The term perspective (in Latin: perspicere "to see through")[1] is used in the graphic arts to designate a representation, generally on a flat surface (such as paper or a canvas), of a motif as it is perceived by the eye, so that its three-dimensional configuration can be intuited.
Geometrically, these representations are obtained from the intersection of a plane with a set of visuals, the straight lines or rays that join the points of the represented object with the point from which it is observed (called the point of view).
In this sense, there are two fundamental types of perspectives, depending on the relative position between the represented model and the point of view:
The appearance of computer programs capable of handling the geometric information of complex models with great agility has diluted the difference that existed between these two types of perspectives in relation to their execution, especially when they were created manually. In any case, the graphic procedures and their theoretical bases associated with these two types of representation systems continue to be part of the academic programs of both secondary education centers and higher schools of architecture, engineering and fine arts around the world.[7].
Overview
Linear perspective always works by representing light from a scene passing through an imaginary rectangle (realized as the plane of the painting), until it reaches the viewer's view, as if the viewer were looking through a window and painting directly on the glass what is seen. If viewed from the same location where the window pane 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, reduced version of the object on the other side of the window.[8] Since each part of the painted object lies in a straight line from the viewer's eye to the equivalent part of the real object depicted, the viewer sees no difference (without considering binocular depth perception) between the scene painted on the window glass and the view of the real scene.
All perspective drawings assume that the viewer is at some distance from the drawing. Objects are scaled relative to that viewing point. An object often does not scale uniformly: a circle frequently appears as an ellipse and a square may appear as a trapezoid. This distortion is known as foreshortening.[9].
Architecture as prospective
Introduction
The term perspective (in Latin: perspicere "to see through")[1] is used in the graphic arts to designate a representation, generally on a flat surface (such as paper or a canvas), of a motif as it is perceived by the eye, so that its three-dimensional configuration can be intuited.
Geometrically, these representations are obtained from the intersection of a plane with a set of visuals, the straight lines or rays that join the points of the represented object with the point from which it is observed (called the point of view).
In this sense, there are two fundamental types of perspectives, depending on the relative position between the represented model and the point of view:
The appearance of computer programs capable of handling the geometric information of complex models with great agility has diluted the difference that existed between these two types of perspectives in relation to their execution, especially when they were created manually. In any case, the graphic procedures and their theoretical bases associated with these two types of representation systems continue to be part of the academic programs of both secondary education centers and higher schools of architecture, engineering and fine arts around the world.[7].
Overview
Linear perspective always works by representing light from a scene passing through an imaginary rectangle (realized as the plane of the painting), until it reaches the viewer's view, as if the viewer were looking through a window and painting directly on the glass what is seen. If viewed from the same location where the window pane 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, reduced version of the object on the other side of the window.[8] Since each part of the painted object lies in a straight line from the viewer's eye to the equivalent part of the real object depicted, the viewer sees no difference (without considering binocular depth perception) between the scene painted on the window glass and the view of the real scene.
Perspective drawings have a horizon line, which is almost always implied. This line, directly opposite the viewer's eye, represents infinitely distant objects, which have been reduced in distance to the infinitesimal thickness of a line. It is analogous to (and named after) the Earth's horizon.[10].
Any perspective representation of a scene that includes parallel lines has one or more vanishing points. A single-point perspective drawing means that the drawing has a single vanishing point, usually (but not necessarily) directly opposite the observer's eye and usually (but not necessarily) on the horizon line. All lines parallel to the viewer's line of sight recede toward the horizon, directed toward the vanishing point. This is the standard railway convergence phenomenon. A drawing with two points of convergence 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 a certain angle to the plane of the drawing.[2].
Perspectives containing many parallel lines are often seen when drawing works of architecture (where lines parallel to the x, y and z axes are frequently used). Because it is rare to have a scene consisting solely of lines parallel to the three Cartesian axes, in practice it is not common to see perspectives with only one, two, or three vanishing points; Even a simple building often has a pointed roof that results in a minimum of six sets of parallel lines, which in turn correspond to up to six vanishing points.[2].
In contrast, nature scenes often do not possess any set of parallel lines and, therefore, have no vanishing points.[11].
On the other hand, orthogonal axonometric perspectives can be considered special cases of conical perspective, in which the views that serve to outline the drawing are parallel to each other (which is equivalent to placing the point of view infinitely far from the model). This results in the absence of vanishing points in the drawing. On the other hand, oblique axonometries do not correspond to any type of optical perspective model, being a mathematical graphic device that allows us to simplify the creation of certain representations of three-dimensional bodies.
Historical development
Contenido
El intento de dar sensación de volumen a las representaciones pictóricas se encuentra más o menos presente desde las primeras muestras del arte paleolítico, como se puede apreciar en las pinturas rupestres de la cueva de Altamira (con unos 35.000 años de antigüedad), en las que se aprovecha el relieve de las rocas de la pared de la cueva para dotar de profundidad a los dibujos.[12] Esta tendencia alcanzó altas cotas de perfección técnica durante la época romana (seguramente basándose en procedimientos empíricos o intuitivos), pero no fue hasta el siglo , durante el Renacimiento italiano, cuando se sentaron las bases geométricas que permitieron convertir el dibujo en perspectiva en una técnica con sólidos fundamentos teóricos. Desde entonces, ha pasado a generalizarse su uso, convirtiéndose en una útil herramienta primero para los pintores, después para los arquitectos y más adelante para los ingenieros, hasta llegar al desarrollo en el último cuarto del siglo de las aplicaciones por ordenador que permiten automatizar la generación de este tipo de vistas, que hasta entonces podían requerir de una laboriosa construcción gráfica.[13].
Background
Early artistic paintings and drawings generally classified many objects and characters hierarchically according to their spiritual or thematic importance, but not according to their distance from the viewer, and did not use foreshortening. The most important figures are often shown as the tallest in a composition, especially of hieratic motifs, leading to the so-called "vertical perspective", common in Ancient Egyptian art, where a group of "closer" figures are shown below the largest figure or figures. In Egyptian paintings, a two-dimensional space of the surface to be painted was conceived, without strictly suggesting an idea of spatial conception. They arranged the characters increasing their size according to their importance, what art historians call hierarchical or theological perspective.[14].
The only method of indicating the relative position of elements in the composition was superposition, which is widely used in works such as the Elgin Marbles, the famous sculptures that decorated the Parthenon in Athens. However, there are numerous studies about the Parthenon itself, which claim that its dimensions (especially the shape and inclination of its columns) were meticulously studied to counteract the effects of perspective on the main lines of the building.[15].
Antiquity and Middle Ages
The first attempts to develop a perspective system are considered to have begun around the century BC. C. in the art of Ancient Greece, as part of the interest in producing the optical illusion of depth in theatrical settings. This fact is described in Aristotle's Poetics "Poetics (Aristotle)") as scenography: the use of flat panels on a stage to give the illusion of depth.[16] The philosophers Anaxagoras and Democritus developed geometric theories of perspective to be used in skenographia. Alcibiades had paintings in his house designed using this technique,[17] so this art was not simply limited to settings.
Plato was one of the first to discuss the problems of perspective:
In his Optics, Euclid introduced a mathematical theory of perspective, but there is some debate about the extent to which it matches the modern mathematical definition. In late ancient periods, artists, especially those from less popular traditions, knew that distant objects could be shown smaller than nearby ones to increase 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 remarkable realism and perspective for their time.[19] It has been claimed that complete systems of perspective were already developed in antiquity, but most scholars do not accept this. Almost none of the many works in which such a system was used have survived. A passage from Philostratus suggests that classical artists and theorists thought in terms of "circles" at equal distance from the viewer, like a classical semicircular theater seen from the stage.[20] In drawings from the Vergilius Vaticanus codex, circa 400 or so, the ceiling beams of the rooms are shown converging at a common vanishing point, but this is not systematically related to the rest of the composition.
Chinese artists used oblique projection from the 18th century to the 19th century. It is not certain how they came to use it; Some authorities suggest that the Chinese acquired this technique from India, which in turn acquired it from Ancient Rome. Oblique projection also appears in Japanese art, such as in the Ukiyo-e paintings of Torii Kiyonaga (1752-1815). In the 19th century, Chinese artists began to combine oblique perspective with the regular decrease in the size of people and objects with distance; no particular point of view is chosen, but a convincing effect is achieved.[21].
In the Late Antique period, the use of perspective techniques declined. The art of the new cultures of the period of the great migrations had no tradition of attempting compositions with large numbers of figures, and early medieval art was slow and inconsistent in relearning the convention of classical models, although the process can already be seen in Carolingian art.[23].
Several paintings during the Middle Ages show attempts at projections in furniture drawings, where parallel lines are successfully represented in isometric projection or by lines that are themselves parallel, but without a single vanishing point.
Medieval artists in Europe, like those in the Islamic world and China, knew the general principle of varying the relative size of elements according to distance, but they had compositional reasons for ignoring it, even more so than classical art. Buildings are often shown obliquely according to a particular convention. The use and sophistication of attempts to convey the sense of distance increased steadily during this period, but without being based on systematic theory. Byzantine art also knew these principles, but maintained the convention of inverted perspective to highlight the main figures.[24] Until the end of the Late Middle Ages, attempts to achieve a certain idea of perspective are found in knightly perspective,[25] where the furthest objects are placed at the top of the composition and the closest ones at the bottom.
Renaissance: mathematical basis
The artist who is considered the predecessor of the Italian Renaissance, the Gothic painter Giotto (1267-1336), was one of the first painters who began to provide three-dimensionality in a coherent but still intuitive way to his compositions.[26] Artists began to seek spatial sensation through the observation of nature. With the works of Fra Angelico (1390-1455) - such as in The Annunciation "The Annunciation (Fra Angelico, Florence)") - and especially with those of Masaccio[27] - in his Trinity "Trinidad (Masaccio)") (c. 1420-1425) -, the sensation of space is achieved through the methodical use of conical perspective, where the parallel lines of an object converge towards a certain vanishing point. The size of the figures reduces depending on the distance, which causes the optical illusion of depth.
