orthographic drawings for 3d art

Design technique

Classification of some 3D projections

A 3D projection (or graphical project) is a blueprint technique used to display a 3-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 2nd geometry to 3D was mastered past Heron of Alexandria in the kickoff century.^[1] Heron could be called the father of 3D. 3D Project is the ground of the concept for Estimator Graphics simulating fluid flows to imitate realistic effects.^[ii] Lucas Films 'ILM group is credited with introducing the concept (and even the term "Particle effect").

In 1982, the first all-digital reckoner generated sequence for a motility picture file was in: Star Trek Two: The Wrath of Khan. A 1984 patent related to this concept was written by William East Masters, "Computer automatic manufacturing process and system" US4665492A using mass particles to fabricate a cup.^[3] 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 and so connected to 1 another to create a visual element. The result is a graphic that contains conceptual properties to interpret that the figure or paradigm as not actually flat (2d), but rather, every bit a solid object (3D) being viewed on a 2nd display.

3D objects are largely displayed on ii-dimensional mediums (i.east. paper and reckoner monitors). As such, graphical projections are a ordinarily used design chemical element; notably, in technology drawing, drafting, and computer graphics. Projections can exist calculated through employment of mathematical analysis 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 apply of imaginary "projectors"; the projected, mental paradigm becomes the technician's vision of the desired, finished picture.^{[ further caption needed ]} Methods provide a uniform imaging process amongst people trained in technical graphics (mechanical drawing, computer aided design, etc.). Past post-obit a method, the technician may produce the envisioned motion-picture show on a planar surface such as cartoon newspaper.

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

• parallel projection
• perspective projection

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Parallel projection [edit]

Parallel projection corresponds to a perspective project with a hypothetical viewpoint; i.e. one where the camera lies an space altitude away from the object and has an infinite focal length, or "zoom".

In parallel projection, the lines of sight from the object to the projection plane are parallel to each other. Thus, lines that are parallel in iii-dimensional space remain parallel in the ii-dimensional projected image. Parallel projection as well corresponds to a perspective project 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 out along axes"), every bit described in Pohlke's theorem. In full general, the resulting image is oblique (the rays are non perpendicular to the paradigm airplane); but in special cases the event is orthographic (the rays are perpendicular to the image airplane). Axonometry should non be confused with axonometric projection, equally in English language literature the latter usually refers only to a specific class of pictorials (come across below).

Orthographic projection [edit]

The orthographic projection is derived from the principles of descriptive geometry and is a two-dimensional representation of a iii-dimensional object. It is a parallel projection (the lines of project 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 chief axes (which is the x, y, or z axis), the mathematical transformation is every bit follows; To projection the 3D signal ${\displaystyle a_{x}}$ , ${\displaystyle a_{y}}$ , ${\displaystyle a_{z}}$ onto the 2D betoken ${\displaystyle b_{x}}$ , ${\displaystyle b_{y}}$ using an orthographic projection parallel to the y centrality (where positive y represents forrad direction - profile view), the following equations tin be used:

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

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

where the vector due south is an capricious 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:

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

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

Multiview projection [edit]

Symbols used to define whether a multiview project is either Third Angle (right) or First Angle (left).

With multiview projections, up to vi pictures (called primary views) of an object are produced, with each projection plane parallel to i of the coordinate axes of the object. The views are positioned relative to each other according to either of two schemes: commencement-angle or third-angle project. In each, the appearances of views may be thought of as being projected onto planes that class a 6-sided box around the object. Although six unlike sides tin can be fatigued, ordinarily three views of a drawing give enough information to brand a 3D object. These views are known as front end view, top view, and finish view. The terms height, programme and department are as well used.

