edit]

In isometric pictorials (for methods, see Isometric projection), the direction of viewing is such that the three axes of space announced as foreshortened, and there is a mutual angle of 120° between them. The baloney caused by foreshortening is compatible, therefore the proportionality of all sides and lengths are preserved, and the axes share a common calibration. This enables measurements to be read or taken straight from the drawing.

Dimetric projection [edit]

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

Trimetric projection [edit]

In trimetric pictorials (for methods, encounter Trimetric projection), the direction of viewing is such that all of the 3 axes of space appear unequally foreshortened. The scale along each of the iii axes and the angles amidst them are determined separately equally dictated by the angle of viewing. Approximations in Trimetric drawings are common.

Limitations of parallel project [edit]

An example of the limitations of isometric projection. The acme divergence between the red and blue balls cannot be determined locally.

The Penrose stairs depicts a staircase which seems to ascend (anticlockwise) or descend (clockwise) still forms a continuous loop.

Objects drawn with parallel project do not appear larger or smaller as they extend closer to or away from the viewer. While advantageous for architectural drawings, where measurements must exist taken straight from the prototype, the result is a perceived distortion, since unlike perspective projection, this is not how our eyes or photography unremarkably work. It also can easily issue in situations where depth and altitude are difficult to guess, as is shown in the illustration to the correct.

In this isometric cartoon, the blueish sphere is two units college than the red one. However, this departure in elevation is not credible if one covers the right half of the moving picture, as the boxes (which serve as clues suggesting tiptop) are so obscured.

This visual ambiguity has been exploited in op art, too as "impossible object" drawings. M. C. Escher'south 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 downwards path, only to then paradoxically fall again as it returns to its source. The water thus appears to disobey the constabulary of conservation of energy. An extreme instance is depicted in the picture show Inception, where by a forced perspective trick an immobile stairway changes its connectivity. The video game Fez uses tricks of perspective to decide where a player tin can and cannot move in a puzzle-like way.

Perspective project [edit]

Perspective of a geometric solid using two vanishing points. In this case, the map of the solid (orthogonal projection) is drawn below the perspective, as if bending the footing plane.

Axonometric projection of a scheme displaying the relevant elements of a vertical motion picture plane perspective. The continuing point (P.S.) is located on the basis plane π, and the bespeak of view (P.V.) is right to a higher place it. P.P. is its projection on the flick aeroplane α. L.O. and L.T. are the horizon and the footing lines (linea d'orizzonte and linea di terra). The bold lines due south and q prevarication on π, and intercept α at Ts and Tq respectively. The parallel lines through P.V. (in ruby-red) intercept L.O. in the vanishing points Fs and Fq: thus i can draw the projections south′ and q′, and hence besides their intersection R′ on R.

Perspective projection or perspective transformation is a linear projection where iii dimensional objects are projected on a movie airplane. This has the issue that distant objects announced smaller than nearer objects.

Information technology too means that lines which are parallel in nature (that is, meet at the signal 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, chosen the vanishing indicate. Photographic lenses and the man heart work in the aforementioned way, therefore perspective projection looks most realistic.^[8] Perspective project is usually categorized into one-point, ii-indicate and three-signal perspective, depending on the orientation of the project plane towards the axes of the depicted object.^[ix]

Graphical project methods rely on the duality between lines and points, whereby ii directly lines decide a signal while two points determine a straight line. The orthogonal projection of the heart point onto the film plane is chosen the principal vanishing signal (P.P. in the scheme on the left, from the Italian term punto principale, coined during the renaissance).^[x]

2 relevant points of a line are:

• its intersection with the picture airplane, and
• its vanishing point, constitute at the intersection between the parallel line from the eye betoken and the motion picture plane.

The principal vanishing point is the vanishing point of all horizontal lines perpendicular to the motion picture airplane. The vanishing points of all horizontal lines lie on the horizon line. If, as is often the example, the movie airplane is vertical, all vertical lines are fatigued vertically, and have no finite vanishing point on the picture plane. Various graphical methods tin be easily envisaged for projecting geometrical scenes. For example, lines traced from the centre point at 45° to the picture aeroplane intersect the latter forth a circle whose radius is the distance of the eye indicate from the plane, thus tracing that circle aids the construction of all the vanishing points of 45° lines; in particular, the intersection of that circumvolve with the horizon line consists of ii distance points. They are useful for cartoon 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 bespeak —which determines the horizon line— the 45° vanishing point on the left side of the drawing completes the characterization of the (as distant) point of view. Two lines are fatigued from the orthogonal projection of each vertex, one at 45° and one at 90° to the movie plane. After intersecting the footing line, those lines go toward the distance bespeak (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 let accurate measurements, perspective projection shows distant objects as smaller to provide additional realism.