Between the years 1416 and 1420, Filippo Brunelleschi (1377-1476), Florentine artist and architect of the Italian Renaissance, in order to represent buildings in perspective, carried out a series of studies with the help of optical instruments. With them, he discovered the geometric principles that govern conical perspective, a form of linear perspective based on the intersection of a plane with an imaginary visual cone "Cone (geometry)") whose vertex would be the eye of the observer. Objects appear smaller the further away they are. Furthermore, pictorially, they have fainter colors, have more diffuse contours and less contrast the further away they are.[26].
In 1434, Brunelleschi demonstrated the geometric method of perspective used by artists today. When painting the outlines of several buildings in Florence on a mirror, when he extended his main lines, he realized that they converged on the straight horizon. According to Giorgio Vasari, he introduced a demonstration of his technique on the incomplete door of the cathedral of Santa Maria del Fiore. He had the viewer look through a small hole in the back of a painting of the Baptistery "Baptistery of St. John (Florence)"), in front of the building itself. Then, he arranged a mirror, facing the viewer, which reflected his painting. To the viewer, the painting of the Baptistery and the building itself were almost indistinguishable.[28].
In the field of sculpture, the bronzes conceived by Lorenzo Ghiberti (1378-1455) for the north door of the baptistery of the Florence Cathedral "Baptistery of Saint John (Florence)") also show a complete mastery of the technique of perspective.[29].
Soon after, almost all artists in Florence and Italy used geometric perspective in their paintings,[30] notably Paolo Uccello, Masolino da Panicale and Donatello. Donatello himself began depicting chessboard-like tiled floors in an engraving about the birth of Christ. Although historically improbable, these pavings obeyed the primary laws of geometric perspective: the lines converged approximately to a vanishing point, and the speed at which the horizontal lines receded as a function of distance was determined graphically. This aspect became an integral part of Quattrocento art.[31].
Later manifestations
During the three centuries that followed the Renaissance, until approximately the end of the century, perspective continued to be a fundamental tool at the disposal of painters, although at different times voices emerged that criticized the mathematical rigor of compositions as a restriction on the expressive freedom of artists. The late-century Italian painter Federico Zuccaro accused this technique of taking away all its grace and spirit from art.[13].
Thus, after the Renaissance, during the Mannerist era, there is no longer an attempt to represent reality in a naturalistic way, it becomes more complicated, illusory perspectives are created with multiple vanishing points or taking the vanishing point outside the painting and the proportions are deliberately distorted in a disjointed and irrational space to achieve an emotional and artistic effect.[13] Shortly after, the Italian astronomer and mathematician Guidobaldo Del Monte (1545-1607), in his work Perspectivas Libri Sex (1600), devises a mathematical formulation of the conic projection more in line with its geometric properties.[6].
At the end of the century the technique of conical perspective reached China and Japan through the first Jesuit missions in Asia, producing a clash with local pictorial traditions, accustomed to respecting the parallelism of lines in their compositions.[43].
Already in the middle of the baroque stage, the form is defined above all by color, light and movement, with which the compositions become complicated, unusual perspectives are adopted and the volumes are distributed asymmetrically. Painters such as the Dutch Johannes Vermeer (1632-1675) or the Spanish Diego Velázquez (1599-1660) incorporated contrasts of luminosity to their paintings to give them their own atmosphere (an effect known as aerial perspective,[44] with which an attempt is made to represent the atmosphere, the air that surrounds the objects, degrading their color as they move away from the viewer, thus providing no just a feeling of depth).
From a theoretical point of view, the culmination of these Renaissance traditions finds its final synthesis in the research on perspective, optics and projective geometry by the French architect, geometer and optician of the 17th century Girard Desargues (1591-1661). It remained restricted to the circle of draftsmen and painters.[6] In 1715, the publication of the treatise on linear perspective by the British mathematician Brook Taylor (1685-1731),[46] allowed the teaching of perspective to artists to be based on the study of the mathematics underlying this technique.
Another painter who made regular use of perspective in his detailed urban views of Venice was the Italian Canaletto (1697-1768),[47] extending the geometric tradition of the previous period to the time of the Enlightenment.
Evolution of axonometric perspective
Orthogonal projections (without vanishing points) have a long history, especially if plan plans are included in this category, of which an example is known from Chaldea from more than 4000 years ago, which represents a temple corresponding to the time of King Gudea.[6].
Subsequently, the development of geometry in ancient Egypt was linked to the making of schematic drawings on papyri, a tradition that passed through classical Greece (where ceramics with axonometric reproductions of architectural elements are not uncommon) to Rome. In this sense, in the ten books of the Roman architect Vitruvius[56] he already writes about the need to make plans before building any work.
The Middle Ages in Europe was a period of stagnation with respect to the technical knowledge acquired by Roman builders and artists. However, a notable exception was the French master builder at the beginning of the century, Villard de Honnecourt, who in his Book of the Stonemason included perspective geometric schemes for fitting the ashlars.[43].
The rapid development of conical perspective in Renaissance Italy meant a certain marginalization of the axonometric system in art, which nevertheless retained an important role in military engineering and technical drawings, as demonstrated by the numerous machine plans contained in Leonardo da Vinci's codices.[43] Renaissance writers, dazzled by the pictorial achievements of conical perspective, barely dealt with the axonometric. Only Luca Pacioli made any reference to its usefulness in representing the Platonic solids in his work Divina Proportio of 1509.
During the century the first works on stereotomy of stone and wood appeared, but above all military architecture gained popularity, with numerous treatises on fortifications such as that of the Italian Francesco di Giorgio (1439-1502), who shared the geometric spirit of Leonardo's work.[43] Together with Jacopo Castriotto, Girolamo Maggi (c. 1523-1572) established the tradition of axonometric drawing in military treaties, explicitly contrasting it with conical perspectives.
The scientific codification of axonometry can be attributed to the Frenchman Desargues and his disciple Abraham Bosse (c. 1602–1676). As in the case of conical perspective, Gaspard Monge and Jean-Victor Poncelet laid the rigorous foundations of orthogonal projections, relating both.
In 1820, British chemist William Farish "William Farish (chemist)") (1759-1837) invented isometric perspective. The reference in the Encyclopedia Britannica of 1835 to this technique made it gain great popularity. Julius Ludwig Weisbach (1806-1871), Karl Wilhelm Pohlke (1810-1876) and Oskar Schlömilch (1823-1901) completed the axiomatic formulation of axonometry in the transition period between the and centuries. In Spain, the engineer Eduardo Torroja (1899-1961)[57] systematized the different types of axonometric perspective in a manual.[6].
Present: computer graphics
The origin of interactive computer graphics dates back to 1963, when Ivan Sutherland presented his doctoral thesis at MIT[58] on a computer system that allowed geometric elements to be handled graphically. This pioneering application would lead to the emergence of CAD, laying the theoretical and practical foundations of the first assisted design programs.[6].
Numerous video games and animated films with three-dimensional settings, as well as the vast majority of computer graphic design applications, use more or less simplified numerical versions to generate perspective images.[59].
Computer programs generally use numerical models in three-dimensional coordinates of the motifs to be represented, formed by surfaces composed of numerous triangular or polygonal scales,[60] normally provided with color and texture. Once the point of view and the drawing plane are located in the same three-dimensional coordinate system of the observed model (as if they were the eye of the observer and a window through which he was looking), the computer program calculates the intersections with the drawing plane of the triples of rays that connect each triangle of the model with the point of view of the scene. Each triple of cut points with the drawing plane generates a triangle projected on the aforementioned plane, which inherits the color and texture of the model's original triangle. According to the coordinates of the triangles projected on the drawing plane, the program code is responsible for giving the computer's graphics card[61] the necessary instructions to control the switching on of the monitor pixels that ultimately make up the generated image.[62].
The constant increase in the computing power of computer equipment and the successive improvements in the algorithms that calculate the geometry and modeling of the displayed objects (with behaviors as complex as those of own and cast shadows; shine and reflections; liquids; fire; transparent objects; the movements of living beings and the textures of hair and skin...) have allowed the creation of applications (especially video games) capable of generating perspective scenes of increasingly greater realism in real time. Given that the mathematical tools necessary to generate these images are available, it seems that it is a matter of time before the computing power required is available for computer-generated images to become practically indistinguishable to the naked eye from real images captured by video equipment or film cameras.[63].
Computer-aided design and most video games (especially applications that use 3-D polygons) use linear algebra, and in particular matrix multiplication, to perform the calculations necessary for generating perspective images. The basic calculations necessary are very simple: to know the coordinates of a view on the plane of the drawing, it is enough to determine the point of intersection between both, which is equivalent to the trivial resolution of a system of three linear equations with three unknowns. In reality, the mathematics[64] underlying the geometry of perspective is very simple, and the complexity of generating realistic images lies both in the large number of elementary objects that must be handled sequentially, and in the high calculation requirements of the sophisticated algorithms that model the behavior of light, influencing the modeling of the objects that make up a scene.
Types of perspective
De los muchos tipos de dibujos en perspectiva cónica, los más habituales son con un punto de fuga, con dos o con tres; característica que sirve para denominarlos, aunque conceptualmente son el mismo tipo de sistema de representación. Por su parte, las perspectivas axonométricas presentan una mayor variedad de tipologías:.
Los sistemas de perspectiva curvilíneos forman parte de las cónicas, dado que todas las visuales que sirven para definir el dibujo pasan por un único vértice común (el punto de vista), con la salvedad de que se utilizan superficies de proyección distintas del plano (como cilindros o esferas).
