Curvilinear perspective

Curvilinear barrel distortion

Curvilinear pincushion distortion

Curvilinear perspective is a graphical projection 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,^[1] 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.^[2]

Background

In 1959, Flocon had acquired a copy of Grafiek en tekeningen by M. C. Escher 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".^[2]

Horizon and vanishing points

A comparison of the same object displayed, on the left using a curvilinear perspective and on the right, using a vanishing point

Curvilinearity in photography: Curvilinear (above) and rectilinear (below) image. Notice the barrel distortion typical for fisheye lenses in the curvilinear image. While this example has been rectilinear-corrected by software, high quality wide-angle lenses are built with optical rectilinear correction.

The system uses curving perspective lines instead of straight converging ones 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 points:

• In five-point (fisheye) 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 Fouquet. Leonardo da Vinci in a lost notebook spoke of curved perspective lines.^[2]

Examples of approximated (not necessarily systematically constructed, but emulated through an empirical method) five-point perspective can also be found in several mannerist paintings such as the curved mirror in Jan van Eyck's Arnolfini's Wedding as well as in the famous self-portrait of Parmigianino 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 $\left(x_1,y_1,z_1\right)$ then its transformation for a single direction to a curvilinear reference system of radius R with linear distortion is,

$d_1=\sqrt{ x_1^2 + y_1^2 + z_1^2}$

$x_2=R\ \dfrac{x_1}{d_1} \left(1+\epsilon \left| \dfrac{x_1}{d_1} \right| \right)$

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 x₁ replaced by r₁ and so,

$r_1=\sqrt{x_1^2+y_1^2}$

$\begin{pmatrix}x_2\\y_2\end{pmatrix}=\dfrac{R}{d_1}\left(1+\epsilon\ \dfrac{r_1}{d_1} \right) \begin{pmatrix}x_1\\y_1\end{pmatrix}$

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.

Examples

• 

  Jean Fouquet, Arrival of Emperor Charles IV at the Basilica St Denis

• 

  Parmigianino, Self-portrait in a Convex Mirror

See also

• Graphical projection
• Perspective projection distortion
• linear perspective
• M. C. Escher
• Curvilinear coordinates

References

1. ^ Albert Flocon and André Barre, La Perspective curviligne, Flammarion, Éditeur, Paris, 1968
2. ^ ^{a b c} Albert Flocon and André Barre, CurvilinearPerspective: From Visual Space to the Constructed Image, (Robert Hansen, translator), University of California Press, Berkely and Los Angeles, California, 1987 ISBN 0520059794

External links

• Drawing Comics - 5-Point Perspective
• House of Stairs by M. C. Escher