The Column ProblemSo far, in our discussion of linear perspective, the basic setup was that the perspectival image is the intersection of the picture plane and the cone of light rays reflecting off the physical scene. As such, the perspectival image is an absolutely faithful imitation of the physical scene. Namely, an appropriately situated viewer of the picture sees an image that is optically identical with what they would see had they been physically in front of the actual scene at a correspondingly prescribed distance. We now examine what might happen when the viewer's eye is not fixed in space, in the context of what's commonly known as "The Column Problem."
Suppose we wish to depict a row of evenly spaced Greek columns parallel to the picture plane (i.e. front facing). Two rules of linear perspective relevant to this construction are as follows:
- ►Physical ellipses poject to pictorial ellipses.
- ►Linear perspectival projection preserves tangency. In other words, if a physical line is tangent to a physical object, then their pictorial projections are also tangent.
Moreover, the projected images of the columns would appear wider and wider, and spaced closer and closer, towards either end of the picture plane.
All of that is perfectly fine with regard to the rules of linear perspective, and would in fact create the correct optical illusion, provided that the spectator views the painting from a uniquely prescribed height and distance with one immobile eye.
In reality, that is almost never the
case, and to a casual viewer strolling by, the columns would just appear distorted.
This phenomenon has been a subject of serious consideration since the Renaissance,
by such eminent figures as Piero della Francesca and Leonardo da Vinci.
In practice, painters who utilize linear perspective sometimes "break the rules" when it comes to circles and columns. Dora Norton opined in Freehand Perspective and Sketching that:
... cylindrical objects, however placed, should be drawn as if for those objects alone... But this does not apply to the straight-line portions of the picture..., nor to the placing of the cylindrical parts, nor to their height. These must be determined in the ordinary way [(i.e. using standard rules of linear perpsective)].
Freehand Perspective and Sketching, Dora M. Norton.
Hence, in a painting which otherwise adheres to the rule of linear perspective with absolute precision, the base of a cylindrical column off to the side may be depicted as a level ellipse, instead of a slanted one.
The basic optical mechanism of the human eye is not unlike that of a camera. The chief difference is that an inverted image of the physical scene is now projected onto the spherical (or more accurately quasi-spherical) surface of the retina. This projection is a type of curvilinear perspective. In the eyeball situation, the projected retinal image is subject to barrel distortion, where every physical line projects to a line which terminates at a vanishing point (whereas in the case of linear perspective, a transversal line has no vanishing point.)