Perspective with a vanishing point
A perspective drawing of this type contains only one vanishing point on the horizon line. It is generally used to represent images of linear motifs, such as roads, railway tracks, hallways, or buildings viewed so that the front is directly in front of the observer. Any object that is composed of lines, either directly parallel to the viewer's line of sight or directly perpendicular (such as railroad ties) can be adequately represented with a vanishing point perspective, where lines converging away from the viewer.[68].
One-point perspective occurs when the plane of the drawing is parallel to two axes of a scene with rectilinear motifs, composed entirely of linear elements that intersect only at right angles. If an axis is parallel to the image plane, then all elements are parallel to the image plane (either horizontally or vertically) or perpendicular to it. All elements that are parallel to the image plane are drawn as parallel lines. All elements that are perpendicular to the image plane converge to a single point (a vanishing point) on the horizon.[2].
Perspective with two vanishing points
Perspective with two vanishing points, which can be placed arbitrarily on the horizon, is often used to draw the same objects as one-point perspective, but when they are rotated: for example, when looking toward the corner of a house, or in the view of two forking roads whose apparent width reduces with distance. One of the vanishing points represents a set of parallel straight lines "Parallelism (mathematics)"), and the second represents another. Seen from a corner, the horizontal edges of one of the walls of a house would converge towards a vanishing point, while those of the other wall would be directed towards the opposite vanishing point.[68].
Two vanishing point perspective occurs when the drawing plane is parallel to one coordinate axis (usually the vertical axis) but not the other two axes. If the scene being viewed consists solely of a cylinder with its base on a horizontal plane, there is no difference in the image of the cylinder between a one-point and two-point perspective.
It has a set of lines parallel to the image plane and two sets oblique to it. Each family of lines parallel oblique to the image plane converges at its own vanishing point, meaning that this configuration will require two vanishing points.[2].
Perspective with three vanishing points
Three-point perspective is often used to depict buildings seen from above or below. In addition to the two vanishing points already described, one for each family of walls, in this case a third vanishing point is located on which the vertical lines of the walls converge. For an object seen from above, this third vanishing point is below the ground. For an object seen from below, such as when the viewer looks up at a tall building, the third vanishing point is located at the zenith.[68].
Three vanishing point perspective occurs when drawing a motif with orthogonal faces, when the image plane is not parallel to any of the three axes of the scene, each corresponding to one of the three vanishing points of the image.
Perspectives with one, two and three vanishing points seem to incorporate different forms of drawing calculation, and could be thought of as being generated by different methods. Mathematically, however, all three are identical; The difference lies merely in the relative orientation of the orthogonal faces of the rectilinear scene with respect to the viewer and the plane of the drawing.[2].
Foreshortening
Foreshortening is the visual effect or optical illusion that makes an object or distance appear shorter than it really is because it is turned toward the viewer. Additionally, objects in images are generally not scaled uniformly: a circle often appears as an ellipse[69] and a square may appear as a trapezoid.
Although foreshortening is an important element in art where visual perspective is represented, it also occurs in other types of two-dimensional representations of three-dimensional scenes. Some other types in which foreshortening can occur include drawings in oblique parallel projection.[70].
In painting, foreshortening in the representation of the human figure was perfected in the Italian Renaissance, and the famous painting of Andrea Mantegna's "Lamentation over the Dead Christ (Mantegna)") (1480) is one of the best-known realizations in a series of works displaying the new technique, which later became a standard part of the training of artists.[32].
Perspective with numerous vanishing points
One-point, two-point, and three-point perspectives depend on the structure of the scene being observed. They only exist for strictly Cartesian scenarios (with three generally orthogonal rectilinear axes). By inserting into a Cartesian scene a set of mutually parallel lines that are not parallel to any of the three principal axes, a new distinct vanishing point is created. Therefore, it is possible to have a perspective with infinite vanishing points if the scene being viewed does not conform to a system of Cartesian axes, but instead consists of infinite pairs of parallel lines, where each pair of lines is not parallel to any other pair.[2].
Curvilinear perspective
Curvilinear perspective,[71] also called infinite point perspective or four-point perspective, is the curvilinear variant of two-point perspective. An image in curvilinear perspective can represent a panoramic[72] of 360° and even beyond 360° to design impossible scenes. It can be used with either a horizontal or vertical horizon line. In this latter configuration you can render both a worm view and an aerial view of a scene at the same time.
The usual method of generating curvilinear perspectives is to project the model onto a curved theoretical surface, rather than onto a plane (although the result is eventually drawn onto a plane). Thus, we speak of a four vanishing point perspective when a cylinder is used that surrounds the observer (the four points are located in front, behind and on both sides, covering 360°); When half a sphere is used, there are five points (up, down, left, right and in front); and with a complete sphere, we speak of six points (a vanishing point located behind is added).[68].
Like all other foreshortened variants of perspective (one-point to six-point perspectives),[68] it begins with a horizon line, followed by four equally spaced vanishing points to delineate four vertical lines. The vanishing points created to generate the curvilinear orthogonals are freely located on the four vertical lines placed on the opposite side of the horizon line. The only dimension not foreshortened in this type of perspective is that of the straight lines parallel to each other, perpendicular to the horizon line, similar to the vertical lines used in two-point perspective.[68].
Perspective without vanishing points
A perspective without vanishing points ("zero vanishing point" perspective) occurs when the viewer is viewing a nonlinear scene, and therefore does not contain parallel lines.[73] The most common example of such a nonlinear view is a natural scene (e.g., a mountain range) that often does not contain any parallel lines. This should not be confused with views of a dihedral system, since a view without explicit vanishing points could have been drawn in such a way that there would have been vanishing points if there had been parallel lines, and thus enjoy the sensation of depth as in any perspective projection.[11].
On the other hand, a parallel projection, such as a dihedral projection, can be approximated to a perspective when the object in question is observed from very far away, because the projection lines tend to become parallel as the point of view approaches infinity. This may explain the confusion of perspectives without vanishing points with orthogonal projections, since natural scenes are often viewed from very far away, and the size of objects within the scene is often insignificant compared to their distance from the viewpoint. The appearance of any small object in said scene would thus resemble its appearance in a parallel projection.[74].
Axonometric perspectives
This type of projections are characterized because the visuals are parallel to each other, which in theory is equivalent to the point of view of the projection being located at infinity. In practice, they have the advantage that they allow the dimensions of the represented model to be measured directly on the three coordinate axes.
They are classified into two main types:[75].
Construction methods
There are several methods for generating insights, including:.
Limitations
Perspective images are calculated assuming a certain relationship between the point of view and the plane where the image is projected, which in turn have a certain position with respect to the model to be drawn. For the resulting image to appear identical to the original scene, a perspective viewer must view the image from the exact viewpoint used in the calculations relating to the image. This causes what would appear as distortions in the image when viewed from a different point. In practice, unless the user chooses an extreme angle, such as looking at the image from the bottom corner of the window, the perspective usually looks more or less correct. This effect is known as Zeeman's paradox.[76].
It has been suggested that a perspective drawing still appears to be in perspective when viewed off-center because it is still perceived as a drawing, lacking the depth of field that the original model does possess.[77].
However, for a typical perspective, the field of view is narrow enough (often only 60 degrees) that the distortions are small enough that the image can be viewed from a point other than the actual calculated viewpoint without appearing significantly distorted. When a larger viewing angle is required, the standard method of projecting rays onto a flat surface is not practical. As a theoretical maximum, the field of view of a planar image should be less than 180 degrees, because as the field of view increases toward 180 degrees, the required width of the image plane approaches infinity.
To create a projected ray image with a large field of view, the image can be projected onto a curved surface. To have a large horizontal field of view in the image, a surface that is a vertical cylinder (i.e., the axis of the cylinder is parallel to the z-axis) will be sufficient (similarly, if the wide field of view desired is only in the vertical direction of the image, a horizontal cylinder will be sufficient). A cylindrical imaging surface will allow a projected image of rays up to a full 360 degrees in the horizontal or vertical dimension of the perspective image (depending on the orientation of the cylinder). Likewise, when using a spherical imaging surface, the field of view can be a full 360 degrees in any direction (note that for a spherical surface, all rays projected from the scene to the eye intersect the surface at right angles).[68].
Just as a standard perspective image must be viewed from the calculated viewpoint for the image to appear identical to the real scene, an image projected onto a cylinder or sphere must be viewed from the calculated viewpoint for it to be exactly identical to the original scene. If an image projected on a cylindrical surface is "unrolled" into a flat image, different types of distortions occur. For example, many of the straight lines in the scene will be drawn as curves. An image projected on a spherical surface can be flattened in several ways:[68].
References
[1] ↑ Lajo Pérez, Rosina (1990). Léxico de arte. Madrid - España: Akal. p. 162. ISBN 978-84-460-0924-5. |fechaacceso= requiere |url= (ayuda).
[6] ↑ a b c d e f g h i j José Ignacio Rojas-Sola; Ana Serrano-Tierz; David Hernández-Díaz (15 de abril de 2011). «UNA REVISIÓN HISTÓRICA: DESDE EL DIBUJO EN INGENIERÍA HACIA LA INGENIERÍA DEL DISEÑO». DYNA. Consultado el 24 de julio de 2018.: https://revistas.unal.edu.co/index.php/dyna/article/view/25758/39341
[13] ↑ a b c Antonio Nevot Luna, Julián Aguirre Estibález, Roberto Rodríguez del Río. Dibujo técnico y matemáticas: una consideración interdisciplinar Aulas de Verano. Serie: Ciencias. Ministerio de Educación, 2007. pp. 105 de 368. ISBN 9788436945416. Consultado el 23 de julio de 2018.: https://books.google.es/books?id=cLYNoqJP3F8C&pg=PA105#v=onepage&q&f=false
[21] ↑ a b c Cucker, Felix (2013). Manifold Mirrors: The Crossing Paths of the Arts and Mathematics. Cambridge University Press. pp. 269–278. ISBN 978-0-521-72876-8. «Dubery and Willats (1983:33) escribieron que La proyección oblicua parece haber llegado a China desde Roma a través de la India alrededor del primer o segundo siglos.». [Wen-Chi regresa a casa, anon, China, siglo XII] es un arquetipo del clásico uso de la perspectiva oblicua en la pintura china.: https://archive.org/details/manifoldmirrorsc0000cuck
[22] ↑ Claramente, hay un intento general de reducir el tamaño de los elementos más distantes, pero de forma no sistemática. Las secciones de la composición están a una escala similar, con la distancia relativa mostrada por superposición, y otros objetos son más altos que los más cercanos, aunque los trabajadores de la izquierda muestran un ajuste de tamaño más fino. Pero este criterio es abandonado a la derecha, donde la figura más importante es mucho más grande que el albañil. Los edificios rectangulares y los bloques de piedra se muestran oblicuamente.
