drawing views of 3d objects

Design technique

Nomenclature of some 3D projections

A 3D projection (or graphical projection) is a design technique used to display a three-dimensional (3D) object on a two-dimensional (second) surface. These projections rely on visual perspective and aspect analysis to project a complex object for viewing capability on a simpler aeroplane. This concept of extending 2D geometry to 3D was mastered past Heron of Alexandria in the first century.^[one] Heron could be called the father of 3D. 3D Projection is the ground of the concept for Estimator Graphics simulating fluid flows to imitate realistic effects.^[two] Lucas Films 'ILM group is credited with introducing the concept (and even the term "Particle effect").

In 1982, the start all-digital computer generated sequence for a move picture file was in: Star Trek II: The Wrath of Khan. A 1984 patent related to this concept was written past William E Masters, "Reckoner automated manufacturing process and system" US4665492A using mass particles to fabricate a loving cup.^[three] The procedure of particle deposition is one technology of 3D printing.

3D projections employ the primary qualities of an object'due south basic shape to create a map of points, that are and 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 image equally non really flat (second), but rather, as a solid object (3D) being viewed on a second 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 technology drawing, drafting, and computer graphics. Projections can exist calculated through employment of mathematical assay and formulae, or by using various geometric and optical techniques.

Overview [edit]

Several types of graphical projection compared

Various projections and how they are produced

Projection is achieved by the use of imaginary "projectors"; the projected, mental image becomes the technician'due south vision of the desired, finished film.^{[ further explanation needed ]} Methods provide a uniform imaging procedure amid people trained in technical graphics (mechanical cartoon, computer aided blueprint, etc.). By following a method, the technician may produce the envisioned picture on a planar surface such every bit drawing paper.

In that location are two graphical project categories, each with its ain method:

• parallel project
• perspective projection

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Parallel projection [edit]

Parallel projection corresponds to a perspective projection with a hypothetical viewpoint; i.e. one where the camera lies an infinite altitude away 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 infinite remain parallel in the two-dimensional projected image. Parallel projection also corresponds to a perspective projection with an infinite focal length (the altitude from a camera's lens and focal point), or "zoom".

Images fatigued in parallel projection rely upon the technique of axonometry ("to mensurate along axes"), as described in Pohlke's theorem. In general, the resulting prototype is oblique (the rays are non perpendicular to the prototype airplane); but in special cases the consequence is orthographic (the rays are perpendicular to the image plane). Axonometry should not be dislocated with axonometric projection, as in English literature the latter ordinarily refers only to a specific class of pictorials (see below).

Orthographic projection [edit]

The orthographic projection is derived from the principles of descriptive geometry and is a ii-dimensional representation of a 3-dimensional object. It is a parallel project (the lines of projection are parallel both in reality and in the project plane). It is the projection type of pick for working drawings.

If the normal of the viewing plane (the camera direction) is parallel to one of the main axes (which is the x, y, or z axis), the mathematical transformation is as follows; To projection the 3D point ${\displaystyle a_{x}}$ , ${\displaystyle a_{y}}$ , ${\displaystyle a_{z}}$ onto the 2D betoken ${\displaystyle b_{x}}$ , ${\displaystyle b_{y}}$ using an orthographic projection parallel to the y centrality (where positive y represents forward direction - profile view), the post-obit equations can be used:

${\displaystyle b_{10}=s_{ten}a_{x}+c_{x}}$

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

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

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

While orthographically projected images represent the 3 dimensional nature of the object projected, they practise non represent the object equally it would be recorded photographically or perceived past a viewer observing information technology 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 about to the virtual viewer. Every bit a outcome, lengths are non foreshortened equally they would exist in a perspective projection.

Multiview projection [edit]

Symbols used to ascertain whether a multiview projection is either Third Angle (right) or Commencement Angle (left).

With multiview projections, up to six pictures (called master 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: get-go-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 effectually the object. Although 6 different sides tin can exist drawn, normally three views of a cartoon requite enough data to brand a 3D object. These views are known as front view, superlative view, and end view. The terms meridian, 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/3

Stone arch fatigued in military perspective

In oblique projections the parallel projection rays are non perpendicular to the viewing plane as with orthographic projection, but strike the projection aeroplane 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 baloney created thereby is usually attenuated by aligning one plane of the imaged object to exist parallel with the aeroplane of projection thereby creating a true shape, full-size image of the chosen airplane. Special types of oblique projections are:

Cavalier project (45°) [edit]

In cavalier projection (sometimes cavalier perspective or high view indicate) a point of the object is represented by 3 coordinates, x, y and z. On the cartoon, it is represented past simply two coordinates, 10″ and y″. On the apartment drawing, two axes, x and z on the figure, are perpendicular and the length on these axes are drawn with a ane:one scale; it is thus like to the dimetric projections, although information technology is not an axonometric projection, every bit the tertiary axis, here y, is fatigued in diagonal, making an capricious angle with the x″ axis, ordinarily thirty or 45°. The length of the third axis is not scaled.

