 Perspective (graphical)

Part of a series on: Graphical projection  Parallel projection
 Orthographic projection
 Multiviews
 Plan, or floor plan
 Section
 Elevation
 Auxiliary
 Axonometric projection (i.e. pictorials)
 Isometric projection
 Dimetric projection
 Trimetric projection
 Multiviews
 Oblique projection
 Cavalier projection
 Cabinet projection
 Orthographic projection
 Perspective projection
 Linear perspective
 Onepoint perspective
 Twopoint perspective
 Threepoint perspective
 Zeropoint perspective
 Curvilinear perspective
 Reverse perspective
 Linear perspective
Views Bird'seye view/Aerial view
 Detail view
 3/4 perspective
 Cutaway drawing
 Exploded view drawing
 Fisheye
 Fixed 3D
 Panorama
 Topdown perspective
 Worm'seye view
 Zoom
Perspective (from Latin perspicere, to see through) in the graphic arts, such as drawing, is an approximate representation, 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 are drawn:
 Smaller as their distance from the observer increases
 Foreshortened: the size of an object's dimensions along the line of sight are relatively shorter than dimensions across the line of sight
Contents
Overview
Linear perspective works by representing the light that passes from a scene through an imaginary rectangle (the painting), to the viewer's eye. It is similar to a viewer 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 a flat, scaled down version of the object on the other side of the window.^{[1]} 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 cannot perceive (sans depth perception) any difference 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. Additionally, 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 typically have an often implied horizon line. 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 onepoint 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 twopoint 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. Such a perspective would thus have no vanishing points.
History
Early history
The earliest art paintings and drawings typically sized objects and characters hieratically 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 "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.
Systematic attempts to evolve a system of perspective are usually considered to have begun around the 5th century B.C. in the art of Ancient Greece, as part of a developing interest in illusionism allied to theatrical scenery and detailed within Aristotle's Poetics as 'skenographia': Using flat panels on a stage to give the illusion of depth.^{[2]} The philosophers Anaxagoras and Democritus worked out geometric theories of perspective for use with skenographia. Alcibiades had paintings in his house designed based on skenographia, thus this art was not confined merely to the stage. Euclid's Optics introduced a mathematical theory of perspective; however, there is some debate over the extent to which Euclid's perspective coincides with a modern mathematical definition of perspective.
By the later periods of antiquity artists, especially those in less popular traditions, were well aware that distant objects could be shown smaller than those close at hand for increased illusionism, but whether this convention was actually used in a work depended on many factors. Some of the paintings found in the ruins of Pompeii show a remarkable realism and perspective for their time;^{[3]} it has been claimed that comprehensive systems of perspective were evolved in antiquity, but most scholars do not accept this. Hardly any of the many works where such a system would have been used have survived. A passage in Philostratus suggests that classical artists and theorists thought in terms of "circles" at equal distance from the viewer, like a classical semicircular theatre seen from the stage.^{[4]} The roof beams in rooms in the Vatican Virgil, from about 400 AD, are shown converging, more or less, on a common vanishing point, but this is not systematically related to the rest of the composition.^{[5]} In the Late Antique period use of perspective techniques declined. The art of the new cultures of the Migration Period had no tradition of attempting compositions of large numbers of figures and Early Medieval art was slow and inconsistent in relearning the convention from classical models, though the process can be seen underway in Carolingian art.
European Medieval artists were aware of the general principle of varying the relative size of elements according to distance, but even more than classical art was perfectly ready to override it for other reasons. Buildings were often shown obliquely according to a particular convention. The use and sophistication of attempts to convey distance increased steadily during the period, but without a basis in a systematic theory. Byzantine art was also aware of these principles, but also had the reverse perspective convention for the setting of principal figures.
Giotto attempted drawings in perspective using an algebraic method to determine the placement of distant lines. The problem with using a linear ratio in this manner is that the apparent distance between a series of evenly spaced lines actually falls off with a sine dependence. To determine the ratio for each succeeding line, a recursive ratio must be used.^{[6]}
One of Giotto's first uses of his algebraic method of perspective was Jesus Before Caiaphas. Although the picture does not conform to the modern, geometrical method of perspective, it does give a considerable illusion of depth, and was a large step forward in Western art.^{[citation needed]}
With the exception of dice,^{[7]} heraldry typically ignores perspective in the treatment of charges, though sometimes in later centuries charges are specified as in perspective.