[30] ↑ "...y estas obras (las perspectivas de Brunelleschi) fueron el medio para despertar las mentes de los otros artesanos, quienes luego se dedicaron a esta técnica con gran celo."Vasari, Vida de los Artistas, Capítulo sobre Brunelleschi.
[31] ↑ (Martin Kemp, 2000, p. 24).
[32] ↑ a b (Martin Kemp, 2000, p. 52).
[33] ↑ "Maese Paolo dal Pozzo Toscanelli, habiendo regresado de sus estudios, invitó a Filippo con otros amigos a cenar en el jardín, y la conversación recaló en asuntos matemáticos, Filippo trabó amistad con él, y de él aprendió geometría."Vasari, Vidas de los Artistas, Capítulo sobre Brunelleschi.
[35] ↑ El-Bizri, Nader (2010). «Classical Optics and the Perspectiva Traditions Leading to the Renaissance». En Hendrix, John Shannon; Carman, Charles H., eds. Renaissance Theories of Vision (Visual Culture in Early Modernity). Farnham, Surrey: Ashgate. pp. 11–30. ISBN 1-409400-24-7.: https://archive.org/details/renaissancetheor00hend
[36] ↑ Hans, Belting (2011). Florence and Baghdad: Renaissance art and Arab science (1st English edición). Cambridge, Massachusetts: Belknap Press of Harvard University Press. pp. 90–92. ISBN 9780674050044. OCLC 701493612.: https://archive.org/details/florencebaghdadr0000belt
[39] ↑ O'Connor, J. J.; Robertson, E. F. (July 1999). «Luca Pacioli». Universidad de Saint Andrews. Archivado desde el original el 22 de septiembre de 2015. Consultado el 23 de septiembre de 2015.: http://www-history.mcs.st-and.ac.uk/Biographies/Pacioli.html
[56] ↑ VITRUBIO POLLIONE, M.L. Compendio de los diez libros de arquitectura de Vitrubio, Traducción de Joseph Castañeda, Imprenta de D. Gabriel Ramírez, Madrid, 1761.
[57] ↑ Texto de Torroja: «Axonometría o perspectiva axonométrica. Sistema general de representación que comprende, como casos particulares, las perspectivas caballera y militar, la proyección isográfica y otros varios» (1897).
[58] ↑ Sketchpad: A Man-Machine Communication System.
[73] ↑ Basant Agrawal (2008). Engineering Drawing. Tata McGraw-Hill Education. pp. 17.3.4. ISBN 978-0-07-066863-8. Archivado desde el original el 23 de diciembre de 2017.: https://books.google.com/books?id=_hdOU8kRb2AC
[76] ↑ Mathographics by Robert Dixon New York: Dover, p. 82, 1991.
[77] ↑ "...la paradoja es puramente conceptual: supone que vemos una representación en perspectiva como una simulación retinal, cuando de hecho la vemos como una pintura bidimensional. En otras palabras, las construcciones de perspectiva crean símbolos visuales, no ilusiones visuales. La clave es que las pinturas carecen de la profundidad de las señales de campo creadas por la visión binocular; siempre somos conscientes de que una pintura es plana en lugar de profunda. Y así es como nuestra mente lo interpreta, ajustando nuestra comprensión de la pintura para compensar nuestra posición."«Archived copy». Archivado desde el original el 6 de enero de 2007. Consultado el 25 de diciembre de 2006. Retrieved on 25 December 2006.: http://www.handprint.com/HP/WCL/perspect1.html
All perspective drawings assume that the viewer is at some distance from the drawing. Objects are scaled relative to that viewing point. An object often does not scale uniformly: a circle frequently appears as an ellipse and a square may appear as a trapezoid. This distortion is known as foreshortening.[9].
Perspective drawings have a horizon line, which is almost always implied. This line, directly opposite the viewer's eye, represents infinitely distant objects, which have been reduced in distance to the infinitesimal thickness of a line. It is analogous to (and named after) the Earth's horizon.[10].
Any perspective representation of a scene that includes parallel lines has one or more vanishing points. A single-point perspective drawing means that the drawing has a single vanishing point, usually (but not necessarily) directly opposite the observer's eye and usually (but not necessarily) on the horizon line. All lines parallel to the viewer's line of sight recede toward the horizon, directed toward the vanishing point. This is the standard railway convergence phenomenon. A drawing with two points of convergence 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 a certain angle to the plane of the drawing.[2].
Perspectives containing many parallel lines are often seen when drawing works of architecture (where lines parallel to the x, y and z axes are frequently used). Because it is rare to have a scene consisting solely of lines parallel to the three Cartesian axes, in practice it is not common to see perspectives with only one, two, or three vanishing points; Even a simple building often has a pointed roof that results in a minimum of six sets of parallel lines, which in turn correspond to up to six vanishing points.[2].
In contrast, nature scenes often do not possess any set of parallel lines and, therefore, have no vanishing points.[11].
On the other hand, orthogonal axonometric perspectives can be considered special cases of conical perspective, in which the views that serve to outline the drawing are parallel to each other (which is equivalent to placing the point of view infinitely far from the model). This results in the absence of vanishing points in the drawing. On the other hand, oblique axonometries do not correspond to any type of optical perspective model, being a mathematical graphic device that allows us to simplify the creation of certain representations of three-dimensional bodies.
Historical development
Contenido
El intento de dar sensación de volumen a las representaciones pictóricas se encuentra más o menos presente desde las primeras muestras del arte paleolítico, como se puede apreciar en las pinturas rupestres de la cueva de Altamira (con unos 35.000 años de antigüedad), en las que se aprovecha el relieve de las rocas de la pared de la cueva para dotar de profundidad a los dibujos.[12] Esta tendencia alcanzó altas cotas de perfección técnica durante la época romana (seguramente basándose en procedimientos empíricos o intuitivos), pero no fue hasta el siglo , durante el Renacimiento italiano, cuando se sentaron las bases geométricas que permitieron convertir el dibujo en perspectiva en una técnica con sólidos fundamentos teóricos. Desde entonces, ha pasado a generalizarse su uso, convirtiéndose en una útil herramienta primero para los pintores, después para los arquitectos y más adelante para los ingenieros, hasta llegar al desarrollo en el último cuarto del siglo de las aplicaciones por ordenador que permiten automatizar la generación de este tipo de vistas, que hasta entonces podían requerir de una laboriosa construcción gráfica.[13].
Background
Early artistic paintings and drawings generally classified many objects and characters hierarchically according to their spiritual or thematic importance, but not according to their distance from the viewer, and did not use foreshortening. The most important figures are often shown as the tallest in a composition, especially of hieratic motifs, leading to the so-called "vertical perspective", common in Ancient Egyptian art, where a group of "closer" figures are shown below the largest figure or figures. In Egyptian paintings, a two-dimensional space of the surface to be painted was conceived, without strictly suggesting an idea of spatial conception. They arranged the characters increasing their size according to their importance, what art historians call hierarchical or theological perspective.[14].
The only method of indicating the relative position of elements in the composition was superposition, which is widely used in works such as the Elgin Marbles, the famous sculptures that decorated the Parthenon in Athens. However, there are numerous studies about the Parthenon itself, which claim that its dimensions (especially the shape and inclination of its columns) were meticulously studied to counteract the effects of perspective on the main lines of the building.[15].
Antiquity and Middle Ages
The first attempts to develop a perspective system are considered to have begun around the century BC. C. in the art of Ancient Greece, as part of the interest in producing the optical illusion of depth in theatrical settings. This fact is described in Aristotle's Poetics "Poetics (Aristotle)") as scenography: the use of flat panels on a stage to give the illusion of depth.[16] The philosophers Anaxagoras and Democritus developed geometric theories of perspective to be used in skenographia. Alcibiades had paintings in his house designed using this technique,[17] so this art was not simply limited to settings.
Plato was one of the first to discuss the problems of perspective:
In his Optics, Euclid introduced a mathematical theory of perspective, but there is some debate about the extent to which it matches the modern mathematical definition. In late ancient periods, artists, especially those from less popular traditions, knew that distant objects could be shown smaller than nearby ones to increase 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 remarkable realism and perspective for their time.[19] It has been claimed that complete systems of perspective were already developed in antiquity, but most scholars do not accept this. Almost none of the many works in which such a system was used have survived. A passage from Philostratus suggests that classical artists and theorists thought in terms of "circles" at equal distance from the viewer, like a classical semicircular theater seen from the stage.[20] In drawings from the Vergilius Vaticanus codex, circa 400 or so, the ceiling beams of the rooms are shown converging at a common vanishing point, but this is not systematically related to the rest of the composition.
Chinese artists used oblique projection from the 18th century to the 19th century. It is not certain how they came to use it; Some authorities suggest that the Chinese acquired this technique from India, which in turn acquired it from Ancient Rome. Oblique projection also appears in Japanese art, such as in the Ukiyo-e paintings of Torii Kiyonaga (1752-1815). In the 19th century, Chinese artists began to combine oblique perspective with the regular decrease in the size of people and objects with distance; no particular point of view is chosen, but a convincing effect is achieved.[21].