Oblique projection [edit]

Potting demote drawn in cabinet projection with an angle of 45° and a ratio of 2/iii

Stone arch drawn 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 plane at an angle other than ninety degrees. In both orthographic and oblique projection, parallel lines in space announced parallel on the projected image. Because of its simplicity, oblique project is used exclusively for pictorial purposes rather than for formal, working drawings. In an oblique pictorial drawing, the displayed angles among the axes every bit well as the foreshortening factors (calibration) are arbitrary. The baloney created thereby is usually adulterate by aligning 1 plane of the imaged object to be parallel with the plane of projection thereby creating a true shape, full-size image of the called plane. Special types of oblique projections are:

Cavalier project (45°) [edit]

In cavalier projection (sometimes condescending 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, 10″ and y″. On the flat cartoon, two axes, x and z on the effigy, are perpendicular and the length on these axes are drawn with a i:one scale; it is thus similar to the dimetric projections, although it is not an axonometric projection, as the 3rd axis, here y, is fatigued in diagonal, making an capricious angle with the x″ axis, normally 30 or 45°. The length of the third centrality is not scaled.

Cabinet project [edit]

The term cabinet project (sometimes chiffonier perspective) stems from its employ in illustrations by the furniture manufacture.^{[ citation needed ]} Like condescending perspective, one face up of the projected object is parallel to the viewing plane, and the third axis is projected every bit going off in an angle (typically 30° or 45° or arctan(2) = 63.four°). Different condescending project, where the third axis keeps its length, with cabinet projection the length of the receding lines is cut in half.

Armed services project [edit]

A variant of oblique projection is called military projection. In this case, the horizontal sections are isometrically drawn so that the floor plans are non distorted and the verticals are drawn at an angle. The military project is given by rotation in the xy-aeroplane and a vertical translation an amount z.^[4]

Axonometric project [edit]

Axonometric projections prove an epitome of an object as viewed from a skew management in order to reveal all three directions (axes) of space in i flick.^[v] Axonometric projections may be either orthographic or oblique. Axonometric instrument drawings are often used to approximate graphical perspective projections, but in that location is attendant baloney in the approximation. Because pictorial projections innately incorporate this distortion, in instrument drawings of pictorials great liberties may then exist taken for economy of attempt and best effect.^{[ clarification needed ]}

Axonometric projection is farther subdivided into three categories: isometric projection, dimetric projection, and trimetric projection, depending on the exact bending at which the view deviates from the orthogonal.^{[six] [7]} A typical characteristic of orthographic pictorials is that ane axis of infinite is usually displayed as vertical.

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

Isometric project [edit]

In isometric pictorials (for methods, see Isometric projection), the management of viewing is such that the three axes of space appear equally foreshortened, and at that place is a common angle of 120° between them. The distortion acquired by foreshortening is compatible, 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 [edit]

In dimetric pictorials (for methods, meet Dimetric project), the direction of viewing is such that two of the three axes of space appear equally foreshortened, of which the attendant calibration and angles of presentation are determined according to the bending of viewing; the scale of the tertiary direction (vertical) is determined 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 three axes of space announced unequally foreshortened. The scale along each of the 3 axes and the angles among them are determined separately as dictated by the angle of viewing. Approximations in Trimetric drawings are mutual.

Limitations of parallel projection [edit]

An case of the limitations of isometric project. The height difference between the red and blue assurance 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 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 direct from the image, the consequence is a perceived distortion, since dissimilar perspective projection, this is not how our eyes or photography normally piece of work. It also can easily result in situations where depth and altitude are hard to gauge, as is shown in the illustration to the right.

In this isometric cartoon, the blueish sphere is two units higher than the cherry 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, also equally "impossible object" drawings. Yard. C. Escher'south Waterfall (1961), while not strictly utilizing parallel projection, is a well-known example, in which a aqueduct of h2o seems to travel unaided along a downward path, just to then paradoxically fall once again equally it returns to its source. The water thus appears to disobey the constabulary of conservation of free energy. An extreme example is depicted in the moving picture 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 tin can and cannot move in a puzzle-like manner.

Perspective projection [edit]

Perspective of a geometric solid using ii vanishing points. In this instance, the map of the solid (orthogonal project) is drawn below the perspective, as if angle the ground plane.