Mathematical formula [edit]

The perspective projection requires a more involved definition as compared to orthographic projections. A conceptual help to understanding the mechanics of this projection is to imagine the 2d project every bit though the object(south) are beingness viewed through a camera viewfinder. The camera's position, orientation, and field of view command the beliefs of the projection transformation. The following variables are defined to describe this transformation:

• ${\displaystyle \mathbf {a} _{x,y,z}}$ – the 3D position of a point A that is to be projected.
• ${\displaystyle \mathbf {c} _{x,y,z}}$ – the 3D position of a indicate C representing the camera.
• ${\displaystyle \mathbf {\theta } _{x,y,z}}$ – The orientation of the camera (represented past Tait–Bryan angles).
• ${\displaystyle \mathbf {e} _{x,y,z}}$ – the brandish surface's position relative to the camera pinhole C.^[11]

Nigh conventions utilize positive z values (the plane existence in front of the pinhole), withal negative z values are physically more than correct, merely the image will be inverted both horizontally and vertically. Which results in:

When ${\displaystyle \mathbf {c} _{ten,y,z}=\langle 0,0,0\rangle ,}$ and ${\displaystyle \mathbf {\theta } _{x,y,z}=\langle 0,0,0\rangle ,}$ the 3D vector ${\displaystyle \langle 1,2,0\rangle }$ is projected to the 2D vector ${\displaystyle \langle i,ii\rangle }$ .

Otherwise, to compute ${\displaystyle \mathbf {b} _{ten,y}}$ we first define a vector ${\displaystyle \mathbf {d} _{10,y,z}}$ as the position of point A with respect to a coordinate arrangement defined by the camera, with origin in C and rotated by ${\displaystyle \mathbf {\theta } }$ with respect to the initial coordinate system. This is accomplished by subtracting ${\displaystyle \mathbf {c} }$ from ${\displaystyle \mathbf {a} }$ then applying a rotation by ${\displaystyle -\mathbf {\theta } }$ to the issue. This transformation is often chosen a photographic camera transform , and can exist expressed as follows, expressing the rotation in terms of rotations about the x, y, and z axes (these calculations presume that the axes are ordered as a left-handed system of axes): ^{[12] [13]}

${\displaystyle {\begin{bmatrix}\mathbf {d} _{x}\\\mathbf {d} _{y}\\\mathbf {d} _{z}\end{bmatrix}}={\begin{bmatrix}ane&0&0\\0&\cos(\mathbf {\theta } _{x})&\sin(\mathbf {\theta } _{x})\\0&-\sin(\mathbf {\theta } _{x})&\cos(\mathbf {\theta } _{ten})\stop{bmatrix}}{\brainstorm{bmatrix}\cos(\mathbf {\theta } _{y})&0&-\sin(\mathbf {\theta } _{y})\\0&one&0\\\sin(\mathbf {\theta } _{y})&0&\cos(\mathbf {\theta } _{y})\end{bmatrix}}{\brainstorm{bmatrix}\cos(\mathbf {\theta } _{z})&\sin(\mathbf {\theta } _{z})&0\\-\sin(\mathbf {\theta } _{z})&\cos(\mathbf {\theta } _{z})&0\\0&0&1\cease{bmatrix}}\left({{\brainstorm{bmatrix}\mathbf {a} _{x}\\\mathbf {a} _{y}\\\mathbf {a} _{z}\\\end{bmatrix}}-{\begin{bmatrix}\mathbf {c} _{10}\\\mathbf {c} _{y}\\\mathbf {c} _{z}\\\end{bmatrix}}}\right)}$

This representation corresponds to rotating by 3 Euler angles (more properly, Tait–Bryan angles), using the xyz convention, which can exist interpreted either as "rotate about the extrinsic axes (axes of the scene) in the order z, y, x (reading correct-to-left)" or "rotate about the intrinsic axes (axes of the photographic camera) in the order x, y, z (reading left-to-correct)". Note that if the photographic camera is not rotated ( ${\displaystyle \mathbf {\theta } _{x,y,z}=\langle 0,0,0\rangle }$ ), so the matrices drop out (equally identities), and this reduces to merely a shift: ${\displaystyle \mathbf {d} =\mathbf {a} -\mathbf {c} .}$

Alternatively, without using matrices (allow us replace ${\displaystyle a_{x}-c_{x}}$ with ${\displaystyle \mathbf {10} }$ and and then on, and abbreviate ${\displaystyle \cos \left(\theta _{\alpha }\right)}$ to ${\displaystyle c_{\alpha }}$ and ${\displaystyle \sin \left(\theta _{\blastoff }\correct)}$ to ${\displaystyle s_{\alpha }}$ ):

