# Perspective (graphical)

Last updated

External video
Linear Perspective: Brunelleschi's Experiment, Smarthistory [1]
How One-Point Linear Perspective Works, Smarthistory [2]
Empire of the Eye: The Magic of Illusion: The Trinity-Masaccio, Part 2, National Gallery of Art [3]

Linear or point-projection perspective (from Latin : 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, generally on a flat surface, of an image as it is seen by the eye. The most characteristic features of linear 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. All objects will recede to points in the distance, usually along the horizon line, but also above and below the horizon line depending on the view used.

## Contents

Italian Renaissance painters and architects including 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

Perspective 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. [4] 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.

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.

### Aerial perspective

Aerial (or atmospheric) perspective depends on distant objects being more obscured by atmospheric factors, so farther objects are less visible to the viewer. As the distance between an object and a viewer increases, the contrast between the object and its background decreases, and the contrast of any markings or details within the object also decreases. The colours of the object also become less saturated and shift towards the background colour.

Aerial perspective can be combined with, but does not depend on, one or more vanishing points.

### 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 ties/sleepers) 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—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.

### Two-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 picture plane 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.

### 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.

### Curvilinear perspective

By superimposing two perpendicular, curved sets of two-point perspective lines, a four-or-above-point curvilinear perspective can be achieved. This perspective can be used with a central horizon line of any orientation, and can depict both a worm's-eye and bird's-eye view at the same time.

Additionally, a central vanishing point can be used (just as with one-point perspective) to indicate frontal (foreshortened) depth. [5]

### Foreshortening

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 ]

In painting, foreshortening in the depiction of the human figure was improved during 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. (Andrea Mantegna is also an author of the Frescoes in the Camera degli Sposi; in which a part called "The oculus" uses foreshortening represented by the figures which look down upon the watchers.)

## History

Rudimentary attempts to create the illusion of depth were made in ancient times, with artists achieving isometric projection by the Middle Ages. Various early Renaissance works depict perspective lines with an implied convergence, albeit without a unifying vanishing point. The first to master perspective was Italian Renaissance architect Filippo Brunelleschi, who developed the adherence of perspective to a vanishing point in the early fifteenth century. His discovery was immediately influential on subsequent Renaissance art and was explored contemporaneously in manuscripts by Leon Battista Alberti, Piero della Francesca and others.

### 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; simple overlapping was also employed to relate distance. [7] Additionally, oblique foreshortening of round elements like shields and wheels is evident in Ancient Greek red-figure pottery. [8]

Systematic attempts to evolve a system of perspective are usually considered to have begun around the fifth century BC in the art of ancient Greece, as part of a developing interest in illusionism allied to theatrical scenery. This was detailed within Aristotle's Poetics as skenographia: using flat panels on a stage to give the illusion of depth. [9] The philosophers Anaxagoras and Democritus worked out geometric theories of perspective for use with skenographia. Alcibiades had paintings in his house designed using skenographia, so this art was not confined merely to the stage. Euclid in his Optics (c.300 BC) argues correctly that the perceived size of an object is not related to its distance from the eye by a simple proportion. [10] In the first-century BC frescoes of the Villa of P. Fannius Synistor, multiple vanishing points are used in a systematic but not fully consistent manner. [6]

Chinese artists made use of oblique projection from the first or second century until the 18th century. It is not certain how they came to use the technique; Dubery and Willats (1983) speculate that the Chinese acquired the technique from India, which acquired it from Ancient Rome, [11] while others credit it as an indigenous invention of Ancient China. [12] [13] [14] Oblique projection is also seen in Japanese art, such as in the Ukiyo-e paintings of Torii Kiyonaga (1752–1815). [11] [lower-alpha 1]

Various paintings and drawings from the Middle Ages show amateur attempts at projections of objects, where parallel lines are successfully represented in isometric projection, or by nonparallel ones without a vanishing point.

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 realism, 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. [15] 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 semi-circular theatre seen from the stage. [16] 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. [17] 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.

Medieval artists in Europe, like those in the Islamic world and China, were aware of the general principle of varying the relative size of elements according to distance, but even more than classical art were 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 used the reverse perspective convention for the setting of principal figures. Ambrogio Lorenzetti painted a floor with convergent lines in his Presentation at the Temple (1342), though the rest of the painting lacks perspective elements. [18] Other artists of the greater proto-Renaissance, such as Melchior Broederlam, strongly anticipated modern perspective in their works but lacked the constraint of a vanishing point.