In the Late Antique period, the use of perspective techniques declined. The art of the new cultures of the period of the great migrations had no tradition of attempting compositions with large numbers of figures, and early medieval art was slow and inconsistent in relearning the convention of classical models, although the process can already be seen in Carolingian art.[23].
Several paintings during the Middle Ages show attempts at projections in furniture drawings, where parallel lines are successfully represented in isometric projection or by lines that are themselves parallel, but without a single vanishing point.
Medieval artists in Europe, like those in the Islamic world and China, knew the general principle of varying the relative size of elements according to distance, but they had compositional reasons for ignoring it, even more so than classical art. Buildings are often shown obliquely according to a particular convention. The use and sophistication of attempts to convey the sense of distance increased steadily during this period, but without being based on systematic theory. Byzantine art also knew these principles, but maintained the convention of inverted perspective to highlight the main figures.[24] Until the end of the Late Middle Ages, attempts to achieve a certain idea of perspective are found in knightly perspective,[25] where the furthest objects are placed at the top of the composition and the closest ones at the bottom.
Renaissance: mathematical basis
The artist who is considered the predecessor of the Italian Renaissance, the Gothic painter Giotto (1267-1336), was one of the first painters who began to provide three-dimensionality in a coherent but still intuitive way to his compositions.[26] Artists began to seek spatial sensation through the observation of nature. With the works of Fra Angelico (1390-1455) - such as in The Annunciation "The Annunciation (Fra Angelico, Florence)") - and especially with those of Masaccio[27] - in his Trinity "Trinidad (Masaccio)") (c. 1420-1425) -, the sensation of space is achieved through the methodical use of conical perspective, where the parallel lines of an object converge towards a certain vanishing point. The size of the figures reduces depending on the distance, which causes the optical illusion of depth.
Between the years 1416 and 1420, Filippo Brunelleschi (1377-1476), Florentine artist and architect of the Italian Renaissance, in order to represent buildings in perspective, carried out a series of studies with the help of optical instruments. With them, he discovered the geometric principles that govern conical perspective, a form of linear perspective based on the intersection of a plane with an imaginary visual cone "Cone (geometry)") whose vertex would be the eye of the observer. Objects appear smaller the further away they are. Furthermore, pictorially, they have fainter colors, have more diffuse contours and less contrast the further away they are.[26].
In 1434, Brunelleschi demonstrated the geometric method of perspective used by artists today. When painting the outlines of several buildings in Florence on a mirror, when he extended his main lines, he realized that they converged on the straight horizon. According to Giorgio Vasari, he introduced a demonstration of his technique on the incomplete door of the cathedral of Santa Maria del Fiore. He had the viewer look through a small hole in the back of a painting of the Baptistery "Baptistery of St. John (Florence)"), in front of the building itself. Then, he arranged a mirror, facing the viewer, which reflected his painting. To the viewer, the painting of the Baptistery and the building itself were almost indistinguishable.[28].
In the field of sculpture, the bronzes conceived by Lorenzo Ghiberti (1378-1455) for the north door of the baptistery of the Florence Cathedral "Baptistery of Saint John (Florence)") also show a complete mastery of the technique of perspective.[29].
Soon after, almost all artists in Florence and Italy used geometric perspective in their paintings,[30] notably Paolo Uccello, Masolino da Panicale and Donatello. Donatello himself began depicting chessboard-like tiled floors in an engraving about the birth of Christ. Although historically improbable, these pavings obeyed the primary laws of geometric perspective: the lines converged approximately to a vanishing point, and the speed at which the horizontal lines receded as a function of distance was determined graphically. This aspect became an integral part of Quattrocento art.[31].
Later manifestations
During the three centuries that followed the Renaissance, until approximately the end of the century, perspective continued to be a fundamental tool at the disposal of painters, although at different times voices emerged that criticized the mathematical rigor of compositions as a restriction on the expressive freedom of artists. The late-century Italian painter Federico Zuccaro accused this technique of taking away all its grace and spirit from art.[13].
Thus, after the Renaissance, during the Mannerist era, there is no longer an attempt to represent reality in a naturalistic way, it becomes more complicated, illusory perspectives are created with multiple vanishing points or taking the vanishing point outside the painting and the proportions are deliberately distorted in a disjointed and irrational space to achieve an emotional and artistic effect.[13] Shortly after, the Italian astronomer and mathematician Guidobaldo Del Monte (1545-1607), in his work Perspectivas Libri Sex (1600), devises a mathematical formulation of the conic projection more in line with its geometric properties.[6].
At the end of the century the technique of conical perspective reached China and Japan through the first Jesuit missions in Asia, producing a clash with local pictorial traditions, accustomed to respecting the parallelism of lines in their compositions.[43].
Already in the middle of the baroque stage, the form is defined above all by color, light and movement, with which the compositions become complicated, unusual perspectives are adopted and the volumes are distributed asymmetrically. Painters such as the Dutch Johannes Vermeer (1632-1675) or the Spanish Diego Velázquez (1599-1660) incorporated contrasts of luminosity to their paintings to give them their own atmosphere (an effect known as aerial perspective,[44] with which an attempt is made to represent the atmosphere, the air that surrounds the objects, degrading their color as they move away from the viewer, thus providing no just a feeling of depth).
From a theoretical point of view, the culmination of these Renaissance traditions finds its final synthesis in the research on perspective, optics and projective geometry by the French architect, geometer and optician of the 17th century Girard Desargues (1591-1661). It remained restricted to the circle of draftsmen and painters.[6] In 1715, the publication of the treatise on linear perspective by the British mathematician Brook Taylor (1685-1731),[46] allowed the teaching of perspective to artists to be based on the study of the mathematics underlying this technique.
Another painter who made regular use of perspective in his detailed urban views of Venice was the Italian Canaletto (1697-1768),[47] extending the geometric tradition of the previous period to the time of the Enlightenment.
Evolution of axonometric perspective
Orthogonal projections (without vanishing points) have a long history, especially if plan plans are included in this category, of which an example is known from Chaldea from more than 4000 years ago, which represents a temple corresponding to the time of King Gudea.[6].
Subsequently, the development of geometry in ancient Egypt was linked to the making of schematic drawings on papyri, a tradition that passed through classical Greece (where ceramics with axonometric reproductions of architectural elements are not uncommon) to Rome. In this sense, in the ten books of the Roman architect Vitruvius[56] he already writes about the need to make plans before building any work.
The Middle Ages in Europe was a period of stagnation with respect to the technical knowledge acquired by Roman builders and artists. However, a notable exception was the French master builder at the beginning of the century, Villard de Honnecourt, who in his Book of the Stonemason included perspective geometric schemes for fitting the ashlars.[43].
The rapid development of conical perspective in Renaissance Italy meant a certain marginalization of the axonometric system in art, which nevertheless retained an important role in military engineering and technical drawings, as demonstrated by the numerous machine plans contained in Leonardo da Vinci's codices.[43] Renaissance writers, dazzled by the pictorial achievements of conical perspective, barely dealt with the axonometric. Only Luca Pacioli made any reference to its usefulness in representing the Platonic solids in his work Divina Proportio of 1509.
During the century the first works on stereotomy of stone and wood appeared, but above all military architecture gained popularity, with numerous treatises on fortifications such as that of the Italian Francesco di Giorgio (1439-1502), who shared the geometric spirit of Leonardo's work.[43] Together with Jacopo Castriotto, Girolamo Maggi (c. 1523-1572) established the tradition of axonometric drawing in military treaties, explicitly contrasting it with conical perspectives.
The scientific codification of axonometry can be attributed to the Frenchman Desargues and his disciple Abraham Bosse (c. 1602–1676). As in the case of conical perspective, Gaspard Monge and Jean-Victor Poncelet laid the rigorous foundations of orthogonal projections, relating both.
In 1820, British chemist William Farish "William Farish (chemist)") (1759-1837) invented isometric perspective. The reference in the Encyclopedia Britannica of 1835 to this technique made it gain great popularity. Julius Ludwig Weisbach (1806-1871), Karl Wilhelm Pohlke (1810-1876) and Oskar Schlömilch (1823-1901) completed the axiomatic formulation of axonometry in the transition period between the and centuries. In Spain, the engineer Eduardo Torroja (1899-1961)[57] systematized the different types of axonometric perspective in a manual.[6].
Present: computer graphics
The origin of interactive computer graphics dates back to 1963, when Ivan Sutherland presented his doctoral thesis at MIT[58] on a computer system that allowed geometric elements to be handled graphically. This pioneering application would lead to the emergence of CAD, laying the theoretical and practical foundations of the first assisted design programs.[6].
Numerous video games and animated films with three-dimensional settings, as well as the vast majority of computer graphic design applications, use more or less simplified numerical versions to generate perspective images.[59].
Computer programs generally use numerical models in three-dimensional coordinates of the motifs to be represented, formed by surfaces composed of numerous triangular or polygonal scales,[60] normally provided with color and texture. Once the point of view and the drawing plane are located in the same three-dimensional coordinate system of the observed model (as if they were the eye of the observer and a window through which he was looking), the computer program calculates the intersections with the drawing plane of the triples of rays that connect each triangle of the model with the point of view of the scene. Each triple of cut points with the drawing plane generates a triangle projected on the aforementioned plane, which inherits the color and texture of the model's original triangle. According to the coordinates of the triangles projected on the drawing plane, the program code is responsible for giving the computer's graphics card[61] the necessary instructions to control the switching on of the monitor pixels that ultimately make up the generated image.[62].
The constant increase in the computing power of computer equipment and the successive improvements in the algorithms that calculate the geometry and modeling of the displayed objects (with behaviors as complex as those of own and cast shadows; shine and reflections; liquids; fire; transparent objects; the movements of living beings and the textures of hair and skin...) have allowed the creation of applications (especially video games) capable of generating perspective scenes of increasingly greater realism in real time. Given that the mathematical tools necessary to generate these images are available, it seems that it is a matter of time before the computing power required is available for computer-generated images to become practically indistinguishable to the naked eye from real images captured by video equipment or film cameras.[63].
