# Linear perspective

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Perspective (from Latin perspicere, to see clearly) in the graphic arts, such as drawing, is an approximate representation on a flat surface (such as paper) of an image as it is perceived by the eye. The two most characteristic features of perspective are:

• Objects are drawn smaller as their distance from the observer increases
• Spatial foreshortening, which is the distortion of items when viewed at an angle

In art, the term "foreshortening" is often used synonymously with perspective, even though foreshortening can occur in other types of non-perspective drawing representations (such as oblique parallel projection).

## What is Perspective?

### Basic concept

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. 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. If the viewer is standing in a different spot, the illusion should be ruined, but unless the viewer chooses an extreme angle, like looking at it from the bottom corner of the window, the perspective normally looks more or less correct.

The person who invented this type of drawing was a usless good for nothing wa*@er who would had too much free time and little to be at.

In practice, however, nearly all perspectives (including those created mathematically), introduce distortions in comparison to the view of the real scene. Distortions can occur from:

• mathematical approximations in calculated perspectives
• type of lens used in perspectives generated through photography
• inaccuracies from freehand sketching

These distortions are usually introduced knowingly in order to simplify construction of the perspective.

### Other related concepts

Some concepts that are commonly associated with perspectives include:

• foreshortening
• horizon line
• vanishing points

All perspective drawings assume a viewer, a certain distance away from the drawing. Objects are scaled relative to that viewer. Additionally, an object is often not scaled evenly --- a circle often appears as an ellipse and a square can appear as a trapezoid. This distortion is referred to as foreshortening.

Perspective drawings typically have an (often implied) horizon line. This line, directly opposite the viewer's eye, represents objects infinitely far away. They have shrunk, in the distance, to the infinitesimal thickness of a line. It is analogous (and named after) the Earth's horizon.

Any perspective representation of a scene that includes parallel lines has one or more vanishing points in a perspective drawing. A one-point perspective drawing means that the drawing has a single vanishing point, usually (though not necessarily) directly opposite the viewer's eye and usually (though not necessarily) on the horizon line. All lines parallel with the viewer's line of sight recede to the horizon towards this vanishing point. This is the standard "receding railroad tracks" phenomenon. A two-point drawing would have lines parallel to two different angles. Any number of vanishing points are possible in a drawing, one for each set of parallel lines that are at an angle relative to the plane of the drawing.

Perspectives consisting of many parallel lines are observed most often when drawing architecture (architecture frequently uses lines parallel to the x, y, and z axes). Because it is rare to have a scene consisting solely of lines parallel to the three Cartesian axes (x, y, and z), it is rare to see perspectives in practice with only one, two, or three vanishing points. Consider that even a simple house frequently has a peaked roof which results in a minimum of five sets of parallel lines, in turn corresponding to up to five vanishing points.

In contrast, perspectives of natural scenes often do not have any sets of parallel lines. Such a perspective would thus have no vanishing points.

## History of Perspective

### Early history

Before perspective, paintings and drawings typically sized objects and characters according to their spiritual or thematic importance, not with distance. Especially in Medieval art, art was meant to be read as a group of symbols, rather than seen as a coherent picture. The only method to show distance was by overlapping characters. Overlapping alone made poor drawings of architecture; medieval paintings of cities are a hodgepodge of lines in every direction.

The optical basis of perspective was defined in the year 1000, when the Arabian mathematician and philosopher Alhazen, in his Perspectiva, first explained that light projects conically into the eye. This was, theoretically, enough to translate objects convincingly onto a painting, but Alhalzen was concerned only with optics, not with painting. Conical translations are also mathematically difficult, so a drawing using them would be incredibly time consuming.

The artist Giotto di Bondone first attempted drawings in perspective using an algebraic method to determine the placement of distant lines. The problem with using a linear ratio in this manner is that the apparent distance between a series of evenly spaced lines actually falls off with a sine dependence. To determine the ratio for each succeeding line, a recursive ratio must be used. This was not discovered until the 20th Century, in part by Erwin Panofsky.

One of Giotto's first uses of his algebraic method of perspective was Jesus Before the Caïf. Although the picture does not conform to the modern, geometrical method of perspective, it does give a decent illusion of depth, and was a large step forward in Western art.

