T3D Re stereo math

[aij.klooswyk@xxxxxxxxx]

I have got several requests to repost here what I had send to Photo-3D 
on angles and so on. I do this with an apology to the double subscribers,
and also to the technical experts on this list. The posting was not 
meant for them and is a little loose in its argumentation.
Moreover, it only partially bears on the present subject, and then on a
very basic level. It ignores working focal length, the subject I feel
I should write on to this list. And it is long (14 kB). 
When I wrote "leave all computational math for a while" I was aware that 
the key word is "computational", and that it is in fact all math. 
(But don't tell the other list :-))

>---Reposting from PHOTO-3D Digest 2902, 
>---Original subject: Re: Re Stereo Base, Re Curious Deep Math Tale, 
>---Re 1/30 rule
>---Posted on Sat, 15 Aug 1998, From: abram.klooswyk@xxxxxx

This long posting is meant to be understandable for formulaphobes, it
contains no formula but requires some imagination. But I am convinced 
that stereographers are better in visualising than average persons.
Basic imagination of lens function is helpful. In my opinion this is 
about general stereo subjects, not specially Tech-3D.

Michael Georgoff wrote (P3D 2890):
> It appears that stereo base is dependent on f-stop.  That is all.
> Over a range of distances, base is the same per each f-stop. 

George Themelis answered (P3D 2891):
>You have just discovered that the formulas for the Depth of Field
>and the Stereoscopic Deviation are very similar.  This is something
>that I discovered myself while "playing" with formulas (as you did)
>and then was happy to confirm that others knew Charles Piper had
>taken advantage of these similarities and worked out a method where 
>you use your Depth of Field Scale on your stereo camera, to
>determine the stereo depth range.

And again (P3D 2893):
>My feeling is that it is independent of FL but dependent on 
>interaxial.

Michael Georgoff again(P3D 2894):
>And I thought optical science and geometry would yield the 
>Holy Grail of Stereo Base Math...

George Themelis (P3D 2898) at last:
>Subject: P3D Last comment on stereo base
>What is the 1/30 rule trying to achieve?
>This rule is trying to limit or optimize the stereo deviation.

--------------------------

To me it seems that Micheal Georgoff has rediscovered 
"Todeschini's Law" (or is about to do it).
F. Todeschini was an Italian stereographer who wrote on stereoscopy
in an Italian photography encyclopedia, and who wrote in 1935 in the
German stereomagazine "Das Raumbild" articles on:
"Die Theorie der kritischen Blende in der Stereoskopie" 
(the theory of the critical f-stop in stereoscopy; The Capitals In
German Always Give Articles Extra Authority). 

In short he says that, because you have to limit the depth range in 
your stereopictures, there is for every case a certain f-stop that
gives a depth of field equal to the maximum depth range.
This has indeed been advocated as a law. 

--(A curious thing is that in Germany not the ol' 1 in 20 rule nor the)
--(ol' l in 25.4 rule nor the ol' 1 in 30 rule was used but the       )
--(ol' 1 in 50 (fifty!) rule. But that's another long and sad story,  )
--(someday I will tell it again, and critise all 1 in x rules and base)
--(formula's.                                                         )

To gain some insight in Micheal Georgoff's "curious, but true, 
artifact" you should leave spreadsheets and all computational math 
for a while and do these things:

1. THINK IN SPACE      2. THINK IN ANGLES      3. USE CAPITALS

Space first. Stereo space reproduction means: there is something in
real life that you want to see back in the stereoviewer. 
Look at a regular stereopicture of the 5P system (OK, Realist, Seton
Rochwite, but expanded to other camera's and defined in Standards)
mounted in a 'Distant' mask. The near point of the view is supposed
to be at the stereowindow at 7 feet and the far point in infinity. 
So your STANDARD STEREO VIEWING SPACE extends from 7 feet to infinity. 
In taking the picture you take care that your objects are also from
7 feet to infinity, this is your PHOTOGRAPHIC SPACE (taking space,
object space, real space, or whatever else seems clear).
When you take closeups from 5 feet you limit the range to 20 feet and
use the 'Medium' mask. Your photographic space is squeezed in number
of feet, but these masks make that your stereo viewing space stays 
the same, 7 feet to infinity. 
So in this case the photographic space is enlarged to fit in the
Standard Stereo Viewing Space. In macro's this Procrustes effect is  
even larger. 