Axonometric projection of a scheme displaying the relevant elements of a vertical film plane perspective. The continuing signal (P.S.) is located on the ground plane π, and the point of view (P.V.) is right above information technology. P.P. is its projection on the movie plane α. 50.O. and L.T. are the horizon and the ground lines (linea d'orizzonte and linea di terra). The assuming lines s and q lie on π, and intercept α at Ts and Tq respectively. The parallel lines through P.5. (in red) intercept Fifty.O. in the vanishing points Fs and Fq: thus ane 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 airplane. This has the upshot that distant objects appear smaller than nearer objects.

Information technology besides ways that lines which are parallel in nature (that is, see at the point at infinity) appear to intersect in the projected image, for instance if railways are pictured with perspective projection, they appear to converge towards a single indicate, chosen the vanishing point. Photographic lenses and the human heart piece of work in the aforementioned mode, therefore perspective project looks near realistic.^[viii] Perspective project is usually categorized into one-bespeak, 2-point and three-betoken perspective, depending on the orientation of the project airplane towards the axes of the depicted object.^[nine]

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

Two relevant points of a line are:

• its intersection with the picture show plane, and
• its vanishing point, institute at the intersection between the parallel line from the centre signal and the picture plane.

The main 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 moving picture plane is vertical, all vertical lines are fatigued vertically, and have no finite vanishing point on the moving picture plane. Diverse graphical methods can exist easily envisaged for projecting geometrical scenes. For instance, lines traced from the eye point at 45° to the motion-picture show plane intersect the latter along a circumvolve whose radius is the distance of the eye point from the aeroplane, 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 ii 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 master vanishing bespeak —which determines the horizon line— the 45° vanishing betoken on the left side of the drawing completes the label of the (equally distant) point of view. Two lines are drawn from the orthogonal projection of each vertex, 1 at 45° and one at 90° to the picture plane. Afterward intersecting the ground line, those lines become toward the distance point (for 45°) or the principal bespeak (for 90°). Their new intersection locates the projection of the map. Natural heights are measured in a higher place the ground line and then projected in the aforementioned way until they meet the vertical from the map.

While orthographic projection ignores perspective to let accurate measurements, perspective projection shows afar objects equally smaller to provide additional realism.

Mathematical formula [edit]

The perspective projection requires a more involved definition equally 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 beingness viewed through a camera viewfinder. The camera's position, orientation, and field of view command the behavior of the project 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.
• ${\displaystyle \mathbf {c} _{ten,y,z}}$ – the 3D position of a point C representing the camera.
• ${\displaystyle \mathbf {\theta } _{ten,y,z}}$ – The orientation of the camera (represented by Tait–Bryan angles).
• ${\displaystyle \mathbf {e} _{x,y,z}}$ – the brandish surface'southward position relative to the camera pinhole C.^[11]

Virtually conventions utilize positive z values (the plane being in forepart of the pinhole), still negative z values are physically more right, 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 ${\displaystyle \langle 1,ii,0\rangle }$ is projected to the 2D vector ${\displaystyle \langle 1,2\rangle }$ .

Otherwise, to compute ${\displaystyle \mathbf {b} _{10,y}}$ we first define a vector ${\displaystyle \mathbf {d} _{10,y,z}}$ as the position of point A with respect to a coordinate system defined past the photographic camera, with origin in C and rotated by ${\displaystyle \mathbf {\theta } }$ with respect to the initial coordinate system. This is achieved by subtracting ${\displaystyle \mathbf {c} }$ from ${\displaystyle \mathbf {a} }$ and so applying a rotation by ${\displaystyle -\mathbf {\theta } }$ to the result. This transformation is often called a camera transform , and tin exist expressed every bit 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): ^{[12] [xiii]}

