Linear or point-projection perspective () is one of two types of

graphical projection 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 complex object f ...

perspective in the

graphic arts A category of fine art, graphic art covers a broad range of visual artistic expression, typically two-dimensional graphics, i.e. produced on a flat surface,parallel projection. Linear perspective is an approximate representation, generally on a flat surface, of an image as it is seen by the eye. Perspective drawing is useful for representing a

three-dimensional In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), poi ...

scene in a

two-dimensional A two-dimensional space is a mathematical space with two dimensions, meaning points have two degrees of freedom: their locations can be locally described with two coordinates or they can move in two independent directions. Common two-dimension ...

medium, like

paper Paper is a thin sheet material produced by mechanically or chemically processing cellulose fibres derived from wood, Textile, rags, poaceae, grasses, Feces#Other uses, herbivore dung, or other vegetable sources in water. Once the water is dra ...

. It is based on the optical fact that for a person an object looks N times (linearly) smaller if it has been moved N times further from the eye than the original distance was. 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 , meaning that an object's dimensions parallel to the line of sight appear shorter than its dimensions

perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...

to 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.

Italian Renaissance The Italian Renaissance ( ) was a period in History of Italy, Italian history between the 14th and 16th centuries. The period is known for the initial development of the broader Renaissance culture that spread across Western Europe and marked t ...

painters and architects including Filippo Brunelleschi,

Leon Battista Alberti Leon Battista Alberti (; 14 February 1404 – 25 April 1472) was an Italian Renaissance humanist author, artist, architect, poet, Catholic priest, priest, linguistics, linguist, philosopher, and cryptography, cryptographer; he epitomised the natu ...

Masaccio Masaccio (, ; ; December 21, 1401 – summer 1428), born Tommaso di Ser Giovanni di Simone, was a Florentine artist who is regarded as the first great List of Italian painters, Italian painter of the Quattrocento period of the Italian Renaiss ...

Paolo Uccello Paolo Uccello ( , ; 1397 – 10 December 1475), born Paolo di Dono, was an Italian Renaissance painter and mathematician from Florence who was notable for his pioneering work on visual Perspective (graphical), perspective in art. In his book ''Liv ...

Piero della Francesca Piero della Francesca ( , ; ; ; – 12 October 1492) was an Italian Renaissance painter, Italian painter, mathematician and List of geometers, geometer of the Early Renaissance, nowadays chiefly appreciated for his art. His painting is charact ...

and

Luca Pacioli Luca Bartolomeo de Pacioli, O.F.M. (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as account ...

studied linear perspective, wrote treatises on it, and incorporated it into their artworks.

Overview

Linear or point-projection perspective works by putting an imaginary flat plane that is close to an object under observation and directly facing an observer's eyes (i.e., the observer is on a normal, or perpendicular line to the plane). Then draw straight lines from every point in the object to the observer. The area on the plane where those lines pass through the plane is a point-projection prospective image resembling what is seen by the observer.

Examples of one-point perspective

File:Perspective-1point.svg File:LOMEX,Rendering of streetscape.tiff File:Railroad-Tracks-Perspective.jpg

Examples of two-point perspective

Perspective1.jpg, A cube drawing using two-point perspective EquitableTrustBuildingBiltmoreNYC1921.jpg Harrington's hardware shop Broadstairs Kent England - inspiration for the 'Four Candles' Two Ronnies sketch 02.jpg The Whitaker, Rawtenstall, formerly Rossendale Museum, Lancashire, England 2008.jpg Big Dam Film Festival Logo.jpg 2-pt-sketchup.jpg

Examples of three-point perspective

3-punktperspektive 1.svg, A cube in three-points ''perspective'' where, compared with the above cube in two-points ''prospective'', there is an additional

vanishing point A vanishing point is a point (geometry), point on the projection plane, image plane of a graphical perspective, perspective rendering where the two-dimensional perspective projections of parallel (geometry), parallel lines in three-dimensional ...

far below the cube, from which straight lines come that are coincident with vertical edges of the cube. South tower of the Notre Dame.jpg Canary wharf looking up.jpg Staircase perspective.jpg, Staircase in multi-points perspective. Each point is a vanishing point from which straight lines come, and these lines are a guide to draw 3-dimensional object.

