Curvilinear perspective
Encyclopedia
Graphical projection
Graphical projection is a protocol by which an image of a three-dimensional object is projected onto a planar surface without the aid of mathematical calculation, used in technical drawing.- Overview :...
used to draw 3D objects on 2D surfaces. It was formally codified in 1968 by the artists and art historians André Barre and Albert Flocon in the book La Perspective curviligne, which was translated into English in 1987 as Curvilinear Perspective: From Visual Space to the Constructed Image and published by the University of California Press
University of California Press
University of California Press, also known as UC Press, is a publishing house associated with the University of California that engages in academic publishing. It was founded in 1893 to publish books and papers for the faculty of the University of California, established 25 years earlier in 1868...
.
Background
In 1959, Flocon had acquired a copy of Grafiek en tekeningen by M. C. EscherM. C. Escher
Maurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...
who strongly impressed him with his use of bent and curved perspective, which influenced the theory Flocon and Barre were developing. They started a long correspondence, in which Escher called Flocon a "kindred spirit".
Horizon and vanishing points
Rectilinear
Rectilinear may refer to:* Rectilinear grid, a tessellation of the Euclidean plane* Rectilinear lens, a photographic lens* Rectilinear locomotion, a form of animal locomotion* Rectilinear polygon, a polygon whose edges meet at right angles...
to approximate the image on the retina of the eye, which is itself spherical, more accurately than the traditional linear perspective, which uses straight lines and gets very strangely distorted at the edges.
It uses either four or five vanishing point
Vanishing point
A vanishing point is a point in a perspective drawing to which parallel lines not parallel to the image plane appear to converge. The number and placement of the vanishing points determines which perspective technique is being used...
s:
- In five-point (fisheyeFisheye lensIn photography, a fisheye lens is a wide-angle lens that takes in a broad, panoramic and hemispherical image. Originally developed for use in meteorology to study cloud formation and called "whole-sky lenses", fisheye lenses quickly became popular in general photography for their unique, distorted...
) perspective: Four vanishing points are placed around in a circle, they are named N, W, S, E, and one vanishing point in the center of the circle. - Four, or infinite-point perspective is the one that (arguably) most approximates the perspective of the human eye, while at the same time being effective for making impossible spaces, while five point is the curvilinear equivalent of one point perspective, so is four point the equivalent of two point perspective.
This technique can, like two-point perspective, use a vertical line as a horizon line, creating both a worms and birds eye view at the same time. It uses four or more points equally spaced along an horizon line, all vertical lines are made perpendicular to the horizon line, while orthogonals are created using a compass set on a line made at a 90-degree angle through each of the four vanishing points.
History
Earlier, less mathematically precise versions can be seen in the work of the miniaturist Jean FouquetJean Fouquet
Jean Fouquet was a preeminent French painter of the 15th century, a master of both panel painting and manuscript illumination, and the apparent inventor of the portrait miniature. He was the first French artist to travel to Italy and experience at first hand the Italian Early...
. Leonardo da Vinci
Leonardo da Vinci
Leonardo di ser Piero da Vinci was an Italian Renaissance polymath: painter, sculptor, architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist and writer whose genius, perhaps more than that of any other figure, epitomized the Renaissance...
in a lost notebook spoke of curved perspective lines.
Examples of approximated (not necessarily systematically constructed, but emulated through an empirical method) five-point perspective can also be found in several mannerist
Mannerism
Mannerism is a period of European art that emerged from the later years of the Italian High Renaissance around 1520. It lasted until about 1580 in Italy, when a more Baroque style began to replace it, but Northern Mannerism continued into the early 17th century throughout much of Europe...
paintings such as the curved mirror in Jan van Eyck
Jan van Eyck
Jan van Eyck was a Flemish painter active in Bruges and considered one of the best Northern European painters of the 15th century....
's Arnolfini's Wedding as well as in the famous self-portrait of Parmigianino
Parmigianino
Girolamo Francesco Maria Mazzola , also known as Francesco Mazzola or more commonly as Parmigianino or sometimes "Parmigiano", was an Italian Mannerist painter and printmaker active in Florence, Rome, Bologna, and his native city of Parma...
seen through a shaving mirror.
The book Vanishing Point: Perspective for Comics from the Ground Up by Jason Cheeseman-Meyer teaches five and four (infinite) point perspective.
Mathematics
If a point has the 3D Cartesian coordinates
then its transformation for a single direction to a curvilinear reference system of radius R with linear distortion is,
For ε < 0 one gets 1D barrel distortion while for ε > 0 there is pincushion distortion.
For radial distortion in two dimensions the transformation for every radial line is similar with
replaced by 
and so,
The result matches the barrel and pincushion distortion seen above. There are more complicated examples of distortion which depend on the particular system under consideration.
See also
- Graphical projectionGraphical projectionGraphical projection is a protocol by which an image of a three-dimensional object is projected onto a planar surface without the aid of mathematical calculation, used in technical drawing.- Overview :...
- Perspective projection distortionPerspective projection distortionPerspective projection distortion is the inevitable misrepresentation of three-dimensional space when drawn or "projected" onto a two-dimensional surface....
- linear perspective
- M. C. EscherM. C. EscherMaurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...
- Curvilinear coordinatesCurvilinear coordinatesCurvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible at each point. This means that one can convert a point given...
External links
- Drawing Comics - 5-Point Perspective
- House of Stairs by M. C. EscherM. C. EscherMaurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...