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

This representation corresponds to rotating by three Euler angles (more than properly, Tait–Bryan angles), using the xyz convention, which tin be interpreted either equally "rotate about the extrinsic axes (axes of the scene) in the social club z, y, ten (reading right-to-left)" or "rotate about the intrinsic axes (axes of the camera) in the lodge ten, y, z (reading left-to-right)". Notation that if the photographic camera is non rotated ( ${\displaystyle \mathbf {\theta } _{x,y,z}=\langle 0,0,0\rangle }$ ), then the matrices drop out (as identities), and this reduces to simply a shift: ${\displaystyle \mathbf {d} =\mathbf {a} -\mathbf {c} .}$

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

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

This transformed point tin then be projected onto the 2d airplane using the formula (hither, x/y is used as the projection airplane; literature also may utilize 10/z):^[14]

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

Or, in matrix grade using homogeneous coordinates, the system

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

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

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

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

The to a higher place equations tin can also be rewritten as:

${\displaystyle {\begin{aligned}\mathbf {b} _{x}&=(\mathbf {d} _{x}\mathbf {due south} _{10})/(\mathbf {d} _{z}\mathbf {r} _{x})\mathbf {r} _{z},\\\mathbf {b} _{y}&=(\mathbf {d} _{y}\mathbf {s} _{y})/(\mathbf {d} _{z}\mathbf {r} _{y})\mathbf {r} _{z}.\cease{aligned}}}$

In which ${\displaystyle \mathbf {south} _{x,y}}$ is the display size, ${\displaystyle \mathbf {r} _{x,y}}$ is the recording surface size (CCD or flick), ${\displaystyle \mathbf {r} _{z}}$ is the distance from the recording surface to the entrance pupil (photographic camera middle), and ${\displaystyle \mathbf {d} _{z}}$ is the distance, from the 3D point beingness projected, to the archway pupil.

Subsequent clipping and scaling operations may be necessary to map the second plane onto any detail brandish media.

Weak perspective projection [edit]

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. Information technology can exist seen every bit a hybrid between an orthographic and a perspective projection, and described either as a perspective projection with private point depths ${\displaystyle Z_{i}}$ replaced past an average constant depth ${\displaystyle Z_{\text{ave}}}$ ,^[15] or simply every bit 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-size compared to the distance 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

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

assuming focal length $f=1$ .

Diagram [edit]

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

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

where

${\displaystyle B_{10}}$ is the screen x coordinate

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

${\displaystyle B_{z}}$ is the focal length—the axial distance from the camera center to the epitome airplane

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

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

You tin use that to practice clipping techniques, replacing the variables with values of the point that's are out of the FOV-angle and the signal inside Photographic camera Matrix.

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

Meet besides [edit]

• 3D computer graphics
• Camera matrix
• Computer graphics
• Cross section (geometry)
• Cross-sectional view
• Curvilinear perspective
• Cutaway drawing
• Descriptive geometry
• Applied science drawing
• Exploded-view drawing
• Homogeneous coordinates
• Homography
• Map projection (including Cylindrical projection)
• Multiview projection
• Perspective (graphical)
• Plan (cartoon)
• Technical drawing
• Texture mapping
• Transform, clipping, and lighting
• Video carte
• 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 cute images are made 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 design 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-one-55860-659-three.
8. ^ D. Hearn, & M. Baker (1997). Computer Graphics, C Version. Englewood Cliffs: Prentice Hall], affiliate nine
9. ^ James Foley (1997). Estimator 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 Calculating Surveys. ten (4): 465–502. CiteSeerX10.1.1.532.4774. doi:ten.1145/356744.356750. S2CID 708008.
12. ^ Riley, K F (2006). Mathematical Methods for Physics and Engineering . Cambridge University Press. pp. 931, 942. doi:x.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, M; Hlavac, V; Boyle, R (1995). Prototype Processing, Analysis & Machine 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 Corresponding Points under Weak-Perspective Projection (PDF) (Technical report). MIT AI Lab.

Farther reading [edit]

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

External links [edit]

• Creating 3D Environments from Digital Photographs

hoganunpoid.blogspot.com

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

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