Examples of curvilinear perspective

Additionally, a central

can be used (just as with one-point perspective) to indicate frontal (foreshortened) depth. cmglee Judge Business School rear.jpg Boston, Boylston Street.jpg 194 - Buenos Aires - Casa Rosada - Janvier 2010.jpg Regensburg Uferpanorama 08 2006.jpg

History

Paintings from the Chauvet cave (museum replica).jpg, Chauvet cave, spatially effective grading of a group of animals through overlap () Ägyptischer Maler um 1500 v. Chr. 001.jpg,

Fresco Fresco ( or frescoes) is a technique of mural painting executed upon freshly laid ("wet") lime plaster. Water is used as the vehicle for the dry-powder pigment to merge with the plaster, and with the setting of the plaster, the painting become ...

from an Egyptian grave, Cubiculum (bedroom) from the Villa of P. Fannius Synistor at Boscoreale MET DP170950.jpg, Fresco from the Villa of Publius Fannius Synistor in Boscoreale near

Pompeii Pompeii ( ; ) was a city in what is now the municipality of Pompei, near Naples, in the Campania region of Italy. Along with Herculaneum, Stabiae, and Villa Boscoreale, many surrounding villas, the city was buried under of volcanic ash and p ...

, 1st ct. BC Song Dynasty Hydraulic Mill for Grain.JPG, A

Song dynasty The Song dynasty ( ) was an Dynasties of China, imperial dynasty of China that ruled from 960 to 1279. The dynasty was founded by Emperor Taizu of Song, who usurped the throne of the Later Zhou dynasty and went on to conquer the rest of the Fiv ...

watercolor painting of a mill in an oblique projection, 12th century Ambrogio_Lorenzetti_Annunciation.jpg, The floor tiles in Lorenzetti's ''Annunciation'' (1344) strongly anticipate modern perspective

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 Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography * Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...

, also from

hieratic Hieratic (; ) is the name given to a cursive writing system used for Ancient Egyptian and the principal script used to write that language from its development in the third millennium BCE until the rise of Demotic in the mid-first millennium BCE ...

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. Additionally, oblique foreshortening of round elements like shields and wheels is evident in

Ancient Greek Ancient Greek (, ; ) includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Greek ...

red-figure pottery Red-figure pottery () is a style of Pottery of ancient Greece, ancient Greek pottery in which the background of the pottery is painted black while the figures and details are left in the natural red or orange color of the clay. It developed in A ...

. 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 Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...

's '' Poetics'' as ''skenographia'': using flat panels on a stage to give the illusion of depth. The philosophers

Anaxagoras Anaxagoras (; , ''Anaxagóras'', 'lord of the assembly'; ) was a Pre-Socratic Greek philosopher. Born in Clazomenae at a time when Asia Minor was under the control of the Persian Empire, Anaxagoras came to Athens. In later life he was charged ...

and

Democritus Democritus (, ; , ''Dēmókritos'', meaning "chosen of the people"; – ) was an Ancient Greece, Ancient Greek Pre-Socratic philosophy, pre-Socratic philosopher from Abdera, Thrace, Abdera, primarily remembered today for his formulation of an ...

worked out geometric theories of perspective for use with ''skenographia''.

Alcibiades Alcibiades (; 450–404 BC) was an Athenian statesman and general. The last of the Alcmaeonidae, he played a major role in the second half of the Peloponnesian War as a strategic advisor, military commander, and politician, but subsequently ...

had paintings in his house designed using ''skenographia'', so this art was not confined merely to the stage.

Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...

in his ''

Optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...

'' () argues correctly that the perceived size of an object is not related to its distance from the eye by a simple proportion. In the first-century BC

fresco Fresco ( or frescoes) is a technique of mural painting executed upon freshly laid ("wet") lime plaster. Water is used as the vehicle for the dry-powder pigment to merge with the plaster, and with the setting of the plaster, the painting become ...

es of the Villa of P. Fannius Synistor, multiple

s are used in a systematic but not fully consistent manner. 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, while others credit it as an indigenous invention of

Ancient China The history of China spans several millennia across a wide geographical area. Each region now considered part of the Chinese world has experienced periods of unity, fracture, prosperity, and strife. Chinese civilization first emerged in the Y ...