Cabinet projection [edit]

The term cabinet projection (sometimes cabinet perspective) stems from its use in illustrations past the furniture industry.^{[ commendation needed ]} Like cavalier perspective, one face of the projected object is parallel to the viewing airplane, and the third axis is projected as going off in an angle (typically 30° or 45° or arctan(2) = 63.four°). Different cavalier projection, where the third axis keeps its length, with chiffonier projection the length of the receding lines is cutting in half.

Armed forces project [edit]

A variant of oblique project is called military projection. In this example, 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.^[4]

Axonometric projection [edit]

Axonometric projections show an image of an object as viewed from a skew management in order to reveal all three directions (axes) of space in one picture.^[5] Axonometric projections may be either orthographic or oblique. Axonometric instrument drawings are ofttimes used to judge graphical perspective projections, only there is attendant distortion in the approximation. Because pictorial projections innately contain this distortion, in instrument drawings of pictorials great liberties may and so be taken for economy of effort and all-time outcome.^{[ clarification needed ]}

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.^{[6] [7]} A typical characteristic of orthographic pictorials is that i axis of space is usually displayed as vertical.

Axonometric projections are besides sometimes known as auxiliary views, as opposed to the primary views of multiview projections.

Isometric projection [edit]

In isometric pictorials (for methods, see Isometric projection), the direction of viewing is such that the 3 axes of space appear as foreshortened, and there is a mutual angle of 120° between them. The distortion acquired 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 straight from the drawing.

Dimetric project [edit]

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

Trimetric projection [edit]

In trimetric pictorials (for methods, see Trimetric project), the direction of viewing is such that all of the 3 axes of space announced unequally foreshortened. The calibration along each of the three axes and the angles among them are determined separately as dictated by the bending of viewing. Approximations in Trimetric drawings are common.

Limitations of parallel project [edit]

An example of the limitations of isometric projection. The pinnacle departure betwixt the red and blue balls cannot be adamant locally.

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

Objects fatigued with parallel projection exercise not appear larger or smaller as they extend closer to or abroad from the viewer. While advantageous for architectural drawings, where measurements must be taken directly from the epitome, the result is a perceived distortion, since unlike perspective projection, this is not how our eyes or photography normally work. Information technology also can easily outcome in situations where depth and altitude are difficult to gauge, as is shown in the illustration to the correct.

In this isometric drawing, the blue sphere is two units higher than the cherry-red i. However, this difference in tiptop is not apparent if 1 covers the right half of the pic, as the boxes (which serve every bit clues suggesting tiptop) are and then obscured.

This visual ambiguity has been exploited in op art, too as "impossible object" drawings. M. C. Escher'southward Waterfall (1961), while not strictly utilizing parallel projection, is a well-known example, in which a channel of h2o seems to travel unaided along a downward path, only to then paradoxically fall over again every bit it returns to its source. The water thus appears to disobey the law of conservation of energy. An farthermost example is depicted in the film Inception, where by a forced perspective trick an immobile stairway changes its connectivity. The video game Fez uses tricks of perspective to determine where a player can and cannot movement in a puzzle-similar mode.

Perspective projection [edit]

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

Axonometric projection of a scheme displaying the relevant elements of a vertical picture airplane perspective. The standing point (P.Southward.) is located on the ground plane π, and the point of view (P.V.) is right in a higher place it. P.P. is its projection on the motion-picture show plane α. Fifty.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.Five. (in red) intercept L.O. in the vanishing points Fs and Fq: thus one can draw the projections due south′ and q′, and hence too their intersection R′ on R.

Perspective projection or perspective transformation is a linear projection where three dimensional objects are projected on a motion picture plane. This has the outcome that afar objects announced 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 paradigm, for instance if railways are pictured with perspective project, they appear to converge towards a single betoken, called the vanishing point. Photographic lenses and the human being centre work in the aforementioned way, therefore perspective projection looks most realistic.^[eight] Perspective projection is usually categorized into one-point, two-point and three-point perspective, depending on the orientation of the project airplane towards the axes of the depicted object.^[ix]

Graphical projection methods rely on the duality between lines and points, whereby 2 direct lines make up one's mind a betoken while ii points determine a straight line. The orthogonal projection of the eye signal onto the picture airplane is chosen the chief vanishing signal (P.P. in the scheme on the left, from the Italian term punto principale, coined during the renaissance).^[10]

Two relevant points of a line are:

• its intersection with the moving picture plane, and
• its vanishing point, institute at the intersection between the parallel line from the eye indicate and the motion-picture show plane.

