T3D From Euclid to Wheatstone

[abram klooswyk <abram.klooswyk@xxxxxx>]

(continued from t3d digest 366, 26 Sep 1998)  

* Binocular Vision
In Euclid's "Optics" there are three theorems on binocular vision, 
and two more which have additions on seeing with both eyes. 
I first quote the famous theorem 25 entirely, including its proof:

"When a sphere is seen by both eyes, if the diameter of the sphere
is equal to the straight line marking the distance of the eyes from
each other, the whole hemisphere will be seen." 

                   G                         E
                 .-|-.-----------------------|
              .-'  |  '-.                    |
            .'     |     '.                  |
           /       |       \                 |
          |       A|________|________________| Z
          |        |        |                |
          \        |        /                |
           '       |       .                 |
            '-.    |    .-'                  |
               '-._|_._'_____________________|      
                   B                         D 
                 
                                                 (ASCII circle copied)

"Let there be a sphere, of which A is the center, and on the sphere
let the circle BG be inscribed about the center A, and let BG be
drawn as its diameter, and at right angles from B and G let lines be
drawn, BD and GE, and let DE be parallel to BG, and upon this (DE),
let D and E represent the eyes. 
I say that the complete hemisphere will be seen. 
Through A let AZ be drawn parallel to each of the lines, BD and GE; 
then ABZD is a parallelogram. Now, if the inscribed figure is 
revolved and then restored to the same position whence it started, 
AZ remaining in its place, it will start from B and B will come over 
G, and the figure formed under AB will be a circle through the center
of the sphere. So a hemisphere will be seen by the eyes D and E."
       
Theorem 26 adds: "If the distance between the eyes is greater than 
the diameter of the sphere, more than the hemisphere will be seen."  
And of course in 27: "If the distance between the eyes is less then
the diameter of the sphere, less than a hemisphere will be seen."
 
There are also theorems on vision of a cylinder and on vision of a
cone with one eye, both have an addition on binocular vision: 
(28) "If a cylinder should be seen by two eyes, it is clear that
what has been said regarding the sphere will be true also in this
case." 
(31) "And it is clear that in the case of a cone, when seen by both
eyes, the same things will occur as in the case of the sphere and the
cylinder, when seen in the same way."

* Discussion
This is ALL Euclid EVER wrote on vision with both eyes. The not quoted
proofs are again purely geometrical like the proof of theorem 25.

Euclid was a great mathematician, "The Elements" is after the Bible
the most translated and printed work *of all times*, as model for
logical reasoning it has probably influenced western science more
than any other work. Therefore it is a little shocking to see that
theorem 25 is in error (and 26 inaccurate).
  
For, let there be an interpupillary distance of 65 mm, and let a 
globe of 65 mm diameter be held so that the plane of the equator is at
eye level, then exactly half of the equator is  seen with both eyes.
I say you cannot see the North pole nor the South pole. 
For looking around the object works only horizontally (with 
horizontal eyes). So a hemisphere will not be seen.

This is not to blame Euclid (would be rather ridiculous after 23 
centuries !), but the error is significant with respect to our 
understanding of the evolution of space concepts. 
It is clear that Euclid knew that in looking at rounded objects the 
left eye sees a the part of the surface that the right cannot see, 
and vice versa. So two eyes see MORE of the surface of such an 
object. Note that theorem 23 and 24, which directly precede the
first 'binocular' theorem, also are on seeing parts of the surface
of a sphere, but with one eye.
But seeing more of a surface with two eyes doesn't mean depth 
perception.

Wheatstone made his stereodrawings applying geometrical techniques
which are based on Euclid's geometry. For his 1838 demonstrations 
with stereodrawings he intentionally used few objects with rounded 
surfaces, but (in his own words):
"For the purpose of illustration I have employed only outline
figures (... so) no room is left to doubt that the entire effect of
relief is owing to the simultaneous perception of two monocular
projections, one on each retina." 

The point is that Wheatstone used mostly projections of wireframes,
and ALL points of those objects are seen by BOTH eyes, there is no
"looking around" or seeing more or less of one side of the object.
Different projections is were stereopsis is about, binocular 
parallax and retinal disparity are other terms used.
It's not just dissimilar images which differ in the amount of 
surface seen, as Brewster suggests several times. 

* World Image
The fact that Euclid was in error especially on the binocular issue 
is not accidental. Studying binocular vision was beyond the scope of 
his time.
Euclid knew that two eyes see more than one, but he meant it in a
quantitative sense: seeing more of a surface. 
Wheatstone for the first time demonstrated a qualitative binocular 
effect: two eyes see depth. 
Between the two men lie over 2000 years and a world of
developments in culture, science and awareness of space.
Concepts in antiquity were so different that assuming early 
stereoscopic imaging is an unjustified anachronism.

The important thing is that looking from cyberspace age we only
faintly can imagine how people in other centuries saw the world.
Euclid's mathematical works could have been sufficient to make 
drawings or paintings with central perspective, but not before 
the 15th century Italian painters actually used central perspective. 
Again, simple stereo drawings as Wheatstone published in 1838 could 
have been made in any age, if only the concepts of binocular 
stereoscopic vision would have been clear. 
But there is no record of awareness of binocular depth perception 
before Leonardo da Vinci, and even he made no stereo pictures. 
The conclusion must be that making binocular stereo images was 
impossible before Wheatstone, because people looked at the world 
in a different way.

Abram Klooswyk


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