### Mathematical basis for perspective

One hundred years later, in the early 1400s, 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 all the lines all 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 intistinguishable.

Soon after, nearly every artist in Florence used geometrical perspective in their paintings, notably Donatello, who started painting 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, but did not publish, the mathematics behind perspective. Decades later, his friend Leon Battista Alberti wrote Della Pittura, 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. 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.

Piero della Francesca elaborated on Della Pittura in his De Prospectiva Pingendi in 1474. Alberti had limited himself to figures on the ground plane and giving an overall basis for perspective. Francesca fleshed it out, explicitely covering solids in any area of the picture plane. 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. 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.

### Artificial and natural

Leonardo da Vinci distrusted Brunelleschi's formulation of perspective because it failed to take into account the appearance of objects held very close to the eye. Leonardo called Brunelleschi's method artificial perspective projection. It is today called classical perspective projection. Projections closer to the image beheld by the human eye, he named natural perspective.

Artificial perspective projection is a perspective projection onto a flat surface, well suited for drawings and paintings, which are typically flat. Natural perspective projection, in contrast, is a perspective projection onto a spherical surface. From a geometric point of view, the differences between artificial and natural perspectives can be thought of as similar to the distortion that occurs when representing the earth (approximately spherical) as a map (typically flat). Both types of projection involve a distortion. The difference between the two distortions is called perspective projection distortion.

## Varieties of Perspective Drawings

Of the many types of perspective drawings, the most common categorizations of artificial perspective are one-, two- and three-point. The names of these categories refer to the number of vanishing points in the perspective drawing. Strictly speaking, these types can only exist for scenes being represented that are rectilinear (composed entirely of straight lines which intersect only at 90 degrees to each other).

### One-point perspective

One vanishing point is typically used for roads, railroad tracks, 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 (like railroad tracks) or directly perpendicular (the railroad slats) can be represented with one-point perspective.

One-point 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.

### Two-point perspective

Two-point perspective can be used to draw the same objects as one-point 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 opposide vanishing point.

Two-point perspective exists when the painting plate is parallel to a Cartesian scene in one axis (usually the z-axis) but not to the other two axes. If the scene being viewed consists solely of a cylinder sitting on a horizontal plane, no difference exists in the image of the cylinder between a one-point and two-point perspective.

### Three-point perspective

Three-point perspective is usually used for buildings seen from above. 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. Looking up at a tall building is another common example of the third vanishing point.

Three-point perspective exists when the perspective is a view of a Cartesian scene where the picture plane is not parallel to any of the scene's three axes. Each of the three vanishing points corresponds with one of the three axes of the scene. Image constructed using multiple vanishing points.

### Zero-point perspective

Due to the fact that vanishing points exist only when parallel lines are present in the scene, a perspective without any vanishing points ("zero-point" perspective) occurs if the viewer is observing a nonlinear scene. The most common example of a nonlinear scene is a natural scene (ie, a mountain range) which frequently does not contain any parallel lines. Other examples include: a random (ie, not aligned in a three-dimensional Cartesian coordinate system) arrangement of spherical objects, a scene composed entirely of three-dimensionally curvilinear strings, or a scene consisting of lines where no two are parallel to each other. Orthographic projections also do not have vanishing points, but they are not perspective constructions and are thus not equivalent to a "zero-point" perspective. Note that a perspective without vanishing points can still create a sense of "depth," as is clearly apparent in a photograph of a mountain range (for example, more distant mountains have smaller scale features).

### Other varieties of linear perspective

One-point, two-point, and three-point 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 infinite-point 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.

### Varieties of nonlinear perspective

Typically, mathematically constructed perspectives are "linear" in that the ratio at which more distant objects decrease in size is constant (ie, graphing the drawn size of a one-foot object versus the distant from viewer will form a straight line). It is conceivable to have non-linear perspectives — those in which the graph of the ratio mentioned above does not form a straight line.