${\displaystyle {\begin{aligned}\mathbf {d} _{x}&=c_{y}(s_{z}\mathbf {y} +c_{z}\mathbf {x} )-s_{y}\mathbf {z} \\\mathbf {d} _{y}&=s_{x}(c_{y}\mathbf {z} +s_{y}(s_{z}\mathbf {y} +c_{z}\mathbf {ten} ))+c_{x}(c_{z}\mathbf {y} -s_{z}\mathbf {x} )\\\mathbf {d} _{z}&=c_{x}(c_{y}\mathbf {z} +s_{y}(s_{z}\mathbf {y} +c_{z}\mathbf {x} ))-s_{x}(c_{z}\mathbf {y} -s_{z}\mathbf {ten} )\finish{aligned}}}$

This transformed point can then exist projected onto the 2nd plane using the formula (here, x/y is used as the projection plane; literature also may use 10/z):^[14]

${\displaystyle {\begin{aligned}\mathbf {b} _{ten}&={\frac {\mathbf {east} _{z}}{\mathbf {d} _{z}}}\mathbf {d} _{x}+\mathbf {e} _{x},\\[5pt]\mathbf {b} _{y}&={\frac {\mathbf {east} _{z}}{\mathbf {d} _{z}}}\mathbf {d} _{y}+\mathbf {e} _{y}.\finish{aligned}}}$

Or, in matrix form using homogeneous coordinates, the arrangement

${\displaystyle {\begin{bmatrix}\mathbf {f} _{10}\\\mathbf {f} _{y}\\\mathbf {f} _{w}\end{bmatrix}}={\begin{bmatrix}ane&0&{\frac {\mathbf {e} _{x}}{\mathbf {e} _{z}}}\\0&ane&{\frac {\mathbf {eastward} _{y}}{\mathbf {e} _{z}}}\\0&0&{\frac {ane}{\mathbf {e} _{z}}}\end{bmatrix}}{\begin{bmatrix}\mathbf {d} _{10}\\\mathbf {d} _{y}\\\mathbf {d} _{z}\end{bmatrix}}}$

in conjunction with an argument using similar triangles, leads to division by the homogeneous coordinate, giving

${\displaystyle {\begin{aligned}\mathbf {b} _{x}&=\mathbf {f} _{10}/\mathbf {f} _{w}\\\mathbf {b} _{y}&=\mathbf {f} _{y}/\mathbf {f} _{w}\end{aligned}}}$

The distance of the viewer from the display surface, ${\displaystyle \mathbf {e} _{z}}$ , directly relates to the field of view, where ${\displaystyle \blastoff =2\cdot \arctan(1/\mathbf {east} _{z})}$ is the viewed angle. (Annotation: This assumes that y'all map the points (-1,-one) and (1,i) to the corners of your viewing surface)

The above equations can besides exist rewritten equally:

${\displaystyle {\brainstorm{aligned}\mathbf {b} _{10}&=(\mathbf {d} _{10}\mathbf {s} _{ten})/(\mathbf {d} _{z}\mathbf {r} _{x})\mathbf {r} _{z},\\\mathbf {b} _{y}&=(\mathbf {d} _{y}\mathbf {southward} _{y})/(\mathbf {d} _{z}\mathbf {r} _{y})\mathbf {r} _{z}.\end{aligned}}}$

In which ${\displaystyle \mathbf {due south} _{x,y}}$ is the display size, ${\displaystyle \mathbf {r} _{ten,y}}$ is the recording surface size (CCD or film), ${\displaystyle \mathbf {r} _{z}}$ is the distance from the recording surface to the entrance pupil (camera center), and ${\displaystyle \mathbf {d} _{z}}$ is the altitude, from the 3D point being projected, to the entrance pupil.

Subsequent clipping and scaling operations may be necessary to map the 2D plane onto whatever particular brandish media.

Weak perspective projection [edit]

A "weak" perspective projection uses the same principles of an orthographic projection, but requires the scaling cistron to be specified, thus ensuring that closer objects appear bigger in the project, and vice versa. It can exist seen as a hybrid between an orthographic and a perspective projection, and described either as a perspective projection with individual signal depths ${\displaystyle Z_{i}}$ replaced by an average abiding depth ${\displaystyle Z_{\text{ave}}}$ ,^[fifteen] or simply as an orthographic projection plus a scaling.^[16]

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 forth the line of sight is pocket-sized compared to the distance from the camera, and the field of view is pocket-size. With these conditions, it can exist assumed that all points on a 3D object are at the same distance ${\displaystyle Z_{\text{ave}}}$ from the photographic camera without significant errors in the project (compared to the full perspective model).

