Perspective transform
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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 project is any undertaking, carried out individually or collaboratively and possibly involving research or design, that is carefully planned to achieve a particular goal. An alternative view sees a project managerially as a sequence of even ...
a complex object for viewing capability on a simpler plane. 3D projections use the
primary qualities Primary or primaries may refer to: Arts, entertainment, and media Music Groups and labels * Primary (band), from Australia * Primary (musician), hip hop musician and record producer from South Korea * Primary Music, Israeli record label Works * ...
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 as not actually flat (2D), but rather, as 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 An engineering drawing is a type of technical drawing that is used to convey information about an object. A common use is to specify the geometry necessary for the construction of a component and is called a detail drawing. Usually, a number o ...
, drafting, and
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
. Projections can be calculated through employment of mathematical analysis and formulae, or by using various
geometric Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
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 In three-dimensional geometry, a parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane, known as the '' projection plane'' or ''image plane'', where the '' rays'', known as ' ...
*
perspective projection Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation ...
File:Orthographic perspective arch.svg, Multiview projection (elevation) File:Isometrie.png,
Isometric projection Isometric projection is a method for visually representing three-dimensional objects in two dimensions in technical and engineering drawings. It is an axonometric projection in which the three coordinate axes appear equally foreshortened and ...
File:Militärperspektive.PNG, Military projection File:Cabinet perspective 45.svg,
Cabinet projection Oblique projection is a simple type of technical drawing of graphical projection used for producing two-dimensional (2D) images of three-dimensional (3D) objects. The objects are not in perspective and so do not correspond to any view of an ...
File:1ptPerspective.svg,
One-point perspective Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
File:2-punktperspektive.svg,
Two-point perspective Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
File:3-punktperspektive 1.svg, Three-point perspective


Parallel projection

In parallel projection, the lines of sight from the object to the
projection plane A projection plane, or plane of projection, is a type of view in which graphical projections from an object intersect.Gary R. Bertoline et al. (2002) ''Technical Graphics Communication''. McGraw–Hill Professional, 2002. , p. 330. 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 Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation ...
with an infinite
focal length The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative foc ...
(the distance from a camera's
lens A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements ...
and
focal point Focal point may refer to: * Focus (optics) * Focus (geometry) * Conjugate points, also called focal points * Focal point (game theory) * Unicom Focal Point UNICOM Focal Point is a portfolio management and decision analysis tool used by the p ...
), or " zoom". Images drawn in parallel projection rely upon the technique of
axonometry Axonometry is a graphical procedure belonging to descriptive geometry that generates a planar image of a three-dimensional object. The term "axonometry" means "to measure along axes", and indicates that the dimensions and scaling of the coord ...
("to measure along axes"), as described in
Pohlke's theorem Pohlke's theorem is the fundamental theorem of axonometry. It was established 1853 by the German painter and teacher of descriptive geometry Karl Wilhelm Pohlke. The first proof of the theorem was published 1864 by the German mathematician Hermann ...
. 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 Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and ...
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 In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
, the equations become: : \begin b_x \\ b_y \end = \begin s_x & 0 & 0 \\ 0 & 0 & s_z \end\begin a_x \\ a_y \\ a_z \end + \begin c_x \\ c_z \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.


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 In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
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 Axonometric projection is a type of orthographic projection used for creating a pictorial drawing of an object, where the object is rotated around one or more of its axes to reveal multiple sides.Gary R. Bertoline et al. (2002) ''Technical Graphi ...
, 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 Oblique projection is a simple type of technical drawing of graphical projection used for producing two-dimensional (2D) images of three-dimensional (3D) objects. The objects are not in perspective and so do not correspond to any view of an ...
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''.


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 projections may be either ''orthographic'' or ''oblique''. Axonometric instrument drawings are often used to approximate graphical perspective projections, but there is attendant distortion in the approximation. Because pictorial projections innately contain this distortion, in instrument drawings of pictorials great liberties may then be taken for economy of effort and best effect. ''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 Isometric projection is a method for visually representing three-dimensional objects in two dimensions in technical and engineering drawings. It is an axonometric projection in which the three coordinate axes appear equally foreshortened and ...
), 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 Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
is uniform, therefore the proportionality of all sides and lengths are preserved, and the axes share a common scale. This enables measurements to be read or taken directly from the drawing.


Dimetric projection

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


Trimetric projection

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


Limitations of parallel projection

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 Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation ...
, 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 Op art, short for optical art, is a style of visual art that uses optical illusions. Op artworks are abstract, with many better-known pieces created in black and white. Typically, they give the viewer the impression of movement, hidden image ...
, as well as "impossible object" drawings.
M. C. Escher Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, and mezzotints. Despite wide popular interest, Escher was for most of his life neglected in t ...
's ''
Waterfall A waterfall is a point in a river or stream where water flows over a vertical drop or a series of steep drops. Waterfalls also occur where meltwater drops over the edge of a tabular iceberg or ice shelf. Waterfalls can be formed in several ...
'' (1961), while not strictly utilizing parallel projection, is a well-known example, in which a channel of water seems to travel unaided along a downward path, only to then paradoxically fall once again as it returns to its source. The water thus appears to disobey the
law of conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means that ...
. An extreme example is depicted in the film ''
Inception ''Inception'' is a 2010 science fiction action film written and directed by Christopher Nolan, who also produced the film with Emma Thomas, his wife. The film stars Leonardo DiCaprio as a professional thief who steals information by infi ...
'', 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 move in a puzzle-like fashion.