Computer-aided design and most video games (especially applications that use 3-D polygons) use linear algebra, and in particular matrix multiplication, to perform the calculations necessary for generating perspective images. The basic calculations necessary are very simple: to know the coordinates of a view on the plane of the drawing, it is enough to determine the point of intersection between both, which is equivalent to the trivial resolution of a system of three linear equations with three unknowns. In reality, the mathematics[64] underlying the geometry of perspective is very simple, and the complexity of generating realistic images lies both in the large number of elementary objects that must be handled sequentially, and in the high calculation requirements of the sophisticated algorithms that model the behavior of light, influencing the modeling of the objects that make up a scene.
Types of perspective
De los muchos tipos de dibujos en perspectiva cónica, los más habituales son con un punto de fuga, con dos o con tres; característica que sirve para denominarlos, aunque conceptualmente son el mismo tipo de sistema de representación. Por su parte, las perspectivas axonométricas presentan una mayor variedad de tipologías:.
Los sistemas de perspectiva curvilíneos forman parte de las cónicas, dado que todas las visuales que sirven para definir el dibujo pasan por un único vértice común (el punto de vista), con la salvedad de que se utilizan superficies de proyección distintas del plano (como cilindros o esferas).
Perspective with a vanishing point
A perspective drawing of this type contains only one vanishing point on the horizon line. It is generally used to represent images of linear motifs, such as roads, railway tracks, hallways, or buildings viewed so that the front is directly in front of the observer. Any object that is composed of lines, either directly parallel to the viewer's line of sight or directly perpendicular (such as railroad ties) can be adequately represented with a vanishing point perspective, where lines converging away from the viewer.[68].
One-point perspective occurs when the plane of the drawing is parallel to two axes of a scene with rectilinear motifs, composed entirely of linear elements that intersect only at right angles. If an axis is parallel to the image plane, then all elements are parallel to the image plane (either horizontally or vertically) or perpendicular to it. All elements that are parallel to the image plane are drawn as parallel lines. All elements that are perpendicular to the image plane converge to a single point (a vanishing point) on the horizon.[2].
Perspective with two vanishing points
Perspective with two vanishing points, which can be placed arbitrarily on the horizon, is often used to draw the same objects as one-point perspective, but when they are rotated: for example, when looking toward the corner of a house, or in the view of two forking roads whose apparent width reduces with distance. One of the vanishing points represents a set of parallel straight lines "Parallelism (mathematics)"), and the second represents another. Seen from a corner, the horizontal edges of one of the walls of a house would converge towards a vanishing point, while those of the other wall would be directed towards the opposite vanishing point.[68].
Two vanishing point perspective occurs when the drawing plane is parallel to one coordinate axis (usually the vertical axis) but not the other two axes. If the scene being viewed consists solely of a cylinder with its base on a horizontal plane, there is no difference in the image of the cylinder between a one-point and two-point perspective.
It has a set of lines parallel to the image plane and two sets oblique to it. Each family of lines parallel oblique to the image plane converges at its own vanishing point, meaning that this configuration will require two vanishing points.[2].
Perspective with three vanishing points
Three-point perspective is often used to depict buildings seen from above or below. In addition to the two vanishing points already described, one for each family of walls, in this case a third vanishing point is located on which the vertical lines of the walls converge. For an object seen from above, this third vanishing point is below the ground. For an object seen from below, such as when the viewer looks up at a tall building, the third vanishing point is located at the zenith.[68].
Three vanishing point perspective occurs when drawing a motif with orthogonal faces, when the image plane is not parallel to any of the three axes of the scene, each corresponding to one of the three vanishing points of the image.
Perspectives with one, two and three vanishing points seem to incorporate different forms of drawing calculation, and could be thought of as being generated by different methods. Mathematically, however, all three are identical; The difference lies merely in the relative orientation of the orthogonal faces of the rectilinear scene with respect to the viewer and the plane of the drawing.[2].
Foreshortening
Foreshortening is the visual effect or optical illusion that makes an object or distance appear shorter than it really is because it is turned toward the viewer. Additionally, objects in images are generally not scaled uniformly: a circle often appears as an ellipse[69] and a square may appear as a trapezoid.
Although foreshortening is an important element in art where visual perspective is represented, it also occurs in other types of two-dimensional representations of three-dimensional scenes. Some other types in which foreshortening can occur include drawings in oblique parallel projection.[70].
In painting, foreshortening in the representation of the human figure was perfected in the Italian Renaissance, and the famous painting of Andrea Mantegna's "Lamentation over the Dead Christ (Mantegna)") (1480) is one of the best-known realizations in a series of works displaying the new technique, which later became a standard part of the training of artists.[32].
Perspective with numerous vanishing points
One-point, two-point, and three-point perspectives depend on the structure of the scene being observed. They only exist for strictly Cartesian scenarios (with three generally orthogonal rectilinear axes). By inserting into a Cartesian scene a set of mutually parallel lines that are not parallel to any of the three principal axes, a new distinct vanishing point is created. Therefore, it is possible to have a perspective with infinite vanishing points if the scene being viewed does not conform to a system of Cartesian axes, but instead consists of infinite pairs of parallel lines, where each pair of lines is not parallel to any other pair.[2].
Curvilinear perspective
Curvilinear perspective,[71] also called infinite point perspective or four-point perspective, is the curvilinear variant of two-point perspective. An image in curvilinear perspective can represent a panoramic[72] of 360° and even beyond 360° to design impossible scenes. It can be used with either a horizontal or vertical horizon line. In this latter configuration you can render both a worm view and an aerial view of a scene at the same time.
The usual method of generating curvilinear perspectives is to project the model onto a curved theoretical surface, rather than onto a plane (although the result is eventually drawn onto a plane). Thus, we speak of a four vanishing point perspective when a cylinder is used that surrounds the observer (the four points are located in front, behind and on both sides, covering 360°); When half a sphere is used, there are five points (up, down, left, right and in front); and with a complete sphere, we speak of six points (a vanishing point located behind is added).[68].
Like all other foreshortened variants of perspective (one-point to six-point perspectives),[68] it begins with a horizon line, followed by four equally spaced vanishing points to delineate four vertical lines. The vanishing points created to generate the curvilinear orthogonals are freely located on the four vertical lines placed on the opposite side of the horizon line. The only dimension not foreshortened in this type of perspective is that of the straight lines parallel to each other, perpendicular to the horizon line, similar to the vertical lines used in two-point perspective.[68].
Perspective without vanishing points
A perspective without vanishing points ("zero vanishing point" perspective) occurs when the viewer is viewing a nonlinear scene, and therefore does not contain parallel lines.[73] The most common example of such a nonlinear view is a natural scene (e.g., a mountain range) that often does not contain any parallel lines. This should not be confused with views of a dihedral system, since a view without explicit vanishing points could have been drawn in such a way that there would have been vanishing points if there had been parallel lines, and thus enjoy the sensation of depth as in any perspective projection.[11].
On the other hand, a parallel projection, such as a dihedral projection, can be approximated to a perspective when the object in question is observed from very far away, because the projection lines tend to become parallel as the point of view approaches infinity. This may explain the confusion of perspectives without vanishing points with orthogonal projections, since natural scenes are often viewed from very far away, and the size of objects within the scene is often insignificant compared to their distance from the viewpoint. The appearance of any small object in said scene would thus resemble its appearance in a parallel projection.[74].
Axonometric perspectives
This type of projections are characterized because the visuals are parallel to each other, which in theory is equivalent to the point of view of the projection being located at infinity. In practice, they have the advantage that they allow the dimensions of the represented model to be measured directly on the three coordinate axes.
They are classified into two main types:[75].
Construction methods
There are several methods for generating insights, including:.
Limitations
Perspective images are calculated assuming a certain relationship between the point of view and the plane where the image is projected, which in turn have a certain position with respect to the model to be drawn. For the resulting image to appear identical to the original scene, a perspective viewer must view the image from the exact viewpoint used in the calculations relating to the image. This causes what would appear as distortions in the image when viewed from a different point. In practice, unless the user chooses an extreme angle, such as looking at the image from the bottom corner of the window, the perspective usually looks more or less correct. This effect is known as Zeeman's paradox.[76].
It has been suggested that a perspective drawing still appears to be in perspective when viewed off-center because it is still perceived as a drawing, lacking the depth of field that the original model does possess.[77].
However, for a typical perspective, the field of view is narrow enough (often only 60 degrees) that the distortions are small enough that the image can be viewed from a point other than the actual calculated viewpoint without appearing significantly distorted. When a larger viewing angle is required, the standard method of projecting rays onto a flat surface is not practical. As a theoretical maximum, the field of view of a planar image should be less than 180 degrees, because as the field of view increases toward 180 degrees, the required width of the image plane approaches infinity.
To create a projected ray image with a large field of view, the image can be projected onto a curved surface. To have a large horizontal field of view in the image, a surface that is a vertical cylinder (i.e., the axis of the cylinder is parallel to the z-axis) will be sufficient (similarly, if the wide field of view desired is only in the vertical direction of the image, a horizontal cylinder will be sufficient). A cylindrical imaging surface will allow a projected image of rays up to a full 360 degrees in the horizontal or vertical dimension of the perspective image (depending on the orientation of the cylinder). Likewise, when using a spherical imaging surface, the field of view can be a full 360 degrees in any direction (note that for a spherical surface, all rays projected from the scene to the eye intersect the surface at right angles).[68].
Just as a standard perspective image must be viewed from the calculated viewpoint for the image to appear identical to the real scene, an image projected onto a cylinder or sphere must be viewed from the calculated viewpoint for it to be exactly identical to the original scene. If an image projected on a cylindrical surface is "unrolled" into a flat image, different types of distortions occur. For example, many of the straight lines in the scene will be drawn as curves. An image projected on a spherical surface can be flattened in several ways:[68].