Equation

${\displaystyle {\brainstorm{aligned}&P_{x}={\frac {X}{Z_{\text{ave}}}}\\[5pt]&P_{y}={\frac {Y}{Z_{\text{ave}}}}\end{aligned}}}$

bold focal length $f=1$ .

Diagram [edit]

To determine which screen 10-coordinate corresponds to a point at ${\displaystyle A_{10},A_{z}}$ multiply the betoken coordinates by:

${\displaystyle B_{x}=A_{x}{\frac {B_{z}}{A_{z}}}}$

where

${\displaystyle B_{x}}$ is the screen ten coordinate

${\displaystyle A_{x}}$ is the model x coordinate

${\displaystyle B_{z}}$ is the focal length—the axial distance from the photographic camera middle to the prototype plane

${\displaystyle A_{z}}$ is the subject distance.

Considering the camera is in 3D, the same works for the screen y-coordinate, substituting y for 10 in the above diagram and equation.

Yous can utilize that to do clipping techniques, replacing the variables with values of the signal that's are out of the FOV-angle and the signal inside Camera Matrix.

This technique, also known as "Inverse Photographic camera", is a Perspective Projection Calculus with known values to calculate the last point on visible angle, projecting from the invisible point, after all needed transformations finished.

See likewise [edit]

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

References [edit]

1. ^ Peddie, Jon. (2013). The history of visual magic in computers : how beautiful images are fabricated in CAD, 3D, VR and AR. London: Springer. p. 25. ISBN978-1-4471-4932-iii. OCLC 849634980.
2. ^ Peddie, Jon. (2013). The history of visual magic in computers : how beautiful images are fabricated in CAD, 3D, VR and AR. London: Springer. pp. 67–69. ISBN978-1-4471-4932-3. OCLC 849634980.
3. ^ Patent 4665492, Figure 2A, 2B and 2C.
4. ^ "Axonometric projections - a technical overview". Retrieved 24 Apr 2015.
5. ^ Mitchell, William; Malcolm McCullough (1994). Digital pattern media. John Wiley and Sons. p. 169. ISBN978-0-471-28666-0.
6. ^ Maynard, Patric (2005). Drawing distinctions: the varieties of graphic expression. Cornell University Press. p. 22. ISBN978-0-8014-7280-0.
7. ^ McReynolds, Tom; David Blythe (2005). Advanced graphics programming using openGL. Elsevier. p. 502. ISBN978-1-55860-659-3.
8. ^ D. Hearn, & Chiliad. Baker (1997). Computer Graphics, C Version. Englewood Cliffs: Prentice Hall], affiliate 9
9. ^ James Foley (1997). Computer Graphics. Boston: Addison-Wesley. ISBN 0-201-84840-6], chapter 6
10. ^ Kirsti Andersen (2007), The geometry of an art, Springer, p. xxix, ISBN9780387259611
11. ^ Ingrid Carlbom, Joseph Paciorek (1978). "Planar Geometric Projections and Viewing Transformations" (PDF). ACM Computing Surveys. 10 (4): 465–502. CiteSeerX10.i.ane.532.4774. doi:x.1145/356744.356750. S2CID 708008.
12. ^ Riley, K F (2006). Mathematical Methods for Physics and Engineering . Cambridge Academy Printing. pp. 931, 942. doi:10.2277/0521679710. ISBN978-0-521-67971-8.
13. ^ Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, Mass.: Addison-Wesley Pub. Co. pp. 146–148. ISBN978-0-201-02918-five.
14. ^ Sonka, M; Hlavac, V; Boyle, R (1995). Epitome Processing, Analysis & Motorcar Vision (2nd ed.). Chapman and Hall. p. 14. ISBN978-0-412-45570-4.
15. ^ Subhashis Banerjee (2002-02-18). "The Weak-Perspective Camera".
16. ^ Alter, T. D. (July 1992). 3D Pose from 3 Respective Points under Weak-Perspective Projection (PDF) (Technical report). MIT AI Lab.

Further reading [edit]

• Kenneth C. Finney (2004). 3D Game Programming All in One . Thomson Course. p. 93. ISBN978-1-59200-136-1. 3D project.
• Koehler; Dr. Ralph (Dec 2000). 2D/3D Graphics and Splines with Source Lawmaking. ISBN978-0759611870.

External links [edit]

• Creating 3D Environments from Digital Photographs

dunnhatomentand.blogspot.com

Source: https://en.wikipedia.org/wiki/3D_projection