Perspective (graphical)
Susan van Zyl 642 posts 
Perspective (from the[[LATIN said like this {LATIN}]] perspicere, to see through) in the graphic arts, such as drawing, is an approximate representation, on a flat surface (such as paper), of an image as it is seen by the eye. The two most characteristic features of perspective are that objects are drawn:
Overview Linear perspective works by representing the light that passes from a scene through an imaginary rectangle (the painting), to the viewer’s eye. It is similar to a viewer looking through a window and painting what is seen directly onto the windowpane. If viewed from the same spot as the windowpane was painted, the painted image would be identical to what was seen through the unpainted window. Each painted object in the scene is a flat, scaled down version of the object on the other side of the window.^{1} Because each portion of the painted object lies on the straight line from the viewer’s eye to the equivalent portion of the real object it represents, the viewer cannot perceive (sans depth perception) any difference between the painted scene on the windowpane and the view of the real scene. All perspective drawings assume the viewer is a certain distance away from the drawing. Objects are scaled relative to that viewer. Additionally, an object is often not scaled evenly: a circle often appears as an ellipse and a square can appear as a trapezoid. This distortion is referred to as foreshortening. Perspective drawings typically have an Any perspective representation of a scene that includes parallel lines has one or more vanishing points in a perspective drawing. A onepoint perspective drawing means that the drawing has a single vanishing point, usually (though not necessarily) directly opposite the viewer’s eye and usually (though not necessarily) on the horizon line. All lines parallel with the viewer’s line of sight recede to the horizon towards this vanishing point. This is the standard “receding railroad tracks” phenomenon. A twopoint drawing would have lines parallel to two different angles. Any number of vanishing points are possible in a drawing, one for each set of parallel lines that are at an angle relative to the plane of the drawing. Perspectives consisting of many parallel lines are observed most often when drawing architecture (architecture frequently uses lines parallel to the x, y, and z axes). Because it is rare to have a scene consisting solely of lines parallel to the three Cartesian axes (x, y, and z), it is rare to see perspectives in practice with only one, two, or three vanishing points; even a simple house frequently has a peaked roof which results in a minimum of six sets of parallel lines, in turn corresponding to up to six vanishing points. In contrast, natural scenes often do not have any sets of parallel lines. Such a perspective would thus have no vanishing points. Renaissance : Mathematical basis Prior to the Renaissance, a clearly modern optical basis of perspective was given in 1021, when Alhazen (alHasan Ibn alHaytham, d. ca. 1041 CE), an Iraqi physicist and mathematician, in his Book of Optics (Kitab almanazir; known in Latin as De aspectibus or Perspectiva), explained that light projects conically into the eye.^{8} Alhazen’s geometrical, physical, physiopsychological optics resolved in this the ancient dispute between the mathematicians (Ptolemaic and Euclidean) and the physicists (Aristotelian) over the nature of vision and light. He also showed that vision is not merely a phenomenon of pure sensation (namely what results from the introduction of light rays into the eyes), but that essentially it involves the faculties of judgement, imagination and memory.^{9} Alhazen’s geometrical model of the cone of vision was theoretically sufficient to translate visible objects within a given setting into a painting, and this was also supported by his experimental affirmation of the visibility of spatial depth; hence of offering a proper ground for the idea of perspective.^{10} Moreover, Alhazen presented a geometrical conception of place as spatial extension (a postulated void), and he refuted the Aristotelian account of topos as a surface of containment. Alhazen’s mathematical definition of place was more akin to Plato’s notion of Khôra or Chora as ‘space’, yet conceived on pure geometric grounds to facilitate the use of projections.^{11} In all of this, Alhazen was concerned with optics, with vision, light and the nature of colour, as well as with experimentation and the use of optical instruments, and not with painting as such. Conical translations are mathematically difficult, so a drawing constructed using them would be incredibly time consuming. However, what Alhazen named a cone of vision (makhrut alshu’a’) corresponded also with the idea of a pyramid of vision, hence, offering a model that can be more easily projected in orthogonal drawings of side views and top views that are needed in the geometric construction of perspective. By the 14th century, Alhazen’s Book of Optics was available in Italian translation, entitled Deli Aspecti. The Renaissance artist Lorenzo Ghiberti relied heavily upon this work, quoting it “verbatim and at length” while framing his account of art and its aesthetic imperatives in the “Commentario terzo.” Alhazen’s work was thus “central to the development of Ghiberti’s thought about art and visual aesthetics” and “may well have been central to the development of artificial perspective in early Renaissance Italian painting.”^{12}[citation needed] In about 1413 a contemporary of Ghiberti, Filippo Brunelleschi, demonstrated the geometrical method of perspective, used today by artists, by painting the outlines of various Florentine buildings onto a mirror. When the building’s outline was continued, he noticed that all of the lines converged on the horizon line. According to Vasari, he then set up a demonstration of his painting of the Baptistry in the incomplete doorway of the Duomo. He had the viewer look through a small hole on the back of the painting, facing the Baptistry. He would then set up a mirror, facing the viewer, which reflected his painting. To the viewer, the painting of the Baptistry and the Baptistry itself were nearly indistinguishable. Soon after, nearly every artist in Florence and in Italy used geometrical perspective in their paintings,^{13} notably Melozzo da Forlì and Donatello. Melozzo first used the perspective from down to up (in Rome, Loreto, Forlì…), and was celebrated for that. Donatello started sculpting elaborate checkerboard floors into the simple manger portrayed in the birth of Christ. Although hardly historically accurate, these checkerboard floors obeyed the primary laws of geometrical perspective: all lines converged to a vanishing point, and the rate at which the horizontal lines receded into the distance was graphically determined. This became an integral part of Quattrocento art. Not only was perspective a way of showing depth, it was also a new method of composing a painting. Paintings began to show a single, unified scene, rather than a combination of several.