### Renaissance

Filippo Brunelleschi conducted a series of experiments between 1415 and 1420, which included making drawings of various Florentine buildings in correct perspective. [20] According to Vasari and Antonio Manetti, in about 1420, Brunelleschi demonstrated his discovery by having people look through a hole in the back of a painting he had made. Through it, they would see a building such as the Florence Baptistery. When Brunelleschi lifted a mirror in front of the viewer, it reflected his painting of the buildings which had been seen previously, so that the vanishing point was centered from the perspective of the participant. [21] Brunelleschi applied the new system of perspective to his paintings around 1425. [22]

Soon after Brunelleschi's demonstrations, nearly every artist in Florence and in Italy used geometrical perspective in their paintings and sculpture, [23] notably Donatello, Masaccio, Lorenzo Ghiberti, Masolino da Panicale, Paolo Uccello, and Filippo Lippi. Not only was perspective a way of showing depth, it was also a new method of creating a composition. Visual art could now depict a single, unified scene, rather than a combination of several. Early examples include Masolino's St. Peter Healing a Cripple and the Raising of Tabitha (c.1423), Donatello's The Feast of Herod (c.1427), as well as Ghiberti's Jacob and Esau and other panels from the east doors of the Florence Baptistery. [24] Masaccio (d. 1428) achieved an illusionistic effect by placing the vanishing point at the viewer's eye level in his Holy Trinity (c.1427), [25] and in The Tribute Money , it is placed behind the face of Jesus. [26] [lower-alpha 2] In the late 15th century, Melozzo da Forlì first applied the technique of foreshortening (in Rome, Loreto, Forlì and others). [28]

As shown by the quick proliferation of accurate perspective paintings in Florence, Brunelleschi likely understood (with help from his friend the mathematician Toscanelli), [29] but did not publish, the mathematics behind perspective. Decades later, his friend Leon Battista Alberti wrote (c.1435), 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. [lower-alpha 3] 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 Book of Optics . [30] This book, translated around 1200 into Latin, had laid the mathematical foundation for perspective in Europe. [31]

Perspective remained, for a while, the domain of Florence. Jan van Eyck, among others, failed to utilize a consistent vanishing point for the converging lines in paintings, as in the Arnolfini Portrait (1434). 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.

Piero della Francesca elaborated on De pictura in his De Prospectiva pingendi in the 1470s, making many references to Euclid. [32] 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. Luca Pacioli's 1509 Divina proportione (Divine Proportion), illustrated by Leonardo da Vinci, summarizes the use of perspective in painting, including much of Della Francesca's treatise. [33] Leonardo applied one-point perspective as well as shallow focus to some of his works. [34]

Two-point perspective was demonstrated as early as 1525 by Albrecht Dürer, who studied perspective by reading Piero and Pacioli's works, in his Unterweisung der messung ("Instruction of the measurement"). [35]

Perspective features heavily in the research of the 17th-century architect, geometer, and optician Girard Desargues on perspective, optics and projective geometry, as well as the theorem named after him.

## Limitations

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". [36] 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. [37]

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).

To create a projected ray image with a large field of view, one can project the image onto a curved surface. 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 z-axis) 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:

• 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

## Notes

1. In the 18th century, Chinese artists began to combine oblique perspective with regular diminution of size of people and objects with distance; no particular vantage point is chosen, but a convincing effect is achieved. [11]
2. Near the end of the 15th century, Leonardo da Vinci placed the vanishing point in his Last Supper behind Christ's other cheek. [27]
3. 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.

## Related Research Articles

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 angle between any two of them is 120 degrees.

Orthographic projection is a means of representing three-dimensional objects in two dimensions. It is a form of parallel projection, in which all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are not orthogonal to the projection plane.

A 3D 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 complex object for viewing capability on a simpler plane. This concept of extending 2D geometry to 3D was mastered by Heron of Alexandria in the first century. Heron could be called the father of 3D. 3D Projection is the basis of the concept for Computer Graphics simulating fluid flows to imitate realistic effects. Lucas Films 'ILM group is credited with introducing the concept.

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.

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 in art. The theoretical basis for descriptive geometry is provided by planar geometric projections. The earliest known publication on the technique was "Underweysung der Messung mit dem Zirckel und Richtscheyt", published in Linien, Nuremberg: 1525, by Albrecht Dürer. Italian architect Guarino Guarini was also a pioneer of projective and descriptive geometry, as is clear from his Placita Philosophica (1665), Euclides Adauctus (1671) and Architettura Civile, anticipating the work of Gaspard Monge (1746–1818), who is usually credited with the invention of descriptive geometry. Gaspard Monge is usually considered the "father of descriptive geometry" due to his developments in geometric problem solving. His first discoveries were in 1765 while he was working as a draftsman for military fortifications, although his findings were published later on.

Oblique projection is a simple type of technical drawing of graphical projection used for producing two-dimensional (2D) images of three-dimensional (3D) objects.

An autostereogram is a single-image stereogram (SIS), designed to create the visual illusion of a three-dimensional (3D) scene from a two-dimensional image. In order to perceive 3D shapes in these autostereograms, one must overcome the normally automatic coordination between accommodation (focus) and horizontal vergence. The illusion is one of depth perception and involves stereopsis: depth perception arising from the different perspective each eye has of a three-dimensional scene, called binocular parallax.

Curvilinear 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 curviligne, which was translated into English in 1987 as Curvilinear Perspective: From Visual Space to the Constructed Image and published by the University of California Press.

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 perpendicular to a picture plane, the construction is known as one-point perspective, and their vanishing point corresponds to the oculus, or "eye point", from which the image should be viewed for correct perspective geometry. Traditional linear drawings use objects with one to three sets of parallels, defining one to three vanishing points.

