closeups & on-film deviation

[P3D John Bercovitz <bercov@xxxxxxxxxx>]

Jim C kind of shamed me into writing a FAQ about the formula for
on-film deviation.  This formula is important when you do closeups.
The FAQ and a couple of Excel files are in my anonymous ftp site.
ftp ftp.netcom.com and go to pub/be/bercov/photography/3D where 
you will find maofd.FAQ and PCmaofd.xls and MACmaofd.xls.  Below is
maofd.FAQ.  Since this is my first crack at it, I would greatly 
appreciate any comments you might have on it.  It is patched together
from previous posts with a little ironing over the seams.
Thanks,
John B

PS: Thanks, Jim; it was about time.  8-)

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This FAQ is a monograph on the subject of maximum allowable on-
film deviation.  It is written for stereophotographers who have 
achieved a level at which they understand common stereo concepts 
such as homologous image points.  It is not intended to be a FAQ 
on the field of stereophotography.

Maximum allowable on-film deviation

For poorly understood reasons, people viewing stereo pictures will 
tolerate only a finite amount of depth within a stereo scene.  If 
this amount of depth is exceeded, the person feels a sense of 
discomfort or strain.  From technical considerations  and from 
long experience, professional stereo photographers have found that 
the allowable amount of depth in a scene corresponds with an 
allowable amount of "on-film deviation".  

So what is "on-film deviation"?  That is probably best illustrated 
by describing how to measure it.  To measure on-film deviation, 
you lay one of the two stereo transparencies (called "film chips") 
on top of a light box and then lay the other chip on top of it.  
Next you find two corresponding image points, one on each chip, 
from the farthest object in the scene.  Align these two points by 
superimposing them.  Now measure the distance between two image 
points of the closest object in the scene with a ruler.  That 
distance is the on-film deviation.

Stereo pictures come in different formats.  Three standard formats 
are five or seven-perf miniature format using lenses of about 
36 mm focal length, dual 35 mm (twin SLR) format using lenses of 
about 50 mm focal length, and medium format using lenses of about 
80 mm focal length.  The maximum allowable on-film deviation will 
be about one thirtieth of the focal length of the format.  So for 
the standard miniature formats (Realist, FED), the maximum 
allowable on-film deviation is 36/30 or 1.2 mm.  For dual 35 mm 
format, it will be 50/30 or 1.7 mm.  For medium format stereo, it 
will be 80/30 or 2.7 mm.

Naturally a person would rather not have to determine whether or 
not he is exceeding the maximum allowable on-film deviation by 
taking a picture and then measuring the deviation.  Many fairly 
simple approximate formulas of varying accuracy and usefulness 
have been offered over the years.  All of them do poorly in 
closeup situations.  Below is the exact solution for all 
situations.  The only constraint on this formula is that the 
lenses must be fairly symmetrical (pupils located very near 
principal points, i.e. not telephoto or retrofocus lenses).  Most 
stereo cameras use fairly symmetrical lenses.

             an*af 
b0  =  d* [ ------- ( 1/f  - 1/a) ]
             af-an 

b0: This is the maximum tolerable separation of the camera's 
lenses.  (The word for this separation of perspective points is 

"stereobase" or "stereobasis".)  If this value is exceeded, the 
limits of on-film  deviation will be exceeded.  If you are using 
a slidebar, it is the amount you are allowed to translate the 
camera.
 
d = maximum allowable on-film deviation.  

an: distance from camera lens to nearest object in scene.
  
af: distance from camera lens to farthest object in scene.

f = the focal length of the camera's lenses.
 
a = the distance at which the camera is focused.
 
The formula has been tested by using it at extreme values which 
the common formulas say will not work.  For instance, using twin 
SLRs (50 mm lenses) for a scene with a near point of 1.8 meters 
and a far point of 2.2 meters, b0 by this formula is 193 mm for an 
easy-to-view 1.0 mm deviation.  This setup has been photographed, 
and the scene is indeed easily viewed by anyone, though all other 
formulas predict that it is not viewable.

Since the formula is somewhat ungainly, Excel spreadsheets have 
been made for the Mac and for the PC.  See MACmaofd.xls and 
PCmaofd.xls.  The spreadsheets also give the focusing distance 
which will make the far and near points equally sharp, and the 
operating focal length of the lenses, since these figures are 
calculated with the same input data.

Acknowledgements

Many thanks, in temporal order, to Bob Mannle, Steve Spicer, and 
David Jacobson for their enormous help.

John B

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Derivation of formula  (for the mathematically inclined)

Dramatis personae:
 
b0: This is the separation we are going to select for the camera's 
lenses.  b0 is also the separation on film of image points of 
objects located at infinity.  In fact, we could use b.sub.infinity 
for this symbol, but that would be cumbersome.
 
bf: This is the separation on film of two image points from the 
farthest object in the scene.
 
bn: This is the separation on film of two image points from the 
nearest object in the scene.
 
d: This is deviation.  It is equal to bf - bn.

dn: This is bn - b0.

df: This is bf - b0. df is not shown below but is used to get (5).

f = The focal length of the camera's lenses.
 
a = distance at which the camera is focused.
 
a' = the distance from the camera lens to the film.  (More 
precisely the distance from the secondary principal point to the 
image plane.)  This varies with the distance at which the camera 
is focused according to the equation 1/f = 1/a + 1/a'.  When a is 
infinity, a' = f; otherwise a' is greater than f (thereby 
increasing d). 
 
af: distance from camera lens to farthest object in scene.
 
an: distance from camera lens to nearest object in scene.

Derivation:

By similar triangles:          (dn/2)/a' = (b0/2)/an        (1)

Rearranging:                          dn = (a'/an)*b0       (2)

By definition:                        bn = b0 + dn          (3)

Substituting 2nd formula into 3rd:    bn = b0 + (a'/an)*b0  (4)

By a similar analysis:                bf = b0 + (a'/af)*b0  (5)

By definition:                         d = bn - bf          (6)

Subbing 4th & 5th formula into 6th:    d = b0*(a'/an - a'/af) 

Solving for b0:                       

            af*an             af*an    1
b0 = d*[--------------] = d*[-------]* -
          af*a'-an*a'         af-an    a'

           an*af 
b0 = d* [ ------- ( 1/f  - 1/a) ]
           af-an 



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