As shown by the quick proliferation of accurate perspective paintings in Florence, Brunelleschi likely understood (with help from his friend the mathematician Toscanelli)^{14}, but did not publish, the mathematics behind perspective. Decades later, his friend Leon Battista Alberti wrote De pictura/Della Pittura (1435/1436), a treatise on proper methods of showing distance in painting. Alberti’s primary breakthrough was not to show the mathematics in terms of conical projections, as it actually appears to the eye. Instead, he formulated the theory based on planar projections, or how the rays of light, passing from the viewer’s eye to the landscape, would strike the picture plane (the painting). He was then able to calculate the apparent height of a distant object using two similar triangles. The mathematics behind similar triangles is relatively simple, having been long ago formulated by Euclid. In viewing a wall, for instance, the first triangle has a vertex at the user’s eye, and vertices at the top and bottom of the wall. The bottom of this triangle is the distance from the viewer to the wall. The second, similar triangle, has a point at the viewer’s eye, and has a length equal to the viewer’s eye from the painting. The height of the second triangle can then be determined through a simple ratio, as proven by Euclid. Alberti was also trained in the science of optics through the school of Padua and under the influence of Biagio Pelacani da Parma who studied Alhazen’s Optics (see what was noted above in this regard with respect to Ghiberti). Piero della Francesca elaborated on Della Pittura in his “De Prospectiva Pigendi” in 1474. Alberti had limited himself to figures on the ground plane and giving an overall basis for perspective. Della Francesca fleshed it out, explicitly covering solids in any area of the picture plane. Della Francesca also started the now common practice of using illustrated figures to explain the mathematical concepts, making his treatise easier to understand than Alberti’s. Della Francesca was also the first to accurately draw the Platonic solids as they would appear in perspective. Perspective remained, for a while, the domain of Florence. Jan van Eyck, among others, was unable to create a consistent structure for the converging lines in paintings, as in London’s The Arnolfini Portrait, because he was unaware of the theoretical breakthrough just then occurring in Italy. However he achieved very subtle effects by manipulations of scale in his interiors. Gradually, and partly through the movement of academies of the arts, the Italian techniques became part of the training of artists across Europe, and later other parts of the world. Present : Computer graphics 3D computer games and raytracers often use a modified version of perspective. Like the painter, the computer program is generally not concerned with every ray of light that is in a scene. Instead, the program simulates rays of light traveling backwards from the monitor (one for every pixel), and checks to see what it hits. In this way, the program does not have to compute the trajectories of millions of rays of light that pass from a light source, hit an object, and miss the viewer. CAD software, and some computer games (especially games using 3D polygons) use linear algebra, and in particular matrix multiplication, to create a sense of perspective. The scene is a set of points, and these points are projected to a plane (computer screen) in front of the view point (the viewer’s eye). The problem of perspective is simply finding the corresponding coordinates on the plane corresponding to the points in the scene. By the theories of linear algebra, a matrix multiplication directly computes the desired coordinates, thus bypassing any descriptive geometry theorems used in perspective drawing. Types of Perspective Of the many types of perspective drawings, the most common categorizations of artificial perspective are one, two and threepoint. The names of these categories refer to the number of vanishing points in the perspective drawing. Onepoint perspective One vanishing point is typically used for roads, railway tracks, hallways, or buildings viewed so that the front is directly facing the viewer. Any objects that are made up of lines either directly parallel with the viewer’s line of sight or directly perpendicular (the railroad slats) can be represented with onepoint perspective. Onepoint perspective exists when the painting plate (also known as the picture plane) is parallel to two axes of a rectilinear (or Cartesian) scene — a scene which is composed entirely of linear elements that intersect only at right angles. If one axis is parallel with the picture plane, then all elements are either parallel to the painting plate (either horizontally or vertically) or perpendicular to it. All elements that are parallel to the painting plate are drawn as parallel lines. All elements that are perpendicular to the painting plate converge at a single point (a vanishing point) on the horizon. Some examples:
Twopoint perspective