Any stereosystem and any stereo base estimate (skip calculation) 
depends on the possibilities of viewing space. 
The 7 feet window of the 5P system is excellent for projection.
So called double depth mounting moves the stereowindow closer. 
Pictures only for stereoscopes can have much closer windows, and
much larger viewing spaces. It is up to you to manipulate and limit
photographic space to fit in a predefined stereo viewing space.

Now an unrelated virtual experiment. 
Suppose you stand at night in a lane, with a large torch in your hand.
Aiming the beam close to your feet there is an egg shaped oval of
light with the closest point at (say) 1.75 feet and the most far
point of the beam at (say) 3.5 feet. When you redirect the beam so
that the closest point now is at 3.5 feet, the far limit comes at 7
feet. And when the next move brings the near limit to 7 feet, the far
point moves to infinity.  At close distances the range is small (1.75
feet), farther away the range gets larger and larger. 
It gets curiouser and curiouser, for it seems that torches behave
like the depth of field scale on your camera.

The figure below (I'm not experienced in ASCII Art) is meant to be an
aerial view of a Realist camera. Lenses "O" (LL, LR: Left Lens and
Right Lens). The rays in the optical axes of the lenses come from
infinity (8) and hit the film (shutters missing) perpendicular at IL
and IR respectively (central Infinity point on film Left and Right). 
Infinity in this example is also the Far point.
There is a spot marked X (say the top of a pole) 7 feet from the
camera, on the optical axis of the right lens. A "straight" line from
point X through the centre of the left lens runs oblique and reaches
the film at NL (Near point Left). (This my best ASCII try so far, 
best viewed with non-proportional font like classic courier.) 
As X is on the optical axis of the right lens, the Near point Right
(NR) is the same as the IR point.

    HL|
      |
      |
      |
    NL|
      |`-.
 Dev--|   `-.
      |     a`-. LL
    IL|---------`O.------------------------------------------8
      |            `-. a
      |               `-.
      |                  `-.
      |          RL        a `-.
 IR=NR|----------O-------H------`X---------------------------8
      |                3.5      7'
      |
      ^                                      fig. 1
     film plane                             (C) Closearts

In this simplified scheme the focal distance is LL-IL and RL-IR,
infinity separation IR-IL, near point separation NR-NL (greater than
infinity separation, its before transposing). So Deviation is NR-NL
minus IR-IL, or the distance IL-NL. 

In the 5P system focal distance is fixed at 35 mm, and standard
maximum deviation at 1.2 mm (equal to the deviation of the window).
What should be clear is that from these two fixed values follows 
that the Angle "a" also is fixed, the angle between the oblique 
line and the optical axes (the scheme shows three occurrences of
angle "a"). 
This Angle was chosen because it also occurs in Stereo Viewing Space.
In viewing stereoprojection, sitting at the "orthostereo" seat, the
angle between the optical axes of your eyes is zero when you look at
infinity or a far away point of the scene, but when you look at a
point at the stereowindow, or at the window itself, this Angle
between the optical axes of your eyes is the SAME as Angle "a", which
was chosen by Rochwite. (Quiz: How many times is Seton Rochwite 
mentioned in the Stereo Realist Manual?)

Next you take a picture with a near point at 3.5 feet (H, half the
distance of X, again on the right optical axis). A line from H
through the left lens reaches the film at HL (second oblique line not
drawn, is for Master Class ASCII artists). 
Where should the Far point of this second picture be?
The requirements of the Stereo Viewing Space fix the maximum Angle 
between the second line and the first one, it should be equal to the 
standard 5P angle ("a"), which is the angle between the first oblique 
line and the optical axes. It turns out that the far point comes at X, 
the near point of the first picture. 
(OK, in the scheme things don't fit when you draw the lines. But in 
reality the differences are next to zero because focal distance
is small compared with the near point distance.)

