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### Perspective (graphical)

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Perspective (graphical)
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objects appear smaller ... object's dimensions are shorter --> appear shorter ("objects" do not actually change in size)
//-->{{redirect|Perspective projection|a more mathematical treatment|Perspective transform}}{{Use dmy dates|date=April 2013}}(File:Staircase perspective.jpg|thumb|upright=1.3|Staircase in two-point perspective){{external media | width= 210px | align= left
| headerimage= (File:Ãšltima Cena - Da Vinci 5.jpg|210px)
| video3=Empire of the Eye: The Magic of Illusion: The Trinity-Masaccio, Part 2, National Gallery of ArtWEB, Empire of the Eye: The Magic of Illusion: The Trinity-Masaccio, Part 2, National Gallery of Art at ArtBabble,weblink 12 May 2013, no,weblink" title="web.archive.org/web/20130501114331weblink">weblink 1 May 2013, dmy-all, }}
Perspective (from "to see through") in the graphic arts is an approximate representation, generally on a flat surface (such as paper), of an image as it is seen by the eye. The two most characteristic features of perspective are that objects appear smaller as their distance from the observer increases; and that they are subject to foreshortening, meaning that an object's dimensions along the line of sight appear shorter than its dimensions across the line of sight.Italian Renaissance painters and architects including Filippo Brunelleschi, Masaccio, Paolo Uccello, Piero della Francesca and Luca Pacioli studied linear perspective, wrote treatises on it, and incorporated it into their artworks, thus contributing to the mathematics of art.

## Overview

(File:Perspectiva-1.svg|thumb|A cube in two-point perspective)(File:Perspectiva-2.svg|thumb|Rays of light travel from the object, through the picture plane, and to the viewer's eye. This is the basis for graphical perspective.)Linear perspective always works by representing the light that passes from a scene through an imaginary rectangle (realized as the plane of the painting), to the viewer's eye, as if a viewer were looking through a window and painting what is seen directly onto the windowpane. If viewed from the same spot as the windowpane was painted, the painted image would be identical to what was seen through the unpainted window. Each painted object in the scene is thus a flat, scaled down version of the object on the other side of the window.BOOK, Perspective Drawing Handbook, D'Amelio, Joseph, 19, Dover, 2003, Because each portion of the painted object lies on the straight line from the viewer's eye to the equivalent portion of the real object it represents, the viewer sees no difference (sans depth perception) between the painted scene on the windowpane and the view of the real scene.All perspective drawings assume the viewer is a certain distance away from the drawing. Objects are scaled relative to that viewer. An object is often not scaled evenly: a circle often appears as an ellipse and a square can appear as a trapezoid. This distortion is referred to as foreshortening.Perspective drawings have a horizon line, which is often implied. This line, directly opposite the viewer's eye, represents objects infinitely far away. They have shrunk, in the distance, to the infinitesimal thickness of a line. It is analogous to (and named after) the Earth's horizon.Any perspective representation of a scene that includes parallel lines has one or more vanishing points in a perspective drawing. A one-point perspective drawing means that the drawing has a single vanishing point, usually (though not necessarily) directly opposite the viewer's eye and usually (though not necessarily) on the horizon line. All lines parallel with the viewer's line of sight recede to the horizon towards this vanishing point. This is the standard "receding railroad tracks" phenomenon. A two-point drawing would have lines parallel to two different angles. Any number of vanishing points are possible in a drawing, one for each set of parallel lines that are at an angle relative to the plane of the drawing.Perspectives consisting of many parallel lines are observed most often when drawing architecture (architecture frequently uses lines parallel to the x, y, and z axes). Because it is rare to have a scene consisting solely of lines parallel to the three Cartesian axes (x, y, and z), it is rare to see perspectives in practice with only one, two, or three vanishing points; even a simple house frequently has a peaked roof which results in a minimum of six sets of parallel lines, in turn corresponding to up to six vanishing points.In contrast, natural scenes often do not have any sets of parallel lines and thus no vanishing points.

### Early history

The earliest art paintings and drawings typically sized many objects and characters hierarchically according to their spiritual or thematic importance, not their distance from the viewer, and did not use foreshortening. The most important figures are often shown as the highest in a composition, also from hieratic motives, leading to the so-called "vertical perspective", common in the art of Ancient Egypt, where a group of "nearer" figures are shown below the larger figure or figures. The only method to indicate the relative position of elements in the composition was by overlapping, of which much use is made in works like the Parthenon Marbles.