Twopoint perspective can be used to draw the same objects as onepoint perspective, rotated: looking at the corner of a house, or looking at two forked roads shrink into the distance, for example. One point represents one set of parallel lines, the other point represents the other. Looking at a house from the corner, one wall would recede towards one vanishing point, the other wall would recede towards the opposite vanishing point. Twopoint perspective exists when the painting plate is parallel to a Cartesian scene in one axis (usually the zaxis) but not to the other two axes. If the scene being viewed consists solely of a cylinder sitting on a horizontal plane, no difference exists in the image of the cylinder between a onepoint and twopoint perspective. Twopoint perspective has one set of lines parallel to the picture plane and two sets oblique to it.Parallel lines oblique to the picture plane converge to a vanishing point,which means that this setup will require two vanishing points. ThreePoint Perspective
Threepoint perspective is usually used for buildings seen from above (or below). In addition to the two vanishing points from before, one for each wall, there is now one for how those walls recede into the ground. This third vanishing point will be below the ground. Looking up at a tall building is another common example of the third vanishing point. This time the third vanishing point is high in space. Threepoint perspective exists when the perspective is a view of a Cartesian scene where the picture plane is not parallel to any of the scene’s three axes. Each of the three vanishing points corresponds with one of the three axes of the scene. Image constructed using multiple vanishing points. Onepoint, twopoint, and threepoint perspectives appear to embody different forms of calculated perspective. The methods required to generate these perspectives by hand are different. Mathematically, however, all three are identical: The difference is simply in the relative orientation of the rectilinear scene to the viewer. Fourpoint perspective Four point perspective, also called infinitepoint perspective, is the curvilinear variant of twopoint perspective. As the result when made into an infinite point version (i.e. when the amount of vanishing points exceeds the minimum amount required), a four point perspective image becomes a panorama that can go to a 360 degree view and beyond – when going beyond the 360 degree view you create an “impossible” room as the artist might depict something new when it’s supposed to show part of what already exists within those 360 degrees. This elongated frame can be used both horisontally and vertically and when used vertically can be described as an image that depicts both a worms and birds eye view of a scene at the same time. As all other foreshortened variants of perspective (respectively one to sixpoint perspective), it starts off with a horizon line, followed by four equally spaced vanishing points to delineate four vertical lines created in a 90 degree relation to the horizon line. The vanishing points made to create the curvilinear orthogonals are thus made ad hoc on the four vertical lines placed on the opposite side of the horizon line. The only dimension not foreshortened in this type of perspective being the rectilinear and parallell lines at a 90 degree angle to the horizon line – similar to the vertical lines used in twopoint perspective. Zeropoint perspective Because vanishing points exist only when parallel lines are present in the scene, a perspective without any vanishing points (“zeropoint” perspective) occurs if the viewer is observing a nonlinear scene. The most common example of a nonlinear scene is a natural scene (e.g., a mountain range) which frequently does not contain any parallel lines. A perspective without vanishing points can still create a sense of “depth,” as is clearly apparent in a photograph of a mountain range (more distant mountains have smaller scale features). Onepoint, twopoint, and threepoint perspective are dependent on the structure of the scene being viewed. These only exist for strict Cartesian (rectilinear) scenes. By inserting into a Cartesian scene a set of parallel lines that are not parallel to any of the three axes of the scene, a new distinct vanishing point is created. Therefore, it is possible to have an infinitepoint perspective if the scene being viewed is not a Cartesian scene but instead consists of infinite pairs of parallel lines, where each pair is not parallel to any other pair. Foreshortening
Foreshortening refers to the visual effect or optical illusion that an object or distance appears shorter than it actually is because it is angled toward the viewer. Although foreshortening is an important element in art where visual perspective is being depicted, foreshortening occurs in other types of twodimensional representations of threedimensional scenes. Some other types where foreshortening can occur include oblique parallel projection drawings. Figure F1 shows two different projections of a stack of two cubes, illustrating oblique parallel projection foreshortening (“A”) and perspective foreshortening (“B”). Foreshortening is an effect which also occurs on American and Canadian automobile Wing mirrors, see Objects in Mirror Are Closer Than They Appear. This technique was often used in Renaissance painting. Methods of construction Several methods of constructing perspectives exist, including:
Example
