Perspective projection
Perspective projection is a type of drawing that graphically approximates on
a planar (two-dimensional) surface (e.g. paper) the images of three-dimensional
objects so as to approximate actual visual perception. It is sometimes also
called perspective view or perspective drawing or simply
perspective.
All perspectives on a planar surface have some degree of distortion, similar to
the distortion created when portraying the earth's surface on a planar map.
Linear perspective
Linear perspective is the art of representing three-dimensional constructions
on a two-dimensional surface. It presupposes a fixed viewpoint and a desire to
create an "objective" recording of one's visual experience - two conditions
which have been the most dominant in the Western art of the past
half-millennium.
Once the observer assumes a single point of view, several conclusions follow
logically. The first and most important is this: objects appear to get smaller
as their distance from the observer increases. That this is not self-evident in
art is apparent from even a casual perusal of the art of other cultures and eras
- most frequently objects are drawn or painted a certain size for reasons that
have nothing to do with their position in space. In medieval Last Judgement
paintings, for example, the relative scale of the various figures is determined
only by their sacred significance; the most important are the largest.
Once the diminishment of scale with distance is noted, it is an easy step to
understanding why the space between parallel lines must also appear to diminish.
A wall retreating from the observer will appear to get progressively shorter,
and the top and bottom edges of the wall will thus appear to move closer
together.
It was an enormous conceptual leap when artists concluded that these lines,
if extended indefinitely, would appear to meet at a single point on the horizon.
This idea, long since verified, was likely made from a theoretical analysis of
the process of seeing rather than direct observation. Van Eyck, for example, was
unable to create a consistent structure for the converging lines in paintings
like London's The Arnolfini Portrait because he was not aware of the theoretical
breakthrough just then occurring in Italy.
Below are descriptions of the three main varieties of perspective technique.
Note that the only difference among these three varieties is the orientation of
linear objects being viewed relative to the viewer.
1. One-Point Perspective
If the viewpoint is pointing directly into a linear object like a building or
a road, one would use one vanishing point, that is the principal focus. All
lines perpendicular to the painting plate would vanish in the
vanishing point.
More precisely, one-point perspective exists when the painting plate (also
known as the picture plane) is parallel to a "Cartesian scene" (see also
Cartesian coordinate system) --- a scene which is composed entirely of linear
elements that intersect only at right angles. Therefore, 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 on the horizon.
2. Two-Point Perspective
If the lines have angles to the painting plate, they would vanish in the
other vanishing points. There are lot of
vanishing points homologous to
different angles. But all
vanishing points should be located in the same
horizontal line with the focus.
In other words, two-point perspective is derived from one-point perspective
by yawing the line of vision so that the line of vision will be at an acute
angle away from the focus. Then the lines which used to be horizontal and
parallel will now be concurrent, intersecting at the horizon. Interpreted
according to projective geometry, the horizontal parallel lines of one-point
perspective are actually concurrent, intersecting at the point at infinity
[1:0:1]. When the head is turned by a slight angle, these lines no longer
intersect at an ideal point, but at an affine point on the horizon, so they are
no longer parallel.
More precisely, two-point perspective exists when the painting plate is
parallel to a "Cartesian scene" (a scene composed entirely of linear elements
intersecting only at right angles) in one axis (usually the z-axis) but not
parallel to the other two axes. Note that 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.
3. Three-point Perspective
If the lines have angles from the painting plate up or down, one would use
the other kind vanishing points.
Those vanishing points must located
in the same vertical line with the focus. Looking at the object from above or
below, the horizontal line with the focus and all other 2nd vanishing points
would left the horizon up or down.
Three-point perspective exists when the image plane is viewing a
"Cartesian scene" (a scene composed entirely of linear elements intersecting at
right angles) and is not parallel to any of the scenes three axes. Elements that
are parallel to each of the three axes will converge to three vanishing points
respectively.
Other varieties of perspective
A key point to note is that one-point, two-point, and three-point perspective
are dependent on the structure of the scene being viewed. These only exist for
strict Cartesian scenes.
Note that 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.
Due to the fact that vanishing points exist only when parallel lines are
present in the scene, a zero-point perspective is also possible if the viewer is
observing a nonlinear scene. One example is a random (ie, not aligned in a
three-dimensional Cartesian coordinate system) arrangement of spherical objects.
Another would be a scene composed entirely of three-dimensionally curvilinear
strings. A third example would be a scene consisting of lines where no two are
parallel to each other.
See also
