Staircase perspective.
Perspective (from
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 perceived by the eye. The two most
characteristic features of perspective are that objects are
drawn:
 Smaller as their distance from the observer increases
 Foreshortened: the size of an object's dimensions along the
line of sight are relatively shorter than dimensions across the
line of sight (see later).
Overview
A cube in twopoint perspective.
Rays of light travel from the object,
through the picture plane, and to the viewer's eye.
This is the basis for graphical perspective.
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. 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 a 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 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 to (and
named after) the Earth's
horizon.
Any perspective representation of a scene that includes
perpendicular 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.
History
Early history
The earliest art paintings and drawings typically sized objects and
characters hieratically according to their spiritual or thematic
importance, not their distance from the viewer, and did not use
foreshortening. The most important
figures are often shown as the highest in a composition, also from
hieratic motives, leading to the "
vertical perspective", common in the
art of Ancient Egypt, where a
group of "nearer" figures are shown below the larger figure or
figures. The only method to indicate the relative position of
elements in the composition was by overlapping, of which much use
is made in works like the
Parthenon
Marbles.
Systematic attempts to evolve a system of perspective are usually
considered to have begun around the 5th century B.C. in the
art of Ancient Greece, as part
of a developing interest in
illusionism
called
skenographia, allied to
theatrical scenery; using flat panels on a stage to give the
illusion of depth. The philosophers
Anaxagoras and
Democritus worked out geometric theories of
perspective for use with skenographia.
Alcibiades had paintings in his house designed
based on skenographia, thus this art was not confined merely to the
stage.
Euclid's
Optics introduced a
mathematical theory of perspective; however, there is some debate
over the extent to which Euclid's perspective coincides with a
modern mathematical definition of perspective.
By the later periods of antiquity artists, especially those in less
popular traditions, were well aware that distant objects could be
shown smaller than those close at hand for increased illusionism,
but whether this convention was actually used in a work depended on
many factors.
Some of the paintings found in the ruins of
Pompeii show a remarkable realism and perspective for their
time; it has been claimed that comprehensive systems of perspective
were evolved in antiquity, but most scholars do not accept
this. Hardly any of the many works where such a system would
have been used have survived. A passage in
Philostratus suggests that classical artists
and theorists thought in terms of "circles" at equal distance from
the viewer, like a classical semicircular theatre seen from the
stage. The roof beams in rooms in the
Vatican Virgil, from about 400CE, are shown
converging, more or less, on a common vanishing point, but this is
not sytematically related to the rest of the composition. In the
Late Antique period use of perspective techniques declined. The art
of the new cultures of the
Migration
Period had no tradition of attempting compositions of large
numbers of figures and Early Medieval art was slow and inconsistent
in relearning the convention from classical models, though the
process can be seen underway in
Carolingian art.
A clearly modern optical basis of perspective was given in 1021,
when
Alhazen, an
Iraqi physicist and
mathematician, in his
Book of Optics, 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
mathematically difficult, so a drawing constructed using them would
be incredibly time consuming.
Medieval art was aware of the general principle of varying the
relative size of elements according to distance, but even more than
classical art was perfectly ready to overide it for other reasons.
Buildings were often shown obliquely according to a particular
convention. The use and sophistication of attempts to convey
distance increased steadily during the period, but without a basis
in a systematic theory.
Byzantine art
was also aware of these principles, but also had the
reverse perspective convention for the
setting of principal figures.
Giotto 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.
One of Giotto's first uses of his algebraic method of perspective
was
Jesus
Before Caiaphas. Although the picture does not conform to
the modern, geometrical method of perspective, it does give a
considerable illusion of depth, and was a large step forward in
Western art.
With the exception of dice,
heraldry
typically ignores perspective in the treatment of
charges, though sometimes in later
centuries charges are specified as
in perspective.
Renaissance : Mathematical basis
One
hundred years later, in about 1413, 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,"...and these works (of
perspective by Brunelleschi) were the means of arousing the minds
of the other craftsmen, who afterwards devoted themselves to this
with great zeal."
Vasari's
Lives of the Artists Chapter on Brunelleschi
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)"Messer Paolo dal
Pozzo Toscanelli, having returned from his studies, invited Filippo
with other friends to supper in a garden, and the discourse falling
on mathematical subjects, Filippo formed a friendship with him and
learned geometry from him."
Vasarai's
Lives of the Artists, Chapter on Brunelleschi,
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.
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. 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, 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:
Image:Perspectivephoto.jpgImage:One point
perspective.jpgImage:RailroadTracksPerspective.jpg
Twopoint perspective
[[File:2ptsketchup.jpgthumb300pxrightWalls in 2pt
perspective.
Walls converge towards 2 vanishing points.
All vertical beams are parallel.
Model by "The Great One" from 3D Warehouse.
Rendered in SketchUp.]]
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
[[File:3ptperspective.jpgthumbright300pxThreepoint
perspective rendered from computer model by "Noel" from Google 3D
Warehouse.
Rendered using IRender nXt.]]
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.
Zeropoint perspective
Due to the fact that 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).
Other varieties of linear perspective
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.
Methods of construction
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
Rays of light travel from the eye to
an object.
Where those rays hit the picture plane, the object is
drawn.
One of the most common, and earliest, uses of geometrical
perspective is a
checkerboard floor.
It is a simple but striking application of onepoint 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
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
Plato was one of the first to discuss the
problems of perspective. "Thus (through perspective) every sort of
confusion is revealed within us; and this is that weakness of the
human mind on which the art of conjuring and of deceiving by light
and shadow and other ingenious devices imposes, having an effect
upon us like magic...And the arts of measuring and numbering and
weighing come to the rescue of the human understandingthere is the
beauty of them and the apparent greater or less, or more or
heavier, no longer have the mastery over us, but give way before
calculation and measure and weight?"
Perspective images are calculated assuming a particular vanishing
point. In order for the resulting image to appear identical to the
original scene, a viewer of the perspective must view the image
from the exact vantage point used in the calculations relative to
the image. This cancels out what would appear to be distortions in
the image when viewed from a different point. These apparent
distortions are more pronounced away from the center of the image
as the angle between a projected ray (from the scene to the eye)
becomes more acute relative to the picture plane. In practice,
unless the viewer chooses an extreme angle, like looking at it from
the bottom corner of the window, the perspective normally looks
more or less correct. This is referred to as "Zeeman's Paradox." It
has been suggested that a drawing in perspective still seems to be
in perspective at other spots because we still perceive it as a
drawing, because it lacks depth of field cues."...the paradox is
purely conceptual: it assumes we view a perspective representation
as a retinal simulation, when in fact we view it as a two
dimensional painting. In other words, perspective constructions
create visual symbols, not visual illusions. The key is that
paintings lack the depth of field cues created by binocular vision;
we are always aware a painting is flat rather than deep. And that
is how our mind interprets it, adjusting our understanding of the
painting to compensate for our position."
http://www.handprint.com/HP/WCL/perspect1.html Retrieved on
December 25, 2006
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:
 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
See also
Notes
 Perspective Drawing Handbook By Joseph D'Amelio, p. 19,
published by Dover Publications
 Panofsky, 122, note 1
 :File:VaticanVergilFol040rDeathOfDido.jpg
Vatican Virgil image
 Panofsky, p. 127

http://commons.wikimedia.org/wiki/Category:Objects_in_heraldry
 Plato's Republic, Book X, 602d.
http://etext.library.adelaide.edu.au/mirror/classics.mit.edu/Plato/republic.11.x.html
 Mathographics by Robert Dixon New York: Dover, p. 82,
1991.
References
Further reading
External links