Perspective projection

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 In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. ...
) 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 A vanishing point is a point on the image plane of a perspective drawing where the two-dimensional perspective projections of mutually parallel lines in three-dimensional space appear to converge. When the set of parallel lines is perpendicul ...
. 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 The horizon is the apparent line that separates the surface of a celestial body from its sky when viewed from the perspective of an observer on or near the surface of the relevant body. This line divides all viewing directions based on whether i ...
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 The field of view (FoV) is the extent of the observable world that is seen at any given moment. In the case of optical instruments or sensors it is a solid angle through which a detector is sensitive to electromagnetic radiation. Human ...
control the behavior of the projection transformation. The following variables are defined to describe this transformation: * \mathbf_ – the 3D position of a point ''A'' that is to be projected. * \mathbf_ – the 3D position of a point ''C'' representing the camera. * \mathbf_ – The orientation of the camera (represented by
Tait–Bryan angles The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> The ...
). * \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: * \mathbf_ – the 2D projection of \mathbf. When \mathbf_=\langle 0,0,0\rangle, and \mathbf_ = \langle 0,0,0\rangle, the 3D vector \langle 1,2,0 \rangle is projected to the 2D vector \langle 1,2 \rangle. Otherwise, to compute \mathbf_ we first define a vector \mathbf_ as the position of point ''A'' with respect to a
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
defined by the camera, with origin in ''C'' and rotated by \mathbf with respect to the initial coordinate system. This is achieved by subtracting \mathbf from \mathbf and then applying a rotation by -\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 In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subject ...
system of axes): : \begin \mathbf_x \\ \mathbf_y \\ \mathbf_z \end=\begin 1 & 0 & 0 \\ 0 & \cos ( \mathbf_x ) & \sin ( \mathbf_x ) \\ 0 & -\sin ( \mathbf_x ) & \cos ( \mathbf_x ) \end\begin \cos ( \mathbf_y ) & 0 & - \sin ( \mathbf_y ) \\ 0 & 1 & 0 \\ \sin ( \mathbf_y ) & 0 & \cos ( \mathbf_y ) \end\begin \cos ( \mathbf_z ) & \sin ( \mathbf_z ) & 0 \\ -\sin ( \mathbf_z ) & \cos ( \mathbf_z ) & 0 \\ 0 & 0 & 1 \end\left( \right) This representation corresponds to rotating by three
Euler angles The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> Th ...
(more properly,
Tait–Bryan angles The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> The ...
), 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 (\mathbf_ = \langle 0,0,0\rangle), then the matrices drop out (as identities), and this reduces to simply a shift: \mathbf = \mathbf - \mathbf. Alternatively, without using matrices (let us replace a_x - c_x with \mathbf and so on, and abbreviate \cos\left(\theta_\alpha\right) to c_\alpha and \sin\left(\theta_\alpha\right) to s_\alpha): : \begin \mathbf_x & = c_y (s_z \mathbf+c_z \mathbf)-s_y \mathbf \\ \mathbf_y & = s_x (c_y \mathbf+s_y (s_z \mathbf+c_z \mathbf))+c_x (c_z \mathbf-s_z \mathbf) \\ \mathbf_z & = c_x (c_y \mathbf+s_y (s_z \mathbf+c_z \mathbf))-s_x (c_z \mathbf-s_z \mathbf) \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''): : \begin \mathbf_x &= \frac \mathbf_x + \mathbf_x, \\ pt \mathbf_y &= \frac \mathbf_y + \mathbf_y. \end Or, in matrix form using
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
, the system : \begin \mathbf_x \\ \mathbf_y \\ \mathbf_w \end=\begin 1 & 0 & \frac \\ 0 & 1 & \frac \\ 0 & 0 & \frac \end\begin \mathbf_x \\ \mathbf_y \\ \mathbf_z \end in conjunction with an argument using similar triangles, leads to division by the homogeneous coordinate, giving : \begin \mathbf_x &= \mathbf_x / \mathbf_w \\ \mathbf_y &= \mathbf_y / \mathbf_w \end The distance of the viewer from the display surface, \mathbf_z, directly relates to the field of view, where \alpha=2 \cdot \arctan(1/\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: : \begin \mathbf_x & = (\mathbf_x \mathbf_x ) / (\mathbf_z \mathbf_x) \mathbf_z, \\ \mathbf_y & = (\mathbf_y \mathbf_y ) / (\mathbf_z \mathbf_y) \mathbf_z. \end In which \mathbf_ is the display size, \mathbf_ is the recording surface size ( CCD or
film A film also called a movie, motion picture, moving picture, picture, photoplay or (slang) flick is a work of visual art that simulates experiences and otherwise communicates ideas, stories, perceptions, feelings, beauty, or atmospher ...
), \mathbf_z is the distance from the recording surface to the
entrance pupil In an optical system, the entrance pupil is the optical image of the physical aperture stop, as 'seen' through the front (the object side) of the lens system. The corresponding image of the aperture as seen through the back of the lens system ...
(
camera center The pinhole camera model describes the mathematical relationship between the coordinates of a point in three-dimensional space and its projection onto the image plane of an ''ideal'' pinhole camera, where the camera aperture is described as a p ...
), and \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_\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_\text from the camera without significant errors in the projection (compared to the full perspective model). ''Equation'' :\begin & P_x = \frac X \\ pt& P_y = \frac Y \end assuming focal length To determine which screen ''x''-coordinate corresponds to a point at A_x,A_z multiply the point coordinates by: :B_x = A_x \frac where :B_x is the screen ''x'' coordinate :A_x is the model ''x'' coordinate :B_z is the
focal length The focal length of an optical system is a measure of how strongly the system converges or diverges light; it is the inverse of the system's optical power. A positive focal length indicates that a system converges light, while a negative foc ...
—the axial distance from the
camera center The pinhole camera model describes the mathematical relationship between the coordinates of a point in three-dimensional space and its projection onto the image plane of an ''ideal'' pinhole camera, where the camera aperture is described as a p ...
to the
image plane In 3D computer graphics, the image plane is that plane in the world which is identified with the plane of the display monitor used to view the image that is being rendered. It is also referred to as screen space. If one makes the analogy of taki ...
:A_z is the subject 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 can use that to do clipping techniques, replacing the variables with values of the point that's are out of the FOV-angle and the point inside 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, after all needed transformations finished.