### Present: computer graphics

3-D computer games and ray-tracers often use a modified version of perspective. Like the painter, the computer program is generally not concerned with every ray of light that is in a scene. Instead, the program simulates rays of light traveling backwards from the monitor (one for every pixel), and checks to see what it hits. In this way, the program does not have to compute the trajectories of millions of rays of light that pass from a light source, hit an object, and miss the viewer. {{Dubious|date=June 2013}}CAD software, and some computer games (especially games using 3-D polygons) use linear algebra, and in particular matrix multiplication, to create a sense of perspective. The scene is a set of points, and these points are projected to a plane (computer screen){{Dubious|date=June 2013}} in front of the view point (the viewer's eye). The problem of perspective is simply finding the corresponding coordinates on the plane corresponding to the points in the scene. By the theories of linear algebra, a matrix multiplication directly computes the desired coordinates, thus bypassing any descriptive geometry theorems used in perspective drawing. {{Dubious|date=June 2013}}.

## Types

Of the many types of perspective drawings, the most common categorizations of artificial perspective are one-, two- and three-point. The names of these categories refer to the number of vanishing points in the perspective drawing.

### One-point perspective

(File:Perspective-1point.svg|thumbnail|upright=0.6|One-point perspective)A drawing has one-point perspective when it contains only one vanishing point on the horizon line. This type of perspective is typically used for images of roads, railway tracks, hallways, or buildings viewed so that the front is directly facing the viewer. Any objects that are made up of lines either directly parallel with the viewer's line of sight or directly perpendicular (the railroad slats) can be represented with one-point perspective. These parallel lines converge at the vanishing point.One-point perspective exists when the picture plane is parallel to two axes of a rectilinear (or Cartesian) scene â€“ a scene which is composed entirely of linear elements that intersect only at right angles. If one axis is parallel with the picture plane, then all elements are either parallel to the picture plane (either horizontally or vertically) or perpendicular to it. All elements that are parallel to the picture plane are drawn as parallel lines. All elements that are perpendicular to the picture plane converge at a single point (a vanishing point) on the horizon.File:Perspectivephoto.jpgFile:Inside Greenwich Foot Tunnel.jpgFile:One point perspective.jpgFile:Finnish national road 4 VierumÃ¤ki.jpgFile:HK Hung Hum Station Corridor.jpgFile:Railroad-Tracks-Perspective.jpgFile:Tuileries Rivoli Perspective.jpg

### Two-point perspective

(File:Perspective-2point.svg|thumbnail|right|upright=0.5|Two-Point Perspective)(File:Perspective1.jpg|right|thumb|A cube drawing using 2-point perspective)A drawing has two-point perspective when it contains two vanishing points on the horizon line. In an illustration, these vanishing points can be placed arbitrarily along the horizon. Two-point perspective can be used to draw the same objects as one-point perspective, rotated: looking at the corner of a house, or at two forked roads shrinking into the distance, for example. One point represents one set of parallel lines, the other point represents the other. Seen from the corner, one wall of a house would recede towards one vanishing point while the other wall recedes towards the opposite vanishing point.Two-point perspective exists when the painting plate is parallel to a Cartesian scene in one axis (usually the z-axis) but not to the other two axes. If the scene being viewed consists solely of a cylinder sitting on a horizontal plane, no difference exists in the image of the cylinder between a one-point and two-point perspective.Two-point perspective has one set of lines parallel to the picture plane and two sets oblique to it. Parallel lines oblique to the picture plane converge to a vanishing point, which means that this set-up will require two vanishing points.(File:2-pt-sketchup.jpg|thumb|upright=1.5|left|Walls in 2-point perspective, converging toward two vanishing points. All vertical elements are parallel. Model from 3D Warehouse, rendered in SketchUp.){{clear}}

### Three-point perspective

(File:Perspective-3point.svg|thumbnail|upright=0.5|Three-Point Perspective)Three-point perspective is often used for buildings seen from above (or below). In addition to the two vanishing points from before, one for each wall, there is now one for how the vertical lines of the walls recede. For an object seen from above, this third vanishing point is below the ground. For an object seen from below, as when the viewer looks up at a tall building, the third vanishing point is high in space.Three-point perspective exists when the perspective is a view of a Cartesian scene where the picture plane is not parallel to any of the scene's three axes. Each of the three vanishing points corresponds with one of the three axes of the scene.One, two and three-point perspectives appear to embody different forms of calculated perspective, and are generated by different methods. Mathematically, however, all three are identical; the difference is merely in the relative orientation of the rectilinear scene to the viewer.File:Rome Palace of Labor.jpg|thumb|upright=1.5|left|The Palazzo del Lavoro in Mussolini's Esposizione Universale Roma complex, photographed in 3-point perspective. All three axes are oblique to the picture plane; the three vanishing points are at the zenith, and on the horizon to the right and left.]]{{clear}}

