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Now, let's flip the idea around. If we can figure out the laws which govern the placement of the marks by the light rays reflecting off the physical scene, then we could start with a blank canvas, place marks on it according to the same laws, and produce an illusion of a three-dimensional physical scene contained within the canvas (whether or not the scene thus illustrated actually exists in the real world).
What are these laws? We name a few here. In the following:
In particular, parallel transversal lines project to parallel pictorial lines.
It is remarkable that with only these simple rules, and a little bit of ingenuity in their application, some great visual results could be obtained. For example:
Besides employing the geometric rules outlined above to depict objects viewed in perspective, there are various mechanical or semi-mechanical means to produce the same effect. In the following illustration by Albrecht Dürer, the "imaginary screen" we used in devising linear perspective is realized physically as a gridded net (grid #1) placed between a draftsman and the physical scene (in this case a reclining nude). Before the draftsman lies a drawing surface with a similar grid system (grid #2) overlaid upon it. Viewing the physical scene with one eye placed at a fixed point, various points of the scene lies directly behind points on grid #1. If, say, the tip of the nose of the woman lies behind the grid point on grid #1 which is two from the top and one from the left, then the draftsman would draw the tip of the nose on the drawing surface at the corresponding point on grid #2. Repeating this process for each point in the physical scene, a perspectival drawing of the scene is obtained.
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:
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.
Leonardo studied this problem of distortion to the roving viewer and concluded that it cannot be avoided, unless the picture plane is situated "at least 20 times as far off as the greatest width or height of the objects represented." In otherwords, if you want to incorporate into your painting a row of columns, and you have not either the means or intention of fixing your viewer's eye at one point in space, and you want to achieve a convincing visual illusion, then you should consider placing any front-facing row of columns far into the background.
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)].
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.) Barrel distortion is one type of curvilinear perspective, the other type being pin-cushion distortion.
In more concrete terms, barrel distortion is demonstrated in Leonardo da Vinci's observation that a long rectangular shape facing the viewer appears to diminish in height towards either end of the field of vision. Likewise, it is barrel distortion at work when a tall building appears narrower and narrower (however slightly) towards the top of our field of vision.
Hence, our visual experience of the world really obeys the laws of curvilinear perpsective, even though we may not think that is the case, since the deviation from linear perspective is often slight, and only clearly noticeable in our peripheral vision.