Renaissance : Mathematical basis
Prior to the Renaissance, Alhazen (alHasan Ibn alHaytham, d. ca. 1041 AD), in his Book of Optics (Kitab almanazir; known in Latin as De aspectibus or Perspectiva, written in 1021), explained that light projects conically into the eye.^{[8]} Alhazen's geometrical, physical, physiopsychological optics resolved in this the ancient dispute between the mathematicians (Ptolemaic and Euclidean) and the physicists (Aristotelian) over the nature of vision and light. He also showed that vision is not merely a phenomenon of pure sensation (namely what results from the introduction of light rays into the eyes), but that it involves the faculties of judgment, imagination and memory.^{[9]} Alhazen's geometrical model of the cone of vision was theoretically sufficient to translate visible objects within a given setting into a painting, and this was also supported by his experimental affirmation of the visibility of spatial depth; hence of offering a proper ground for the idea of perspective.^{[10]} Moreover, Alhazen presented a geometrical conception of place as spatial extension (a postulated void), and he refuted the Aristotelian account of topos as a surface of containment. Alhazen's mathematical definition of place was more akin to Plato's notion of Khôra or Chora as 'space', yet conceived on pure geometric grounds to facilitate the use of projections.^{[11]} In all of this, Alhazen was concerned with optics, with vision, light and the nature of colour, as well as with experimentation and the use of optical instruments, and not with painting as such. Conical translations are mathematically difficult, so a drawing constructed using them would be incredibly time consuming. However, what Alhazen named a cone of vision (makhrut alshu'a') corresponded also with the idea of a pyramid of vision, hence, offering a model that can be more easily projected in orthogonal drawings of side views and top views that are needed in the geometric construction of perspective.
By the 14th century, Alhazen's Book of Optics was available in Italian translation, entitled Deli Aspecti. The Renaissance artist Lorenzo Ghiberti relied heavily upon this work, quoting it "verbatim and at length" while framing his account of art and its aesthetic imperatives in the “Commentario terzo.” Alhazen’s work was thus "central to the development of Ghiberti’s thought about art and visual aesthetics" and "may well have been central to the development of artificial perspective in early Renaissance Italian painting."^{[12]}^{[citation needed]}
In about 1413 a contemporary of Ghiberti, Filippo Brunelleschi, demonstrated the geometrical method of perspective, used today by artists, by painting the outlines of various Florentine buildings onto a mirror. When the building's outline was continued, he noticed that all of the lines converged on the horizon line. According to Vasari, he then set up a demonstration of his painting of the Baptistery in the incomplete doorway of the Duomo. He had the viewer look through a small hole on the back of the painting, facing the Baptistery. He would then set up a mirror, facing the viewer, which reflected his painting. To the viewer, the painting of the Baptistery and the Baptistery itself were nearly indistinguishable.
Soon after, nearly every artist in Florence and in Italy used geometrical perspective in their paintings,^{[13]} notably Masolino da Panicale and Donatello. Donatello started sculpting elaborate checkerboard floors into the simple manger portrayed in the birth of Christ. Although hardly historically accurate, these checkerboard floors obeyed the primary laws of geometrical perspective: the lines converged approximately to a vanishing point, and the rate at which the horizontal lines receded into the distance was graphically determined. This became an integral part of Quattrocento art. Melozzo da Forlì first used the technique of upward foreshortening (in Rome, Loreto, Forlì and others), and was celebrated for that. Not only was perspective a way of showing depth, it was also a new method of composing a painting. Paintings began to show a single, unified scene, rather than a combination of several.
As shown by the quick proliferation of accurate perspective paintings in Florence, Brunelleschi likely understood (with help from his friend the mathematician Toscanelli),^{[14]} but did not publish, the mathematics behind perspective. Decades later, his friend Leon Battista Alberti wrote De pictura (1435/1436), a treatise on proper methods of showing distance in painting. Alberti's primary breakthrough was not to show the mathematics in terms of conical projections, as it actually appears to the eye. Instead, he formulated the theory based on planar projections, or how the rays of light, passing from the viewer's eye to the landscape, would strike the picture plane (the painting). He was then able to calculate the apparent height of a distant object using two similar triangles. The mathematics behind similar triangles is relatively simple, having been long ago formulated by Euclid. In viewing a wall, for instance, the first triangle has a vertex at the user's eye, and vertices at the top and bottom of the wall. The bottom of this triangle is the distance from the viewer to the wall. The second, similar triangle, has a point at the viewer's eye, and has a length equal to the viewer's eye from the painting. The height of the second triangle can then be determined through a simple ratio, as proven by Euclid. Alberti was also trained in the science of optics through the school of Padua and under the influence of Biagio Pelacani da Parma who studied Alhazen's Optics (see what was noted above in this regard with respect to Ghiberti).