Reverse perspective, also called inverse perspective, inverted perspective, divergent perspective, or Byzantine perspective, is a form of perspective drawing in which the objects depicted in a scene are placed between the projective point and the viewing plane. Objects farther away from the viewing plane are drawn as larger, and closer objects are drawn as smaller, in contrast to the more conventional linear perspective for which closer objects appear larger. Lines that are parallel in three-dimensional space are drawn as diverging against the horizon, rather than converging as they do in linear perspective. Technically, the vanishing points are placed outside the painting with the illusion that they are "in front of" the painting.

2.5D perspective refers to one of two things:

In painting, photography, graphical perspective and descriptive geometry, a picture plane is an image plane located between the "eye point" and the object being viewed and is usually coextensive to the material surface of the work. It is ordinarily a vertical plane perpendicular to the sightline to the object of interest.

The term composition means "putting together". It can be thought of as the organization of the elements of art according to the principles of art. Composition can apply to any work of art, from music through writing and into photography, that is arranged using conscious thought.

Anamorphosis is a distorted projection requiring the viewer to occupy a specific vantage point, use special devices, or both to view a recognizable image. It is used in painting, photography, sculpture and installation, toys, and film special effects. The word is derived from the Greek prefix ana‑, meaning "back" or "again", and the word morphe, meaning "shape" or "form". Extreme anamorphosis has been used by artists to disguise caricatures, erotic and scatological scenes, and other furtive images from a casual spectator, while revealing an undistorted image to the knowledgeable viewer.

A parallel 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 lines of sight or projection lines, are parallel to each other. It is a basic tool in descriptive geometry. The projection is called orthographic if the rays are perpendicular (orthogonal) to the image plane, and oblique or skew if they are not.

A bird's-eye view is an elevated view of an object from above, with a perspective as though the observer were a bird, often used in the making of blueprints, floor plans, and maps. It can be an aerial photograph, but also a drawing.

Phantograms, also known as Phantaglyphs, Op-Ups, free-standing anaglyphs, levitated images, and book anaglyphs, are a form of optical illusion. Phantograms use perspectival anamorphosis to produce a 2D image that is distorted in a particular way so as to appear, to a viewer at a particular vantage point, three-dimensional, standing above or recessed into a flat surface. The illusion of depth and perspective is heightened by stereoscopy techniques; a combination of two images, most typically but not necessarily an anaglyph. With common (red–cyan) 3D glasses, the viewer's vision is segregated so that each eye sees a different image.

Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts.

A projection plane, or plane of projection, is a type of view in which graphical projections from an object intersect. Projection planes are used often in descriptive geometry and graphical representation. A picture plane in perspective drawing is a type of projection plane.

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 coordinate axes play a crucial role. The result of an axonometric procedure is a uniformly-scaled parallel projection of the object. In general, the resulting parallel projection is oblique ; but in special cases the result is orthographic, which in this context is called an orthogonal axonometry.

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12. "Seeing History: Is perspective learned or natural?". Eclectic Light. 10 January 2018. Over the same period, the development of sophisticated and highly-detailed visual art in Asia arrived at a slightly different solution, now known as the oblique projection. Whereas Roman and subsequent European visual art effectively had multiple and incoherent vanishing points, Asian art usually lacked any vanishing point, but aligned recession in parallel. An important factor here is the use of long scrolls, which even now make fully coherent perspective projection unsuitable.
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22. "...and these works (of perspective by Brunelleschi) were the means of arousing the minds of the other craftsmen, who afterwards devoted themselves to this with great zeal."
Vasari's Lives of the Artists Chapter on Brunelleschi
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28. "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
29. El-Bizri, Nader (2010). "Classical Optics and the Perspectiva Traditions Leading to the Renaissance". In Hendrix, John Shannon; Carman, Charles H. (eds.). . Farnham, Surrey: Ashgate. pp.  11–30. ISBN   978-1-409400-24-0.
30. Hans, Belting (2011). Florence and Baghdad: Renaissance art and Arab science (1st English ed.). Cambridge, Massachusetts: Belknap Press of Harvard University Press. pp. 90–92. ISBN   9780674050044. OCLC   701493612.
31. Livio, Mario (2003). The Golden Ratio. New York: Broadway Books. p. 126. ISBN   0-7679-0816-3.
32. O'Connor, J. J.; Robertson, E. F. (July 1999). "Luca Pacioli". University of St Andrews. Archived from the original on 22 September 2015. Retrieved 23 September 2015.
33. Goldstein, Andrew M. (17 November 2011). "The Male "Mona Lisa"?: Art Historian Martin Kemp on Leonardo da Vinci's Mysterious "Salvator Mundi"". Blouin Artinfo.
34. MacKinnon, Nick (1993). "The Portrait of Fra Luca Pacioli". The Mathematical Gazette . 77 (479): 206. doi:10.2307/3619717. JSTOR   3619717.
35. Mathographics by Robert Dixon New York: Dover, p. 82, 1991.
36. "...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."
"Archived copy". Archived from the original on 6 January 2007. Retrieved 25 December 2006.CS1 maint: archived copy as title (link) Retrieved on 25 December 2006