See also

*
3D computer graphics 3D computer graphics, or “3D graphics,” sometimes called CGI, 3D-CGI or three-dimensional computer graphics are graphics that use a three-dimensional representation of geometric data (often Cartesian) that is stored in the computer for th ...
* Camera matrix *
Computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
*
Cross section (geometry) In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. Th ...
* Cross-sectional view *
Curvilinear perspective Curvilinear perspective, also five-point perspective, is a graphical projection used to draw 3D objects on 2D surfaces. It was formally codified in 1968 by the artists and art historians André Barre and Albert Flocon in the book ''La Perspective c ...
* Cutaway drawing *
Descriptive geometry Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and ...
*
Engineering drawing An engineering drawing is a type of technical drawing that is used to convey information about an object. A common use is to specify the geometry necessary for the construction of a component and is called a detail drawing. Usually, a number o ...
*
Exploded-view drawing An exploded-view drawing is a diagram, picture, schematic or technical drawing of an object, that shows the relationship or order of assembly of various parts. It shows the components of an object slightly separated by distance, or suspended i ...
*
Homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
*
Homography In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In gener ...
*
Map projection In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and l ...
(including
Cylindrical projection In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitu ...
) * Multiview projection *
Perspective (graphical) Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
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Plan (drawing) Plans are a set of drawings or two-dimensional diagrams used to describe a place or object, or to communicate building or fabrication instructions. Usually plans are drawn or printed on paper, but they can take the form of a digital file. Pla ...
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Technical drawing Technical drawing, drafting or drawing, is the act and discipline of composing drawings that visually communicate how something functions or is constructed. Technical drawing is essential for communicating ideas in industry and engineering ...
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Tesseract In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of e ...
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Texture mapping Texture mapping is a method for mapping a texture on a computer-generated graphic. Texture here can be high frequency detail, surface texture, or color. History The original technique was pioneered by Edwin Catmull in 1974. Texture mappi ...
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Transform, clipping, and lighting Transform, clipping, and lighting (T&L or TCL) is a term used in computer graphics. Overview Transformation is the task of producing a two-dimensional view of a three-dimensional scene. Clipping means only drawing the parts of the scene that ...
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Video card A graphics card (also called a video card, display card, graphics adapter, VGA card/VGA, video adapter, display adapter, or mistakenly GPU) is an expansion card which generates a feed of output images to a display device, such as a computer m ...
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Viewing frustum In 3D computer graphics, the view frustum (also called viewing frustum) is the region of space in the modeled world that may appear on the screen; it is the field of view of a perspective virtual camera system. The view frustum is typicall ...
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Virtual globe A virtual globe is a three-dimensional (3D) software model or representation of Earth or another world. A virtual globe provides the user with the ability to freely move around in the virtual environment by changing the viewing angle and posit ...


References


Further reading

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External links


Creating 3D Environments from Digital Photographs
{{Authority control 3D computer graphics 3D imaging Display devices Euclidean solid geometry Functions and mappings Graphical projections Linear algebra Projective geometry