Piero della Francesca elaborated on Della Pittura in his De Prospectiva Pingendi in the 1470s. Alberti had limited himself to figures on the ground plane and giving an overall basis for perspective. Della Francesca fleshed it out, explicitly covering solids in any area of the picture plane. Della Francesca also started the now common practice of using illustrated figures to explain the mathematical concepts, making his treatise easier to understand than Alberti's. Della Francesca was also the first to accurately draw the Platonic solids as they would appear in perspective.
Perspective remained, for a while, the domain of Florence. Jan van Eyck, among others, was unable to create a consistent structure for the converging lines in paintings, as in London's The Arnolfini Portrait, because he was unaware of the theoretical breakthrough just then occurring in Italy. However he achieved very subtle effects by manipulations of scale in his interiors. Gradually, and partly through the movement of academies of the arts, the Italian techniques became part of the training of artists across Europe, and later other parts of the world.
Present : Computer graphics
3D computer games and raytracers 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.
CAD software, and some computer games (especially games using 3D 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) 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.
Types of perspective
Of the many types of perspective drawings, the most common categorizations of artificial perspective are one, two and threepoint. The names of these categories refer to the number of vanishing points in the perspective drawing.
Onepoint perspective
One vanishing point is typically used for 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 onepoint perspective.
Onepoint perspective exists when the painting plate (also known as the picture plane) is parallel to two axes of a rectilinear (or Cartesian) scene — 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 painting plate (either horizontally or vertically) or perpendicular to it. All elements that are parallel to the painting plate are drawn as parallel lines. All elements that are perpendicular to the painting plate converge at a single point (a vanishing point) on the horizon.
Some examples:
Twopoint perspective
Twopoint perspective can be used to draw the same objects as onepoint perspective, rotated: looking at the corner of a house, or looking at two forked roads shrink into the distance, for example. One point represents one set of parallel lines, the other point represents the other. Looking at a house from the corner, one wall would recede towards one vanishing point, the other wall would recede towards the opposite vanishing point.
Twopoint perspective exists when the painting plate is parallel to a Cartesian scene in one axis (usually the zaxis) 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 onepoint and twopoint perspective.
Twopoint 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 setup will require two vanishing points.
Threepoint perspective
Threepoint perspective is usually 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 those walls recede into the ground. This third vanishing point will be below the ground. Looking up at a tall building is another common example of the third vanishing point. This time the third vanishing point is high in space.
Threepoint 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. Image constructed using multiple vanishing points.
Onepoint, twopoint, and threepoint perspectives appear to embody different forms of calculated perspective. The methods required to generate these perspectives by hand are different. Mathematically, however, all three are identical: The difference is simply in the relative orientation of the rectilinear scene to the viewer.
Fourpoint perspective
Fourpoint perspective, also called infinitepoint perspective, is the curvilinear variant of twopoint perspective. As the result when made into an infinite point version (i.e. when the amount of vanishing points exceeds the minimum amount required), a four point perspective image becomes a panorama that can go to a 360 degree view and beyond  when going beyond the 360 degree view the artist might depict an "impossible" room as the artist might depict something new when it's supposed to show part of what already exists within those 360 degrees. This elongated frame can be used both horizontally and vertically and when used vertically can be described as an image that depicts both a worm's and bird'seye view of a scene at the same time.
As all other foreshortened variants of perspective (respectively one to sixpoint perspective), it starts off with a horizon line, followed by four equally spaced vanishing points to delineate four vertical lines created in a 90 degree relation to the horizon line.
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 being the rectilinear and parallel lines at a 90 degree angle to the horizon line  similar to the vertical lines used in twopoint perspective.
Zeropoint perspective
Because vanishing points exist only when parallel lines are present in the scene, a perspective without any vanishing points ("zeropoint" perspective) occurs if the viewer is observing a nonlinear scene. The most common example of a nonlinear scene is a natural scene (e.g., a mountain range) which frequently does not contain any parallel lines. A perspective without vanishing points can still create a sense of "depth," as is clearly apparent in a photograph of a mountain range (more distant mountains have smaller scale features).