References
[1] ↑ Lajo Pérez, Rosina (1990). Léxico de arte. Madrid - España: Akal. p. 162. ISBN 978-84-460-0924-5. |fechaacceso= requiere |url= (ayuda).
[6] ↑ a b c d e f g h i j José Ignacio Rojas-Sola; Ana Serrano-Tierz; David Hernández-Díaz (15 de abril de 2011). «UNA REVISIÓN HISTÓRICA: DESDE EL DIBUJO EN INGENIERÍA HACIA LA INGENIERÍA DEL DISEÑO». DYNA. Consultado el 24 de julio de 2018.: https://revistas.unal.edu.co/index.php/dyna/article/view/25758/39341
[13] ↑ a b c Antonio Nevot Luna, Julián Aguirre Estibález, Roberto Rodríguez del Río. Dibujo técnico y matemáticas: una consideración interdisciplinar Aulas de Verano. Serie: Ciencias. Ministerio de Educación, 2007. pp. 105 de 368. ISBN 9788436945416. Consultado el 23 de julio de 2018.: https://books.google.es/books?id=cLYNoqJP3F8C&pg=PA105#v=onepage&q&f=false
[21] ↑ a b c Cucker, Felix (2013). Manifold Mirrors: The Crossing Paths of the Arts and Mathematics. Cambridge University Press. pp. 269–278. ISBN 978-0-521-72876-8. «Dubery and Willats (1983:33) escribieron que La proyección oblicua parece haber llegado a China desde Roma a través de la India alrededor del primer o segundo siglos.». [Wen-Chi regresa a casa, anon, China, siglo XII] es un arquetipo del clásico uso de la perspectiva oblicua en la pintura china.: https://archive.org/details/manifoldmirrorsc0000cuck
[22] ↑ Claramente, hay un intento general de reducir el tamaño de los elementos más distantes, pero de forma no sistemática. Las secciones de la composición están a una escala similar, con la distancia relativa mostrada por superposición, y otros objetos son más altos que los más cercanos, aunque los trabajadores de la izquierda muestran un ajuste de tamaño más fino. Pero este criterio es abandonado a la derecha, donde la figura más importante es mucho más grande que el albañil. Los edificios rectangulares y los bloques de piedra se muestran oblicuamente.
[30] ↑ "...y estas obras (las perspectivas de Brunelleschi) fueron el medio para despertar las mentes de los otros artesanos, quienes luego se dedicaron a esta técnica con gran celo."Vasari, Vida de los Artistas, Capítulo sobre Brunelleschi.
[31] ↑ (Martin Kemp, 2000, p. 24).
[32] ↑ a b (Martin Kemp, 2000, p. 52).
[33] ↑ "Maese Paolo dal Pozzo Toscanelli, habiendo regresado de sus estudios, invitó a Filippo con otros amigos a cenar en el jardín, y la conversación recaló en asuntos matemáticos, Filippo trabó amistad con él, y de él aprendió geometría."Vasari, Vidas de los Artistas, Capítulo sobre Brunelleschi.
[35] ↑ El-Bizri, Nader (2010). «Classical Optics and the Perspectiva Traditions Leading to the Renaissance». En Hendrix, John Shannon; Carman, Charles H., eds. Renaissance Theories of Vision (Visual Culture in Early Modernity). Farnham, Surrey: Ashgate. pp. 11–30. ISBN 1-409400-24-7.: https://archive.org/details/renaissancetheor00hend
[36] ↑ Hans, Belting (2011). Florence and Baghdad: Renaissance art and Arab science (1st English edición). Cambridge, Massachusetts: Belknap Press of Harvard University Press. pp. 90–92. ISBN 9780674050044. OCLC 701493612.: https://archive.org/details/florencebaghdadr0000belt
[39] ↑ O'Connor, J. J.; Robertson, E. F. (July 1999). «Luca Pacioli». Universidad de Saint Andrews. Archivado desde el original el 22 de septiembre de 2015. Consultado el 23 de septiembre de 2015.: http://www-history.mcs.st-and.ac.uk/Biographies/Pacioli.html
[56] ↑ VITRUBIO POLLIONE, M.L. Compendio de los diez libros de arquitectura de Vitrubio, Traducción de Joseph Castañeda, Imprenta de D. Gabriel Ramírez, Madrid, 1761.
[57] ↑ Texto de Torroja: «Axonometría o perspectiva axonométrica. Sistema general de representación que comprende, como casos particulares, las perspectivas caballera y militar, la proyección isográfica y otros varios» (1897).
[58] ↑ Sketchpad: A Man-Machine Communication System.
[73] ↑ Basant Agrawal (2008). Engineering Drawing. Tata McGraw-Hill Education. pp. 17.3.4. ISBN 978-0-07-066863-8. Archivado desde el original el 23 de diciembre de 2017.: https://books.google.com/books?id=_hdOU8kRb2AC
[76] ↑ Mathographics by Robert Dixon New York: Dover, p. 82, 1991.
[77] ↑ "...la paradoja es puramente conceptual: supone que vemos una representación en perspectiva como una simulación retinal, cuando de hecho la vemos como una pintura bidimensional. En otras palabras, las construcciones de perspectiva crean símbolos visuales, no ilusiones visuales. La clave es que las pinturas carecen de la profundidad de las señales de campo creadas por la visión binocular; siempre somos conscientes de que una pintura es plana en lugar de profunda. Y así es como nuestra mente lo interpreta, ajustando nuestra comprensión de la pintura para compensar nuestra posición."«Archived copy». Archivado desde el original el 6 de enero de 2007. Consultado el 25 de diciembre de 2006. Retrieved on 25 December 2006.: http://www.handprint.com/HP/WCL/perspect1.html
Melozzo da Forlì first used the technique of foreshortening upwards (in Rome, Loreto "Loreto (Italy)"), Forlì and other places, becoming famous for it.[32] Perspective was not only a way of showing depth, it was also a new method of pictorial composition. Paintings began to show a single, unified scene, rather than a combination of several different frames.
As demonstrated by the rapid proliferation of paintings with rigorous perspectives in Florence, Brunelleschi probably understood (with the help of his mathematician friend Toscanelli),[33] but did not publish, the underlying mathematics behind perspective.
Decades later, his friend Leon Battista Alberti (1404-1472) wrote De pictura (1435/1436), a treatise on proper methods for showing distance in painting. Alberti's main advance was not to show mathematics in terms of conic projections, since in reality they seemed implicit to the eye. Instead, he formulated a theory based on plane projections, or how rays of light, passing from the viewer's eye from the landscape, would strike the image plane (the painting). He was then able to calculate the apparent height of a distant object using two similar triangles. The mathematics associated with similar triangles is relatively simple, and had already been formulated by Euclid a long time ago.[34] When looking at a wall, for example, the first triangle has one vertex "Vertex (geometry)") in the observer's eye and the other two vertices at the top and bottom of the wall. The bottom of this triangle is the distance from the observer to the wall. The second similar triangle has a point in the observer's eye, and has a length equal to that between the viewer's eye and the painting. The height of the second triangle can be determined through a simple relationship, as Euclid demonstrates.
He reflected on the images that are inscribed inside the "window" formed by a cube open on one side and praised "the faces that in the paintings give the impression of coming out of the painting, as if they were sculpted." To do this, "a painter must be instructed, to the extent possible, in all the liberal arts, but (...) above all, in geometry", thus defining the premises of a theory of perspective. Alberti was also trained in the science of optics through the Padua school and under the influence of Biagio Pelacani da Parma who studied Alhacén's Book of Optics[35] (see what was noted above in this regard with respect to Ghiberti). Alhacén's "Book of Optics", translated around 1200 into Latin, laid the mathematical foundations for perspective in Europe.[36].
The codification of the European humanist perspective was developed in Italian Umbria "Umbria (Italy)", in the middle of the century, under the influence of the work of Piero della Francesca (1415-1492): from mere intuition and technical means, the perspective becomes mathematical theory. He was also the first painter to carry out a scientific study of light in painting "Light (painting)").
He elaborated his theory on Della Pittura in De Prospectiva Pingendi in the 1470s.[37] Alberti had limited himself to figures on the ground plane and gave a general basis for perspective. Della Francesca developed this theory, explicitly covering solids in any area of the picture plane. He also began the common practice of using illustrated figures to explain mathematical concepts, making his treatise easier to understand than Alberti's. Della Francesca was also the first to accurately draw the Platonic solids, which appeared in perspective.[38].
In 1509, Luca Pacioli's De divina proportione (On Divine Proportion) appeared, illustrated by Leonardo da Vinci, summarizing the use of perspective in painting.[39].
Halfway between the 19th century and 20th century, perspective was perfected under the contribution of Leonardo da Vinci himself (1452-1519) in his Treatise on Painting (published in 1680) with color perspective, where the colors blur as the distance increases; and waning perspective, where objects or figures lose sharpness with distance.[40].
The perspective remained, for a time, in the domain of Florence. Jan van Eyck (1390-1441), among others, was unable to create a consistent structure for converging lines in paintings, as in the Portrait of Giovanni Arnolfini and his Wife[41] from London, because he was not aware of the theoretical advance that was occurring at the time in Italy. However, he achieved very subtle effects through scale manipulations in his interiors.
Albrecht Dürer (1471-1528), a key figure of the Renaissance in Germany both for his pictorial work and for his studies on the art of drawing, continued the development of perspective techniques.[42] Among his engravings are detailed images of the practice of the theoretical methods used to transfer real models to the drawing plane.
Gradually, and partly through the arts academies movement, Italian techniques became part of the training of artists throughout Europe, and later in other parts of the world.
The second half of the century saw the rise of the neoclassical style, with painters of an academic nature such as the Frenchman Jacques-Louis David (1748-1825), in whose works perspective served as a backdrop for themes of a historicist or allegorical nature,[48] but without the prominence of the previous stage.