To draw a square in perspective, the artist starts by drawing a horizon line (black) and determining where the vanishing point (green) should be. The higher up the horizon line is, the lower the viewer will appear to be looking, and vice versa. The more offcenter the vanishing point, the more tilted the square will be. Because the square is made up of right angles, the vanishing point should be directly in the middle of the horizon line. A rotated square is drawn using twopoint perspective, with each set of parallel lines leading to a different vanishing point. The foremost edge of the (orange) square is drawn near the bottom of the painting. Because the viewer’s picture plane is parallel to the bottom of the square, this line is horizontal. Lines connecting each side of the foremost edge to the vanishing point are drawn (in grey). These lines give the basic, one point “railroad tracks” perspective. The closer it is the horizon line, the farther away it is from the viewer, and the smaller it will appear. The farther away from the viewer it is, the closer it is to being perpendicular to the picture plane. A new point (the eye) is now chosen, on the horizon line, either to the left or right of the vanishing point. The distance from this point to the vanishing point represents the distance of the viewer from the drawing. If this point is very far from the vanishing point, the square will appear squashed, and far away. If it is close, it will appear stretched out, as if it is very close to the viewer. A line connecting this point to the opposite corner of the square is drawn. Where this (blue) line hits the side of the square, a horizontal line is drawn, representing the farthest edge of the square. The line just drawn represents the ray of light traveling from the farthest edge of the square to the viewer’s eye. This step is key to understanding perspective drawing. The light that passes through the picture plane obviously can not be traced. Instead, lines that represent those rays of light are drawn on the picture plane. In the case of the square, the side of the square also represents the picture plane (at an angle), so there is a small shortcut: when the line hits the side of the square, it has also hit the appropriate spot in the picture plane. The (blue) line is drawn to the opposite edge of the foremost edge because of another shortcut: since all sides are the same length, the foremost edge can stand in for the side edge. Original formulations used, instead of the side of the square, a vertical line to one side, representing the picture plane. Each line drawn through this plane was identical to the line of sight from the viewer’s eye to the drawing, only rotated around the yaxis ninety degrees. It is, conceptually, an easier way of thinking of perspective. It can be easily shown that both methods are mathematically identical, and result in the same placement of the farthest side (see Panofsky). Limitations For a typical perspective, however, the field of view is narrow enough (often only 60 degrees) that the distortions are similarly minimal enough that the image can be viewed from a point other than the actual calculated vantage point without appearing significantly distorted. When a larger angle of view is required, the standard method of projecting rays onto a flat picture plane becomes impractical. As a theoretical maximum, the field of view of a flat picture plane must be less than 180 degrees (as the field of view increases towards 180 degrees, the required breadth of the picture plane approaches infinity). In order to create a projected ray image with a large field of view, one can project the image onto a curved surface. In order to have a large field of view horizontally in the image, a surface that is a vertical cylinder (i.e., the axis of the cylinder is parallel to the zaxis) will suffice (similarly, if the desired large field of view is only in the vertical direction of the image, a horizontal cylinder will suffice). A cylindrical picture surface will allow for a projected ray image up to a full 360 degrees in either the horizontal or vertical dimension of the perspective image (depending on the orientation of the cylinder). In the same way, by using a spherical picture surface, the field of view can be a full 360 degrees in any direction (note that for a spherical surface, all projected rays from the scene to the eye intersect the surface at a right angle). Just as a standard perspective image must be viewed from the calculated vantage point for the image to appear identical to the true scene, a projected image onto a cylinder or sphere must likewise be viewed from the calculated vantage point for it to be precisely identical to the original scene. If an image projected onto a cylindrical surface is “unrolled” into a flat image, different types of distortions occur: For example, many of the scene’s straight lines will be drawn as curves. An image projected onto a spherical surface can be flattened in various ways, including:

Sally Sargent 498 posts 
Hi Susan, Once again you did a nice job in presenting perspective in this forum. I enjoyed the examples of one and two point perspective as well as the examples available. The examples of the horizon line as eye level were excellent. Once again I’m used to teaching perspective more simplistically using silly formulas and tricks to accomplish tiling and window placements. Perhaps a useful piece of information would be placement of people and objects in the perspective plane. Another idea for an article might be using perspective to accomplish foreshortening in figure drawing. I very much enjoy reading your articles. Keep up the good work, and thank you. Sally 