. Oblique projection is also seen in Japanese art, such as in the

Ukiyo-e is a genre of Japanese art that flourished from the 17th through 19th centuries. Its artists produced woodblock printing, woodblock prints and Nikuhitsu-ga, paintings of such subjects as female beauties; kabuki actors and sumo wrestlers; scenes ...

paintings of Torii Kiyonaga (1752–1815). 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. 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 Philostratus or Lucius Flavius Philostratus (; ; 170s – 240s AD), called "the Athenian", was a Greek sophist of the Roman imperial period. His father was a minor sophist of the same name. He flourished during the reign of Septimius Severus ...

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. 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. 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 Byzantine art comprises the body of artistic products of the Eastern Roman Empire, as well as the nations and states that inherited culturally from the empire. Though the empire itself emerged from the decline of Rome, decline of western Rome and ...

was also aware of these principles, but also used the

reverse perspective Reverse perspective, also called inverse perspective,. inverted perspective, divergent perspective, or Byzantine perspective, is a form of perspective (graphical), perspective drawing where the objects depicted in a scene are placed between the ...

convention for the setting of principal figures.

Ambrogio Lorenzetti Ambrogio Lorenzetti (; – after 9 August 1348) was an Italian painter of the Sienese school. He was active from approximately 1317 to 1348. He painted ''The Allegory of Good and Bad Government'' in the Sala dei Nove (Salon of Nine or Council Ro ...

painted a floor with convergent lines in his '' Presentation at the Temple'' (1342), though the rest of the painting lacks perspective elements.

Renaissance

It is generally accepted that Filippo Brunelleschi conducted a series of experiments between 1415 and 1420, which included making drawings of various Florentine buildings in correct perspective. According to

Vasari Giorgio Vasari (30 July 1511 – 27 June 1574) was an Italian Renaissance painter, architect, art historian, and biographer who is best known for his work '' Lives of the Most Excellent Painters, Sculptors, and Architects'', considered the ide ...

and Antonio Manetti, in about 1420, Brunelleschi demonstrated his discovery of perspective by having people look through a hole on his painting from the backside. Through it, they would see a building such as the

Florence Baptistery The Florence Baptistery, also known as the Baptistery of Saint John (), is a religious building in Florence, Italy. Dedicated to the patron saint of the city, John the Baptist, it has been a focus of religious, civic, and artistic life since its ...

for which the painting was made. When Brunelleschi lifted a mirror between the building and the painting, the mirror reflected the painting to an observer looking through the hole, so that the observer can compare how similar the building and the painting of it are. (The

is centered from the perspective of an experiment participant.) Brunelleschi applied this new system of perspective to his paintings around 1425. This scenario is indicative, but faces several problems that are still debated. First of all, nothing can be said for certain about the correctness of his perspective construction of the Baptistery of San Giovanni because Brunelleschi's panel is lost. Second, no other perspective painting or drawing by Brunelleschi is known. (In fact, Brunelleschi was not known to have painted at all.) Third, in the account written by Antonio Manetti in his ''Vita di Ser Brunellesco'' at the end of the 15th century on Brunelleschi's panel, there is not a single occurrence of the word "experiment". Fourth, the conditions listed by Manetti are contradictory with each other. For example, the description of the eyepiece sets a visual field of 15°, much narrower than the visual field resulting from the urban landscape described. Soon after Brunelleschi's demonstrations, nearly every interested artist in Florence and in Italy used geometrical perspective in their paintings and sculpture, notably

Donatello Donato di Niccolò di Betto Bardi ( – 13 December 1466), known mononymously as Donatello (; ), was an Italian Renaissance sculpture, Italian sculptor of the Renaissance period. Born in Republic of Florence, Florence, he studied classical sc ...

Lorenzo Ghiberti Lorenzo Ghiberti (, , ; 1378 – 1 December 1455), born Lorenzo di Bartolo, was an Italian Renaissance sculptor from Florence, a key figure in the Early Renaissance, best known as the creator of two sets of bronze doors of the Florence Baptister ...

Masolino da Panicale Lordship of Perugia , death_date = , death_place = Florence, Republic of Florence , nationality = Italian , field = Painting, fresco , training = , movement = Italian Renaissance , works = frescoes in ...