### Four-point perspective

Four-point perspective, also called infinite-point perspective, is the curvilinear (see curvilinear perspective) variant of two-point perspective. A four-point perspective image can represent a 360Â° panorama, and even beyond 360Â° to depict impossible scenes. This perspective can be used with either a horizontal or a vertical horizon line: in the latter configuration it can depict both a worm's-eye and bird's-eye view of a scene at the same time.Like all other foreshortened variants of perspective (one-point to six-point perspectives), it starts off with a horizon line, followed by four equally spaced vanishing points to delineate four vertical lines. The vanishing points made to create the curvilinear orthogonals are thus made ad hoc on the four vertical lines placed on the opposite side of the horizon line. The only dimension not foreshortened in this type of perspective is that of the rectilinear and parallel lines perpendicular to the horizon line â€“ similar to the vertical lines used in two-point perspective.One-point, two-point, and three-point perspective are dependent on the structure of the scene being viewed. These only exist for strict Cartesian (rectilinear) scenes. By inserting into a Cartesian scene a set of parallel lines that are not parallel to any of the three axes of the scene, a new distinct vanishing point is created. Therefore, it is possible to have an infinite-point perspective if the scene being viewed is not a Cartesian scene but instead consists of infinite pairs of parallel lines, where each pair is not parallel to any other pair.

### Zero-point perspective

In its usual sense, zero-point perspective is not truly "zero-point". Rather, because vanishing points exist only when parallel lines are present in the scene, a perspective with no vanishing points ("zero-point" perspective) occurs if the viewer is observing a non-linear scene containing no parallel lines.BOOK, Basant Agrawal, Engineering Drawing,weblink 2008, Tata McGraw-Hill Education, 978-0-07-066863-8, 17.3.4, no,weblink 23 December 2017, dmy-all, The most common example of such a nonlinear scene is a natural scene (e.g., a mountain range) which frequently does not contain any parallel lines. This is not to be confused with elevation, since a view without explicit vanishing points may still have been drawn such that, there would have been vanishing points had there been parallel lines, and thus enjoy the sense of depth as a perspective projection.On the other hand, parallel projection such as elevation can be approximated by viewing the object in question from very far away, because projection lines from the point of view approaches parallel when the point of view (POV) approaches infinity. This may account for the confusion over zero-point perspective, since natural scenes often are viewed from very far away, and the size of objects within the scene would be insignificant compared to their distance to the POV. Any given small objects in said scene would thus mimic the look of parallel projection.

### Foreshortening

(File:Perspective-foreshortening.svg|thumbnail|Two different projections of a stack of two cubes, illustrating oblique parallel projection foreshortening ("A") and perspective foreshortening ("B"))File:Andrea Mantegna - The Lamentation over the Dead Christ - WGA13981.jpg|thumb|Andrea Mantegna, The Lamentation over the Dead Christ]]Foreshortening is the visual effect or optical illusion that causes an object or distance to appear shorter than it actually is because it is angled toward the viewer. Additionally, an object is often not scaled evenly: a circle often appears as an ellipse and a square can appear as a trapezoid.Although foreshortening is an important element in art where visual perspective is being depicted, foreshortening occurs in other types of two-dimensional representations of three-dimensional scenes. Some other types where foreshortening can occur include oblique parallel projection drawings. Foreshortening also occurs when imaging rugged terrain using a synthetic aperture radar system.{{Citation needed|date=July 2012}}In painting, foreshortening in the depiction of the human figure was perfected in the Italian Renaissance, and the Lamentation over the Dead Christ by Andrea Mantegna (1480s) is one of the most famous of a number of works that show off the new technique, which thereafter became a standard part of the training of artists.