A panorama is a perspective projected onto a cylinder. The actual drawing can be drawn onto a cylinder (typically on the interior surface and viewed from the inside the cylinder) or onto a flat surface, equivalent to "unrolling" the cylinder. A panorama (projection onto a cylinder) removes one of the differences between artificial perspective projection (projection onto a flat surface) and natural perspective projection (projection onto a spherical surface). A standard Mercator map projection is similar to a panorama.

## Methods of Constructing Perspectives

Several methods of constructing perspectives exist, including:

• Freehand sketching (common in art)
• Graphically constructing (once common in architecture)
• Using a perspective grid
• Computing a perspective transform (common in 3D computer applications)
• Mimicry using tools such as a proportional divider (sometimes called a variscaler)

### Example: a square in perspective

One of the most common, and earliest, uses of geometrical perpective is a checkerboard floor. It is a simple but striking application of one-point perspective. Many of the properties of perspective drawing are used while drawing a checkerboard. The checkerboard floor is, essentially, just a combination of a series of squares. Once a single square is drawn, it can be widened or subdivided into a checkerboard. Where necessary, lines and points will be referred to by their colors in the diagram.

To draw a square in perspective, the artist starts by drawing a horizon line (black) and determining where the vanishing point (green) should be. The higher up the horizon line, the lower the viewer will appear to be looking, and vice versa. The more off-center 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 two-point 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 perpindicular 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 furthest edge of the square. The line just drawn represents the ray of light travelling from the viewer's eye to the furthest edge of the square. 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 y-axis 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 furthest side (see Panofsky).

## Foreshortening

Foreshortening refers to the visual effect or optical illusion that an object or distance is 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 two-dimensional representations of three-dimensional 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").

## Other Perspective Topics

The following topics are not critical to understanding perspective, but provide some additional information related to perspectives.

### Limitations of perspective

Perspective images are calculated assuming a particular vantage point. In order for the resulting image to appear identical to the original scene, a viewer of the perspective must view the image from the exact vantage point used in the calculations relative to the image. This cancels out what would appear to be distortions in the image when viewed from a different point. These apparent distortions are more pronounced away from the center of the image as the angle between a projected ray (from the scene to the eye) becomes more acute relative to the picture plane.

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 theorectical 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 z-axis) will suffice (similarly, if the desired large field of view is only in the vertical direction of the image, a horizontal cylinder will suffice). A cylindrical picture surface will allow for a projected ray image up to a full 360 degrees in either the horizontal or vertical dimension of the perspective image (depending on the orientation of the cylinder). In the same way, by using a spherical picture surface, the field of view can be a full 360 degrees in any direction (note that for a spherical surface, all projected rays from the scene to the eye intersect the surface at a right angle).

Just as a standard perspective image must be viewed from the calculated vantage point for the image to appear identical to the true scene, a projected image onto a cylinder or sphere must likewise be viewed from the calculated vantage point for it to be precisely identical to the original scene. If an image projected onto a cylindrical surface is "unrolled" into a flat image, different types of distortions occur: For example, many of the scenes straight lines will be drawn as curves. An image projected onto a spherical surface can be flattened in various ways, including:

• an image equivalent to an unrolled cylinder
• a portion of the sphere can be flattened into an image equivalent to a standard perspective
• an image similar to a fisheye photograph

### The myth of one-, two- and three-point perspectives

One-point, two-point, and three-point 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. For example, the three images illustrating one-, two- and three-point perspective in the above section can be generated in two ways with identical results:

• the "standard" way would be to alter the viewer's position in each perspective with a stationary cube
• identical to this is to simply rotate the cube in space in front of a stationary viewer

A practical use of this fact is an alternative quick and accurate method of generating a "two-point" perspective by hand: The two vanishing points of the perspective can be generated by simply mapping a rotated grid (of any arbitrary angle) onto a standard "one-point" perspective grid (the grids should be of the same unit spacing to facilitate the construction of the actual drawing).

### Geometric transforms

A perspective drawing, whether roughly sketched (ie, intuitively by freehand) or precisely calculated (i.e., using matrix multiplication on a computer or other means), is usually a combination of two geometric transforms:

• A perspective transform: a perspective projection onto a typically flat picture plane (or painting plate) of a scene from the viewpoint of an observer
• A similarity transform: a scaling of the picture plane from the first transform onto an actual drawing of a usually smaller size.