Now suppose a string attached to the right Realist lens, extending
along the optical axis. Suppose in the camera you have a bright
shining light spot, a Tiny Torch in stead of a film, exactly at the 
place of the deviation of the first picture, so from IL to NL. 
Then (in the dark) the string would be lit from X to infinity. 
When you move the torch nearer to HL, the lit part of the string 
comes closer, eventually to reach from H to X.
Now this obviously is very similar to the Large Torch Virtual 
Experiment described above. When you move the light beam then the 
Angle it subtends obviously also doesn't change. This the natural 
behaviour of torches with beams having fixed angles. This "curious 
effect" ties the torch to deviation and depth range.

What about changing stereo base? Obviously, when you cut the Realist 
in half (some people would like to do it) and double the separation 
of the lenses, deviation is also doubled, a relation simple to
visualize.
But, since Viewing Space needs always the same deviation, you need to 
double the near point distance,of scenes extending to infinity,
to set the balance.

------------------------------

(continued)
What about depth of field scales? 
There is a lens having an aperture PQ and centre O. The point U is 
supposed to be sharply imaged on the film in point U*. 
A nearer point X will have its focus behind the film in X*, on the 
film it gives an blurred circle CC.
A farther away point Y is imaged before the film in Y*, it will give
another blurred dot on the film.

       |              |
       |              |                  ....---- X - - - U - - - - Y
       |              |         ...---'''     ..-'
       |           .. P ..---'''          .-''
       |  ...---'''   |                .-'
       C '            |             .-' 
  ..-' |         ---- O ----     .-'
X*     U*    Y*       |       .-'
  '-.  |              |    .-'
     ' C --..         | .-'
       |     ''-..    |'
       |          ''- Q 
       |              |         
       |              |         
       ^              ^              fig. 2
       film plane     lens plane    (C) Closearts

(This has to be my last ASCII try for some time. Again, higher
artists could add the lines UP, UQ, PU* and QU*, and similar lines 
joining Y with Y*.)
When the size of dot CC is equal to a predefined limit of definition, 
that means it is supposed to be indistinguishable from a real sharp 
point, XY is the depth of field. When you make the aperture smaller, 
leaving the points X, U, Y where they are, the dot CC on the film 
gets smaller. 

But suppose you keep the dot at the same seize and the lens aperture
too, but you change the focused distance by moving the lens farther
away from the film. 
Another Tiny Torch Virtual Experiment can illustrate this point. 
A string along the line X-U-Y is lit by back projection from a tiny
torch at CC. The part of string which is more or less sharply lit
will vary with the distance of the lens from the film. Again this is
easier to visualize when you think of the Angles involved. 
It is fun to do this experiment in reality with a small torch and a 
magnifying glass in a darkened room, directing the beam obliquely, 
a round cardboard aperture taped to the torch will help. 

Intuitively you will understand the relations between depth of field, 
stereo range, base estimation and torches.
You couldn't expect to get the Holy Grail without intuition?

So Standard Stereo Viewing Space dictates a standard angle between 
infinity and stereowindow, from that follows a deviation limit,
and from that constrains on stereo base and a stereo photography 
depth range limit. This limit is found back in depth of field 
ranges because all is governed by angles.

There is another analogy: stereo acuity and what I have called 
"photographic stereo acuity". Stereo acuity generally is measured
by devices or apparatus with determine the minimum range of depth
a person can discerne. This is not expressed as a length (minimum 
distance between the points) but as an angle. It turns out that this
angle is more or less independent of viewing distance, just what you
would expect from the Famous Torch Virtual Experiments. 
When you replace the Realist for a moment by eyes (some people hate 
to do this), stereo acuity angle resembles Angle "a" fig.1.

Several years ago you could find estimates of "stereo infinity"
in stereo photography literature. This meant the most far point 
which could be seen to be closer than infinity, in viewing stereo 
pictures. I omit details, but I have generalised that case to 
(photographic) stereo acuity at any distance. This again is an 
angle, which depends on several varying technical factors, but is 
supposed to be independent of the distances in Stereo Viewing Space.

Finally, I must say that I dislike all stereo base estimates 
or calculations that want you to fill up totally all available 
viewing space. Some day I'll give the reasons.
 
Abram Klooswyk       abram.klooswyk@xxxxxx
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