DISTORTION
Figure 1
Figure 1 shows the principle of an Artificial Perspective projection. The Artificial Perspective projection appears upon the Projection Plane. Imagine a human eye is placed at the Station Point. This human eye views both the object and the image of the object as if the Projection Plane did not exist. That is to say, to the eye, from the Station Point (only) the image of the object is indistinguishable from the object itself.
In the special case - and the only instance - in which an Artificial Perspective image appears the same to the eye as does a Natural Perspective image, the eye must view the Artificial Perspective image from precisely the spatial Station Point, i.e., from the Station Point as it is located in actual space.
Figure 2
To illustrate, Figure 2 shows the object in Figure 1 to be pivoted about the imagined eye to a new position, while maintaining its original face toward the eye. The new object image will appear the same to the eye as at the original position. But take note that the Artificial Projection image of the object in its new position is different from the original Artificial Projection image. In other words, the two Artificial Projection images are different because the angles of the respective projectors as they intersect the Picture Plane are different. The pivoted image thus exhibits an obvious distortion over the unpivoted image. However, both images appear identical to the eye because the eye is located at the Station Point and the angle of viewing from the Station Point to the new image cancels out the new angles of the projectors. Or to say it another way, by looking at the distorted image in a distorted (skewed) direction the image appears undistorted to the eye.
It is not immediately apparent on a day-to-day basis that there is distortion in the image beheld by the eye because we experience nothing else visually throughout our lives, but on reflection there is an obvious contradiction that has not been fully explained: Why do straight parallel lines in space appear to intersect twice? The following thought experiment illustrates the contradiction:
- Imagine that on an infinite plane there is an infinitely long and infinitely straight railroad, and that you are standing between the parallel rails. As you peer down the track in one direction the rails appear to intersect (on the horizon). As you peer in the opposite direction they again appear to intersect. You look at your feet and the rails are far apart. It logically follows that if they intersect, as it would appear, one or both of the rails are curved. To determine which, you put your eye onto one rail and sight down it. You discover it is straight in the sighted direction. You then sight down the same rail in the opposite direction. You discover it is straight also in the opposite direction. It logically follows, then, that the other rail must be curved. You similarly test the other rail and discover it too is straight. How can this happen?
Figure 3
The explanation lies in an extension of Alhazen's discovery: the conical projections of spatially, parallel, straight lines create intersecting straight-line images on surfaces of projection. However, for the spatially parallel lines to appear straight and also to appear to intersect twice, the surface of projection must be spherical and viewed from the center. See Fig. 3. (The brick wall in the background serves as a representative object to be projected.) This would produce a true simulation of that which the eye beholds.
To understand the process, be reminded that (a) perspective projections of straight lines that are in space, through the center of a sphere, projectively create great circles upon the (picture) sphere, and (b) these great circles appear as straight lines when viewed from the center of the sphere. These are the reasons why the Natural Perspective Projection of the rails of our thought experiment actually produce two intersecting (at opposite poles of the sphere) circular images upon the Picture Sphere (retina) and yet appear straight when viewed from the center. For this to be consistent with our normal vision requires that the retina, although too small to be easily detected, be spherical in shape - a phenomenon even ophthalmologists often miss. And of course, the image is inverted before striking the retina, but which has no barring on the theory.
Therefore: to produce the image which the retina beholds, using projective geometry techniques, substitute a spherical surface of projection for a plane of projection, i.e., use a "picture sphere" rather than a "picture plane" and place the eye, or "station point," at the center of the picture sphere.
Figure 4
For a step-by-step example, see Figures 4-8.
Figure 4 shows that the perspective projection of the horizontal top line of a wall intersects at the retina (actually, in true life, at the focal point slightly in front of the retina and from where it is inverted). The intersecting lines define a plane that intersects the sphere of projection in a great circle.
Figure 5
Figure 5 shows the projection of the horizontal bottom line of the wall and its great circle of intersection.
Figure 6
Figure 6 shows that the two great circles of intersection intersect. The two points of intersection are two (vanishing) points on the horizon line of the plane of the wall as seen from the center of the picture sphere. The points of intersection are analogous to the two points of intersection as one peers down a railroad track in both directions.
Figure 7
Figure 7 shows a similar determination of the two vanishing points for the spatially vertical lines of the wall. Note that this correct “four point” perspective is analogous to the distorted “three point” perspective of classical perspective projection.
Figure 8
Observe at this stage that a horizon line of the wall is not yet determined. Figure 8 demonstrates that a horizon line is determined by vanishing points only.
CONCLUSION
A spherical projection surface is required in order to represent without distortion the image beheld by the eye. (This conclusively demonstrates that the retina itself is spherical in shape.) As it is impossible for an image on a sphere to be rectified without distortion onto a plane surface, the image that the eye beholds may never be represented without distortion on a plane surface. An analogy is the impossibility of representing the surface of the earth without distortion on a flat surface (map). Note that it logically follows that all photography distorts the image that is beheld by the eye because the film surface is flat in the manner of the Picture Plane. While artifactual characteristics of a camera lens may aggravate the distortion, that distortion is only ancillary to the fundamental cause of distortion described here. This is demonstrated with a pinhole camera which has no lens but which produces the same distortion as described herein.