The principal vanishing signal is the vanishing betoken of all horizontal lines perpendicular to the pic plane. The vanishing points of all horizontal lines lie on the horizon line. If, every bit is oftentimes the case, the film plane is vertical, all vertical lines are fatigued vertically, and have no finite vanishing bespeak on the picture plane. Various graphical methods tin exist easily envisaged for projecting geometrical scenes. For case, lines traced from the eye point at 45° to the picture airplane intersect the latter along a circle whose radius is the altitude of the eye point from the airplane, 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 two altitude 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, afterward choosing the principal vanishing bespeak —which determines the horizon line— the 45° vanishing point on the left side of the cartoon completes the label of the (equally distant) signal of view. Two lines are drawn from the orthogonal projection of each vertex, i at 45° and i at 90° to the motion picture airplane. After intersecting the basis line, those lines go toward the distance betoken (for 45°) or the main 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 style until they encounter the vertical from the map.

While orthographic projection ignores perspective to permit accurate measurements, perspective project shows afar objects as smaller to provide additional realism.

Mathematical formula [edit]

The perspective projection requires a more than involved definition equally compared to orthographic projections. A conceptual aid to agreement the mechanics of this projection is to imagine the 2nd projection as though the object(s) are being viewed through a camera viewfinder. The photographic camera'southward position, orientation, and field of view control the behavior of the projection transformation. The following variables are defined to describe this transformation:

• ${\displaystyle \mathbf {a} _{10,y,z}}$ – the 3D position of a point A that is to be projected.
• ${\displaystyle \mathbf {c} _{x,y,z}}$ – the 3D position of a signal C representing the camera.
• ${\displaystyle \mathbf {\theta } _{ten,y,z}}$ – 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]

Virtually conventions use positive z values (the plane being in forepart of the pinhole), however negative z values are physically more correct, but the image will be inverted both horizontally and vertically. Which results in:

When ${\displaystyle \mathbf {c} _{x,y,z}=\langle 0,0,0\rangle ,}$ 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 ${\displaystyle \langle one,2\rangle }$ .

Otherwise, to compute ${\displaystyle \mathbf {b} _{x,y}}$ nosotros starting time define a vector ${\displaystyle \mathbf {d} _{x,y,z}}$ equally the position of betoken A with respect to a coordinate system defined by the camera, with origin in C and rotated past ${\displaystyle \mathbf {\theta } }$ with respect to the initial coordinate system. This is achieved past subtracting ${\displaystyle \mathbf {c} }$ from ${\displaystyle \mathbf {a} }$ and so applying a rotation by ${\displaystyle -\mathbf {\theta } }$ to the upshot. This transformation is oftentimes called a camera transform , and can be expressed every bit follows, expressing the rotation in terms of rotations about the x, y, and z axes (these calculations presume that the axes are ordered every bit a left-handed system of axes): ^{[12] [thirteen]}

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

This representation corresponds to rotating by iii Euler angles (more properly, Tait–Bryan angles), using the xyz convention, which can exist interpreted either every bit "rotate about the extrinsic axes (axes of the scene) in the order z, y, 10 (reading correct-to-left)" or "rotate virtually the intrinsic axes (axes of the camera) in the order 10, y, z (reading left-to-right)". Note that if the 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 just a shift: ${\displaystyle \mathbf {d} =\mathbf {a} -\mathbf {c} .}$

Alternatively, without using matrices (let u.s.a. replace ${\displaystyle a_{x}-c_{ten}}$ with ${\displaystyle \mathbf {x} }$ and so on, and abridge ${\displaystyle \cos \left(\theta _{\blastoff }\right)}$ to ${\displaystyle c_{\alpha }}$ and ${\displaystyle \sin \left(\theta _{\blastoff }\right)}$ to ${\displaystyle s_{\alpha }}$ ):

${\displaystyle {\brainstorm{aligned}\mathbf {d} _{ten}&=c_{y}(s_{z}\mathbf {y} +c_{z}\mathbf {ten} )-s_{y}\mathbf {z} \\\mathbf {d} _{y}&=s_{10}(c_{y}\mathbf {z} +s_{y}(s_{z}\mathbf {y} +c_{z}\mathbf {10} ))+c_{x}(c_{z}\mathbf {y} -s_{z}\mathbf {x} )\\\mathbf {d} _{z}&=c_{x}(c_{y}\mathbf {z} +s_{y}(s_{z}\mathbf {y} +c_{z}\mathbf {10} ))-s_{ten}(c_{z}\mathbf {y} -s_{z}\mathbf {x} )\end{aligned}}}$