On a theoretical level, figures such as the French Gaspard Monge (1746-1818), creator of descriptive geometry, and Jean-Victor Poncelet (1788-1867), recoverer of projective geometry, laid the foundations that linked the geometry of perspective to technique and to other branches of mathematics such as algebra. Finally, Otto Wilhelm Fiedler (1832-1912) rigorously defined the central projection system in his doctoral thesis of 1859, establishing the mathematical foundations of the conical perspective as they are known today.[6].
After the French Revolution, the Romantic movement emerged, a prelude to the avant-garde artistic currents of the first part of the century. During this period, a more naturalistic concept of art was imposed, less linked to the geometric rigor of the scenes prevailing until then. Furthermore, the improvement of photography in the last decades of the century contributed to facilitating the creation of perspective views, somewhat trivializing the laborious task of manually creating perspective pictorial images.[49].
The appearance of cubism by Pablo Picasso (1881-1873) represented an absolute break with the academic concept of perspective, with its purpose of simultaneously reflecting the different facets of a three-dimensional object developed on the plane of the picture.[50].
However, later movements throughout the century, such as the surrealism represented by Salvador Dalí (1904-1989),[51] or the hyperrealism practiced by Antonio López (1936),[52] have used perspective as a first-order expressive resource. In this sense, we can cite the works of the Dutch engraver M. C. Escher (1898-1972), who explored the theoretical limits of perspective producing paradoxical effects with geometrically impossible objects.[53].
The advance in projective geometry during the centuries led to the development of analytical geometry and algebraic geometry, which some time later would form part of the mathematical support linked to relativity[54] and quantum mechanics.[55].
In the academic field, the Italian scientist Quintino Sella (1827-1884) was one of the first to systematically apply the axonometric perspective to his crystallography studies, and the teaching of this discipline came to the United States at the hands of the French engineer Claudius Crozet (1789-1864).
The generalization of machine tools from the century onwards allowed the manufacturing of precision mechanical components to be industrialized, generating new needs for which technical plans in axonometric perspective fit perfectly. Thus, engineers such as Joseph Whitworth (1803–1887) or Charles Renard (1847-1905) respectively standardized the thread pitches and the thickness of the cables, and both the documentation of their patents and the graphic description of their products frequently used this type of drawings.[6].
The first third of the century saw the birth of national standardization organizations, culminating in the creation of the ISO in 1947, establishing technical standards that have served to internationally unify the criteria with which technical axonometric perspectives are carried out.
In addition to well-known CAD programs such as Autocad, Microstation, SolidEdge, CATIA, SolidWorks, Pro/Engineer or Euclid,[65] there are already classic programs such as Autodesk 3ds Max, Mental Ray or Google Sketchup[66] specialized in the generation of three-dimensional images. Many of these applications use graphics routine packages such as OpenGL or Direct3D, which optimize the performance of computer graphics cards.[67].
Melozzo da Forlì first used the technique of foreshortening upwards (in Rome, Loreto "Loreto (Italy)"), Forlì and other places, becoming famous for it.[32] Perspective was not only a way of showing depth, it was also a new method of pictorial composition. Paintings began to show a single, unified scene, rather than a combination of several different frames.
As demonstrated by the rapid proliferation of paintings with rigorous perspectives in Florence, Brunelleschi probably understood (with the help of his mathematician friend Toscanelli),[33] but did not publish, the underlying mathematics behind perspective.
Decades later, his friend Leon Battista Alberti (1404-1472) wrote De pictura (1435/1436), a treatise on proper methods for showing distance in painting. Alberti's main advance was not to show mathematics in terms of conic projections, since in reality they seemed implicit to the eye. Instead, he formulated a theory based on plane projections, or how rays of light, passing from the viewer's eye from the landscape, would strike the image plane (the painting). He was then able to calculate the apparent height of a distant object using two similar triangles. The mathematics associated with similar triangles is relatively simple, and had already been formulated by Euclid a long time ago.[34] When looking at a wall, for example, the first triangle has one vertex "Vertex (geometry)") in the observer's eye and the other two vertices at the top and bottom of the wall. The bottom of this triangle is the distance from the observer to the wall. The second similar triangle has a point in the observer's eye, and has a length equal to that between the viewer's eye and the painting. The height of the second triangle can be determined through a simple relationship, as Euclid demonstrates.
He reflected on the images that are inscribed inside the "window" formed by a cube open on one side and praised "the faces that in the paintings give the impression of coming out of the painting, as if they were sculpted." To do this, "a painter must be instructed, to the extent possible, in all the liberal arts, but (...) above all, in geometry", thus defining the premises of a theory of perspective. Alberti was also trained in the science of optics through the Padua school and under the influence of Biagio Pelacani da Parma who studied Alhacén's Book of Optics[35] (see what was noted above in this regard with respect to Ghiberti). Alhacén's "Book of Optics", translated around 1200 into Latin, laid the mathematical foundations for perspective in Europe.[36].
The codification of the European humanist perspective was developed in Italian Umbria "Umbria (Italy)", in the middle of the century, under the influence of the work of Piero della Francesca (1415-1492): from mere intuition and technical means, the perspective becomes mathematical theory. He was also the first painter to carry out a scientific study of light in painting "Light (painting)").
He elaborated his theory on Della Pittura in De Prospectiva Pingendi in the 1470s.[37] Alberti had limited himself to figures on the ground plane and gave a general basis for perspective. Della Francesca developed this theory, explicitly covering solids in any area of the picture plane. He also began the common practice of using illustrated figures to explain mathematical concepts, making his treatise easier to understand than Alberti's. Della Francesca was also the first to accurately draw the Platonic solids, which appeared in perspective.[38].
In 1509, Luca Pacioli's De divina proportione (On Divine Proportion) appeared, illustrated by Leonardo da Vinci, summarizing the use of perspective in painting.[39].
Halfway between the 19th century and 20th century, perspective was perfected under the contribution of Leonardo da Vinci himself (1452-1519) in his Treatise on Painting (published in 1680) with color perspective, where the colors blur as the distance increases; and waning perspective, where objects or figures lose sharpness with distance.[40].
The perspective remained, for a time, in the domain of Florence. Jan van Eyck (1390-1441), among others, was unable to create a consistent structure for converging lines in paintings, as in the Portrait of Giovanni Arnolfini and his Wife[41] from London, because he was not aware of the theoretical advance that was occurring at the time in Italy. However, he achieved very subtle effects through scale manipulations in his interiors.
Albrecht Dürer (1471-1528), a key figure of the Renaissance in Germany both for his pictorial work and for his studies on the art of drawing, continued the development of perspective techniques.[42] Among his engravings are detailed images of the practice of the theoretical methods used to transfer real models to the drawing plane.
Gradually, and partly through the arts academies movement, Italian techniques became part of the training of artists throughout Europe, and later in other parts of the world.
The second half of the century saw the rise of the neoclassical style, with painters of an academic nature such as the Frenchman Jacques-Louis David (1748-1825), in whose works perspective served as a backdrop for themes of a historicist or allegorical nature,[48] but without the prominence of the previous stage.
On a theoretical level, figures such as the French Gaspard Monge (1746-1818), creator of descriptive geometry, and Jean-Victor Poncelet (1788-1867), recoverer of projective geometry, laid the foundations that linked the geometry of perspective to technique and to other branches of mathematics such as algebra. Finally, Otto Wilhelm Fiedler (1832-1912) rigorously defined the central projection system in his doctoral thesis of 1859, establishing the mathematical foundations of the conical perspective as they are known today.[6].
After the French Revolution, the Romantic movement emerged, a prelude to the avant-garde artistic currents of the first part of the century. During this period, a more naturalistic concept of art was imposed, less linked to the geometric rigor of the scenes prevailing until then. Furthermore, the improvement of photography in the last decades of the century contributed to facilitating the creation of perspective views, somewhat trivializing the laborious task of manually creating perspective pictorial images.[49].
The appearance of cubism by Pablo Picasso (1881-1873) represented an absolute break with the academic concept of perspective, with its purpose of simultaneously reflecting the different facets of a three-dimensional object developed on the plane of the picture.[50].
However, later movements throughout the century, such as the surrealism represented by Salvador Dalí (1904-1989),[51] or the hyperrealism practiced by Antonio López (1936),[52] have used perspective as a first-order expressive resource. In this sense, we can cite the works of the Dutch engraver M. C. Escher (1898-1972), who explored the theoretical limits of perspective producing paradoxical effects with geometrically impossible objects.[53].
The advance in projective geometry during the centuries led to the development of analytical geometry and algebraic geometry, which some time later would form part of the mathematical support linked to relativity[54] and quantum mechanics.[55].
In the academic field, the Italian scientist Quintino Sella (1827-1884) was one of the first to systematically apply the axonometric perspective to his crystallography studies, and the teaching of this discipline came to the United States at the hands of the French engineer Claudius Crozet (1789-1864).
The generalization of machine tools from the century onwards allowed the manufacturing of precision mechanical components to be industrialized, generating new needs for which technical plans in axonometric perspective fit perfectly. Thus, engineers such as Joseph Whitworth (1803–1887) or Charles Renard (1847-1905) respectively standardized the thread pitches and the thickness of the cables, and both the documentation of their patents and the graphic description of their products frequently used this type of drawings.[6].
The first third of the century saw the birth of national standardization organizations, culminating in the creation of the ISO in 1947, establishing technical standards that have served to internationally unify the criteria with which technical axonometric perspectives are carried out.
In addition to well-known CAD programs such as Autocad, Microstation, SolidEdge, CATIA, SolidWorks, Pro/Engineer or Euclid,[65] there are already classic programs such as Autodesk 3ds Max, Mental Ray or Google Sketchup[66] specialized in the generation of three-dimensional images. Many of these applications use graphics routine packages such as OpenGL or Direct3D, which optimize the performance of computer graphics cards.[67].