Other varieties of linear perspective
Onepoint, twopoint, and threepoint 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 infinitepoint 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.
Foreshortening
Foreshortening refers to the visual effect or optical illusion that an object or distance appears shorter than it actually is because it is angled toward the viewer.
Although foreshortening is an important element in art where visual perspective is being depicted, foreshortening occurs in other types of twodimensional representations of threedimensional scenes. Some other types where foreshortening can occur include oblique parallel projection drawings.
Figure F1 shows two different projections of a stack of two cubes, illustrating oblique parallel projection foreshortening ("A") and perspective foreshortening ("B").
Foreshortening also occurs when imaging rugged terrain using Synthetic Aperture Radar systems.
This technique was often used in Renaissance painting.
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)
Example
One of the most common, and earliest, uses of geometrical perspective is a checkerboard floor. It is a simple but striking application of onepoint 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 offcenter 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 twopoint 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 yaxis 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 (see Panofsky).
Limitations
Plato was one of the first to discuss the problems of perspective. "Thus (through perspective) every sort of confusion is revealed within us; and this is that weakness of the human mind on which the art of conjuring and of deceiving by light and shadow and other ingenious devices imposes, having an effect upon us like magic... And the arts of measuring and numbering and weighing come to the rescue of the human understanding—there is the beauty of them—and the apparent greater or less, or more or heavier, no longer have the mastery over us, but give way before calculation and measure and weight?"^{[15]}
Perspective images are calculated assuming a particular vanishing point. In order for the resulting image to appear identical to the original scene, a viewer of the perspective must view the image from the exact vantage point used in the calculations relative to the image. This cancels out what would appear to be distortions in the image when viewed from a different point. These apparent distortions are more pronounced away from the center of the image as the angle between a projected ray (from the scene to the eye) becomes more acute relative to the picture plane. In practice, unless the viewer chooses an extreme angle, like looking at it from the bottom corner of the window, the perspective normally looks more or less correct. This is referred to as "Zeeman's Paradox."^{[16]} It has been suggested that a drawing in perspective still seems to be in perspective at other spots because we still perceive it as a drawing, because it lacks depth of field cues.^{[17]}
For a typical perspective, however, the field of view is narrow enough (often only 60 degrees) that the distortions are similarly minimal enough that the image can be viewed from a point other than the actual calculated vantage point without appearing significantly distorted. When a larger angle of view is required, the standard method of projecting rays onto a flat picture plane becomes impractical. As a theoretical maximum, the field of view of a flat picture plane must be less than 180 degrees (as the field of view increases towards 180 degrees, the required breadth of the picture plane approaches infinity).
In order to create a projected ray image with a large field of view, one can project the image onto a curved surface. In order to have a large field of view horizontally in the image, a surface that is a vertical cylinder (i.e., the axis of the cylinder is parallel to the zaxis) will suffice (similarly, if the desired large field of view is only in the vertical direction of the image, a horizontal cylinder will suffice). A cylindrical picture surface will allow for a projected ray image up to a full 360 degrees in either the horizontal or vertical dimension of the perspective image (depending on the orientation of the cylinder). In the same way, by using a spherical picture surface, the field of view can be a full 360 degrees in any direction (note that for a spherical surface, all projected rays from the scene to the eye intersect the surface at a right angle).
Just as a standard perspective image must be viewed from the calculated vantage point for the image to appear identical to the true scene, a projected image onto a cylinder or sphere must likewise be viewed from the calculated vantage point for it to be precisely identical to the original scene. If an image projected onto a cylindrical surface is "unrolled" into a flat image, different types of distortions occur: For example, many of the scene's straight lines will be drawn as curves. An image projected onto a spherical surface can be flattened in various ways, including:
 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
See also
Notes
 ^ Perspective Drawing Handbook By Joseph D'Amelio, p. 19, published by Dover Publications
 ^ "Skenographia in Fifth Century". CUNY. http://depthome.brooklyn.cuny.edu/classics/dunkle/tragedy/intr15.htm. Retrieved 20071227.
 ^ "Pompeii. House of the Vettii. Fauces and Priapus". SUNY Buffalo. http://wings.buffalo.edu/AandL/Maecenas/italy_except_rome_and_sicily/pompeii/ac880907.html. Retrieved 20071227.