, 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'' (), Donatello's '' The Feast of Herod'' (), as well as Ghiberti's '' Jacob and Esau'' and other panels from the east doors of the Florence Baptistery. Masaccio (d. 1428) achieved an illusionistic effect by placing the vanishing point at the viewer's eye level in his ''

Holy Trinity The Trinity (, from 'threefold') is the Christian doctrine concerning the nature of God, which defines one God existing in three, , consubstantial divine persons: God the Father, God the Son (Jesus Christ) and God the Holy Spirit, three ...

'' (), and in '' The Tribute Money'', it is placed behind the face of Jesus. In the late 15th century,

Melozzo da Forlì Melozzo da Forlì ( – 8 November 1494) was an Italian Renaissance painter and architect. His fresco paintings are notable for the use of foreshortening. He was the most important member of the Forlì painting school. Biography Melozzo was s ...

first applied the technique of foreshortening (in Rome, Loreto,

Forlì Forlì ( ; ; ; ) is a ''comune'' (municipality) and city in Emilia-Romagna, Northern Italy, and is, together with Cesena, the capital of the Province of Forlì-Cesena.The city is situated along the Via Emilia, to the east of the Montone river, ...

and others). This overall story is based on qualitative judgments, and would need to be faced against the material evaluations that have been conducted on Renaissance perspective paintings. Apart from the paintings of

, which are a model of the genre, the majority of 15th century works show serious errors in their geometric construction. This is true of Masaccio's ''Trinity'' fresco and of many works, including those by renowned artists like Leonardo da Vinci. As shown by the quick proliferation of accurate perspective paintings in Florence, Brunelleschi likely understood (with help from his friend the mathematician Toscanelli), but did not publish the mathematics behind perspective. Decades later, his friend

wrote ' (), 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. 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 Ḥasan Ibn al-Haytham ( Latinized as Alhazen; ; full name ; ) was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq.For the description of his main fields, see e.g. ("He is one of the princ ...

's ''

Book of Optics The ''Book of Optics'' (; or ''Perspectiva''; ) is a seven-volume treatise on optics and other fields of study composed by the medieval Arab scholar Ibn al-Haytham, known in the West as Alhazen or Alhacen (965–c. 1040 AD). The ''Book ...

''. This book, translated around 1200 into Latin, had laid the mathematical foundation for perspective in Europe. Piero della Francesca elaborated on ''De pictura'' in his '' De Prospectiva pingendi'' in the 1470s, making many references to Euclid. 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 In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edge ...

as they would appear in perspective.

's 1509 '' Divina proportione'' (''Divine Proportion''), illustrated by

Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 1452 - 2 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested o ...

, summarizes the use of perspective in painting, including much of Della Francesca's treatise. Leonardo applied one-point perspective as well as shallow focus to some of his works. Two-point perspective was demonstrated as early as 1525 by

Albrecht Dürer Albrecht Dürer ( , ;; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer or Duerer, was a German painter, Old master prin ...

, who studied perspective by reading Piero and Pacioli's works, in his ''Unterweisung der Messung'' ("Instruction of the Measurement").

Limitations

Perspective images are created with reference to a particular center of vision for the picture plane. In order for the resulting image to appear identical to the original scene, a viewer must view the image from the exact vantage point used in the calculations relative to the image. When viewed from a different point, this cancels out what would appear to be distortions in the image. For example, a sphere drawn in perspective will be stretched into an ellipse. 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. Artists may choose to "correct" perspective distortions, for example by drawing all spheres as perfect circles, or by drawing figures as if centered on the direction of view. In practice, unless the viewer observes the image from an extreme angle, like standing far to the side of a painting, the perspective normally looks more or less correct. This is referred to as "Zeeman's Paradox". Retrieved on 25 December 2006

Notes

References

Sources

External links

Teaching Perspective in Art and Mathematics through Leonardo da Vinci's Work
at

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university A university () is an educational institution, institution of tertiary edu ...

Metaphysical Perspective in Ancient Roman-Wall Painting

How to Draw a Two Point Perspective Grid
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Creating Comics
{{DEFAULTSORT:Perspective (Graphical) Perspective projection Technical drawing Functions and mappings Composition in visual art Italian inventions