This transformed point can then be projected onto the 2D airplane using the formula (here, x/y is used as the projection aeroplane; literature likewise may use x/z):^[14]

${\displaystyle {\brainstorm{aligned}\mathbf {b} _{ten}&={\frac {\mathbf {east} _{z}}{\mathbf {d} _{z}}}\mathbf {d} _{10}+\mathbf {e} _{x},\\[5pt]\mathbf {b} _{y}&={\frac {\mathbf {e} _{z}}{\mathbf {d} _{z}}}\mathbf {d} _{y}+\mathbf {e} _{y}.\cease{aligned}}}$

Or, in matrix form using homogeneous coordinates, the arrangement

${\displaystyle {\brainstorm{bmatrix}\mathbf {f} _{x}\\\mathbf {f} _{y}\\\mathbf {f} _{w}\end{bmatrix}}={\begin{bmatrix}1&0&{\frac {\mathbf {east} _{x}}{\mathbf {e} _{z}}}\\0&1&{\frac {\mathbf {e} _{y}}{\mathbf {e} _{z}}}\\0&0&{\frac {1}{\mathbf {e} _{z}}}\stop{bmatrix}}{\brainstorm{bmatrix}\mathbf {d} _{x}\\\mathbf {d} _{y}\\\mathbf {d} _{z}\terminate{bmatrix}}}$

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

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

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

The above equations can also exist rewritten as:

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

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

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

Weak perspective projection [edit]

A "weak" perspective project uses the same principles of an orthographic project, just requires the scaling factor to be specified, thus ensuring that closer objects appear bigger in the projection, and vice versa. Information technology can be seen as a hybrid between an orthographic and a perspective projection, and described either every bit a perspective projection with private point depths ${\displaystyle Z_{i}}$ replaced by an boilerplate constant depth ${\displaystyle Z_{\text{ave}}}$ ,^[15] or but as an orthographic projection plus a scaling.^[16]

The weak-perspective model thus approximates perspective projection while using a simpler model, like 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 altitude from the camera, and the field of view is small. With these conditions, it tin be 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 projection (compared to the full perspective model).

Equation

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

assuming focal length $f=1$ .

Diagram [edit]

To determine which screen ten-coordinate corresponds to a bespeak at ${\displaystyle A_{x},A_{z}}$ multiply the signal coordinates by:

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

where

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

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

${\displaystyle B_{z}}$ is the focal length—the axial altitude from the camera center to the prototype plane

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

Because the camera is in 3D, the aforementioned works for the screen y-coordinate, substituting y for x in the above diagram and equation.

You tin can employ that to do clipping techniques, replacing the variables with values of the indicate that's are out of the FOV-angle and the point inside Photographic camera Matrix.

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

Run into also [edit]

• 3D computer graphics
• Camera matrix
• Reckoner graphics
• Cross section (geometry)
• Cross-sectional view
• Curvilinear perspective
• Cutaway cartoon
• Descriptive geometry
• Engineering drawing
• Exploded-view cartoon
• Homogeneous coordinates
• Homography
• Map projection (including Cylindrical projection)
• Multiview project
• Perspective (graphical)
• Program (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 made 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 April 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). Avant-garde graphics programming using openGL. Elsevier. p. 502. ISBN978-1-55860-659-3.
8. ^ D. Hearn, & K. Bakery (1997). Computer Graphics, C Version. Englewood Cliffs: Prentice Hall], chapter ix
9. ^ James Foley (1997). Computer Graphics. Boston: Addison-Wesley. ISBN 0-201-84840-half-dozen], 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 Calculating Surveys. 10 (4): 465–502. CiteSeerXten.1.1.532.4774. doi:10.1145/356744.356750. S2CID 708008.
12. ^ Riley, K F (2006). Mathematical Methods for Physics and Engineering . Cambridge University 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. ISBN978-0-201-02918-5.
14. ^ Sonka, Chiliad; Hlavac, V; Boyle, R (1995). Image Processing, Analysis & Machine Vision (2nd ed.). Chapman and Hall. p. fourteen. ISBN978-0-412-45570-four.
15. ^ Subhashis Banerjee (2002-02-18). "The Weak-Perspective Camera".
16. ^ Alter, T. D. (July 1992). 3D Pose from 3 Corresponding Points nether Weak-Perspective Projection (PDF) (Technical written report). MIT AI Lab.

Further reading [edit]

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

External links [edit]

• Creating 3D Environments from Digital Photographs

crosbystold1998.blogspot.com

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

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