 ^ Panofsky, 122, note 1
 ^ File:VaticanVergilFol040rDeathOfDido.jpg Vatican Virgil image
 ^ Panofsky, p. 127
 ^ http://commons.wikimedia.org/wiki/Category:Objects_in_heraldry
 ^ Falco, Charles M. (12–15 February 2007), Ibn alHaytham and the Origins of Modern Image Analysis, International Conference on Information Sciences, Signal Processing and its Applications
 ^ Nader ElBizri (2005), "A Philosophical Perspective on Alhazen's Optics", Arabic Sciences and Philosophy: A Historical Journal (Cambridge University Press) 15: 189–218
 ^ Nader ElBizri (2004), "La perception de la profondeur: Alhazen, Berkeley et MerleauPonty", OriensOccidens, CNRS (Centre National de la Recherche Scientifique) 5: 171–184
 ^ Nader ElBizri (2007), "In Defense of the Sovereignty of Philosophy: alBaghdadi's Critique of Ibn alHaytham's Geometrisation of Place", Arabic Sciences and Philosophy: A Historical Journal (Cambridge University Press) 17: 57–80
 ^ A. Mark Smith (2001), "The Latin Source of the FourteenthCentury Italian Translation of Alhacen's De aspectibus (Vat. Lat. 4595)", Arabic Sciences and Philosophy: A Historical Journal (Cambridge University Press) 11: 27–43 [28]
 ^ "...and these works (of perspective by Brunelleschi) were the means of arousing the minds of the other craftsmen, who afterwords devoted themselves to this with great zeal."
Vasari's Lives of the Artists Chapter on Brunelleschi  ^ "Messer Paolo dal Pozzo Toscanelli, having returned from his studies, invited Filippo with other friends to supper in a garden, and the discourse falling on mathematical subjects, Filippo formed a friendship with him and learned geometry from him."
Vasarai's Lives of the Artists, Chapter on Brunelleschi  ^ Plato's Republic, Book X, 602d. http://etext.library.adelaide.edu.au/mirror/classics.mit.edu/Plato/republic.11.x.html
 ^ Mathographics by Robert Dixon New York: Dover, p. 82, 1991.
 ^ "...the paradox is purely conceptual: it assumes we view a perspective representation as a retinal simulation, when in fact we view it as a two dimensional painting. In other words, perspective constructions create visual symbols, not visual illusions. The key is that paintings lack the depth of field cues created by binocular vision; we are always aware a painting is flat rather than deep. And that is how our mind interprets it, adjusting our understanding of the painting to compensate for our position."
http://www.handprint.com/HP/WCL/perspect1.html Retrieved on December 25, 2006
References
 Panofsky, Erwin (1960). Renaissance and Renascences in Western Art. Stockholm: Almqvist & Wiksell. ISBN 0064300269.
Further reading
 Kemp, Martin (1992). The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat. Yale University Press.
 PérezGómez, Alberto, and Pelletier, Louise (1997). Architectural Representation and the Perspective Hinge. Cambridge, Mass.: MIT Press.
 Damisch, Hubert (1994). The Origin of Perspective, Translated by John Goodman. Cambridge, Mass.: MIT Press.
 Hyman, Isabelle, comp (1974). Brunelleschi in Perspective. Englewood Cliffs, New Jersey: PrenticeHall.
 Vasari, Giorgio (1568). The Lives of the Artists. Florence, Italy. http://easyweb.easynet.co.uk/giorgio.vasari/.
 Andersen, Kirsti (2007). The Geometry of an Art: The History of the Mathematical Theory of Perspective from Alberti to Monge. Springer.
External links
 Teaching Perspective in Art and Mathematics through Leonardo da Vinci's Work at Convergence
 Mathematics of Perspective Drawing
 Drawing Comics  Perspective
 Quadrilateral Perspective by Yvonne Tessuto Tavares
 The Perspective Page A short interactive introduction to the geometry of perspective drawing
 Perspective Drawing Video
 The Despotic Eye: An Illustration of Metabletic Phenomenology and Its Implications by Robert D. Romanyshyn. An essay on the link between Western culture and linear perspective drawing, published in Janus Head
 Perspective in Ancient RomanWall Painting (c.50BC)
Categories: Perspective projection
 Technical drawing
 Functions and mappings
 Parallel projection
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