## Methods of construction

Several methods of constructing perspectives exist, including:
• Freehand sketching (common in art)
• Graphically constructing (once common in architecture)
• Using a perspective grid
• Computing a perspective transform (common in 3D computer applications)
• Mimicry using tools such as a proportional divider (sometimes called a variscaler)
• Copying a photograph

## Example

(File:Drawing Square in Perspective 2.svg|thumb|Rays of light travel from the object to the eye, intersecting with a notional picture plane.)(File:Drawing Square in Perspective 1.svg|thumb|Determining the geometry of a square floor tile on the perspective drawing)One of the most common, and earliest, uses of geometrical perspective is a checkerboard floor. It is a simple but striking application of one-point perspective. Many of the properties of perspective drawing are used while drawing a checkerboard. The checkerboard floor is, essentially, just a combination of a series of squares. Once a single square is drawn, it can be widened or subdivided into a checkerboard. Where necessary, lines and points will be referred to by their colors in the diagram.To draw a square in perspective, the artist starts by drawing a horizon line (black) and determining where the vanishing point (green) should be. The higher up the horizon line is, the lower the viewer will appear to be looking, and vice versa. The more off-center the vanishing point, the more tilted the square will be. Because the square is made up of right angles, the vanishing point should be directly in the middle of the horizon line. A rotated square is drawn using two-point perspective, with each set of parallel lines leading to a different vanishing point.The foremost edge of the (orange) square is drawn near the bottom of the painting. Because the viewer's picture plane is parallel to the bottom of the square, this line is horizontal. Lines connecting each side of the foremost edge to the vanishing point are drawn (in grey). These lines give the basic, one point "railroad tracks" perspective. The closer it is the horizon line, the farther away it is from the viewer, and the smaller it will appear. The farther away from the viewer it is, the closer it is to being perpendicular to the picture plane.A new point (the eye) is now chosen, on the horizon line, either to the left or right of the vanishing point. The distance from this point to the vanishing point represents the distance of the viewer from the drawing. If this point is very far from the vanishing point, the square will appear squashed, and far away. If it is close, it will appear stretched out, as if it is very close to the viewer.A line connecting this point to the opposite corner of the square is drawn. Where this (blue) line hits the side of the square, a horizontal line is drawn, representing the farthest edge of the square. The line just drawn represents the ray of light traveling from the farthest edge of the square to the viewer's eye. This step is key to understanding perspective drawing. The light that passes through the picture plane obviously can not be traced. Instead, lines that represent those rays of light are drawn on the picture plane. In the case of the square, the side of the square also represents the picture plane (at an angle), so there is a small shortcut: when the line hits the side of the square, it has also hit the appropriate spot in the picture plane. The (blue) line is drawn to the opposite edge of the foremost edge because of another shortcut: since all sides are the same length, the foremost edge can stand in for the side edge.Original formulations used, instead of the side of the square, a vertical line to one side, representing the picture plane. Each line drawn through this plane was identical to the line of sight from the viewer's eye to the drawing, only rotated around the y-axis ninety degrees. It is, conceptually, an easier way of thinking of perspective. It can be easily shown that both methods are mathematically identical, and result in the same placement of the farthest side.

## Limitations

• An image equivalent to an unrolled cylinder
• A portion of the sphere can be flattened into an image equivalent to a standard perspective
• An image similar to a fisheye photograph

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{{notelist}}

## References

{{Reflist|28em}}

• BOOK, Andersen, Kirsti, Kirsti Andersen, The Geometry of an Art: The History of the Mathematical Theory of Perspective from Alberti to Monge, Springer, 2007,
• BOOK, Damisch, Hubert, The Origin of Perspective, Translated by John Goodman, MIT Press, Cambridge, Massachusetts, 1994,
• BOOK, Hyman, Isabelle, comp, Brunelleschi in Perspective, Prentice-Hall, Englewood Cliffs, New Jersey, 1974,
• BOOK, Kemp, Martin, The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat, Yale University Press, 1992,
• BOOK, PÃ©rez-GÃ³mez, Alberto, and Pelletier, Louise, Architectural Representation and the Perspective Hinge, MIT Press, Cambridge, Massachusetts, 1997,
• BOOK, Vasari, Giorgio, Giorgio Vasari, The Lives of the Artists, 1568, Florence, Italy,weblink
• BOOK, Gill, Robert W, 1974, Perspective From Basic to Creative, Australia, Thames & Hudson,

{{Commons category|Perspective}}{{Commons|Evolution of Perspective}}
{{visualization}}{{Cinematic techniques}}{{Mathematical art}}

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