Rev'd Nov 3, 1996

This text is in two parts.  The first part 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.

The second part considers the best f/stop to use in a given 
situation and is just as applicable to flat photography as to 3D.

There is an accompanying Excel spreadsheet which contains the
formulas and which you can use to automatically do the calculations.

Part I

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.  (If you are doing macro stereo, and hence
have the lens focussed way out, you should use the effective focal
length of the lens and not the focal length written on the lens.
The effective focal length is easy to obtain - just measure how far 
you moved the lens from the infinity position to achieve focus and 
add that to the focal length written on the lens.)

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) if your
subject is near the camera.  Most stereo cameras do 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

==================================================================

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 

==================================================================

Part II

Below are listed all of the inputs and outputs in the spreadsheet.
The only output which is specific to stereophotography is the 
"maximum allowable stereobase".

Inputs

I1) lens focal length.  This is the focal length of the lens on 
the camera (which should be the same as the focal length of the 
lenses on the viewer).
I2) nearest point in scene.  Enter the distance from the camera 
to the nearest point.
I3) farthest point in the scene.  Enter the distance from the 
camera to the farthest point.
I4) maximum allowable on-film deviation.  You calculate this by 
dividing your format's lens focal length by 30 or by entering a 
value you find acceptable from personal experience.
I5) f/number written on the lens barrel.  This is different from 
the operating f/number (which is an output of the spreadsheet).
This is what you set your f-stop lever or barrel on your lens to.
I6) film & lens resolution in lines per mm.  This is the resolution 
of the film including any fixed errors of the lens.  (The loss of 
resolution due to diffraction was calculated separately.)  60 
lines per mm is a very good figure, however 40 lpmm is probably
more realistic for most systems.

Outputs

O1) distance setting for best average focus.  This figure shows 
where you should focus your lens so that the nearest and farthest 
points in the scene are equally sharp on the film.  After 
calculating this distance, I like to actually set an object at 
this distance temporarily and focus on it with my lens wide open.
O2) effective focal length of the lens.  Since you move the lens 
outward towards the objects when you focus, this tells you how 
long your lens is pretending to be in this situation.  For 
instance, if you have a 35 mm lens and your acceptable on-film 
deviation is 1.7 mm and your near point is 100 mm and your far 
point is 140 mm, the lens will be out at 50 mm.  So you would want 
to use a viewer with 50 mm lenses in it to get the correct 
magnification.
O3) maximum allowable stereobase.  This is the maximum amount you 
can shift your camera on your slide bar without exceeding the 
maximum allowable on-film deviation you input above.
O4) effective f/number.  This is the effective f/number of the lens 
considering you had to focus the lens out and that that lengthened 
the operating focal length.  Use this to calculate lighting required.
05) geometric resolution.  This is the resolution at the eye due to 
the geometry of the lens setup.  If you open up the lens too much,
the geometric resolution will go someplace in a handbasket.
O6) diffraction-limited resolution.  This is the system resolution 
due to diffraction effects.  If you stop down a lot, you'll have a 
small aperture and your resolution will go someplace in a handbasket.
06) film + lens res.  This is the linear resolution of the film and 
the lens converted to resolution at the eye in minutes of arc.
O7) summation of all three resolutions in quadrature.  This is a 
really a suspicious equation I just grabbed out of the air.  I 
don't stand behind it (or anywhere near it, for that matter).  It 
will give you the general idea of your total resolution, however, 
giving roughly the right weight to all the various causes of loss 
of resolution, IMHO (only).  It is at least a figure of merit.

Observations

If you're doing really close work with ordinary camera lenses, 
you're going to lose a tremendous amount of resolution by stopping 
down to increase depth of field.  Play with the f/number input and 
you will see what happens to the resolution due to geometry and due 
to diffraction.  You can balance these resolution losses by 
jockeying the f/number.  When you get all done, it would be nice to 
have reasonably good resolution.  Under sharp, high-contrast subject 
conditions, 1 minute of arc would be right.  If you have a lower-
contrast subject, you might get away with 2 or even 3 minutes of arc.

Additional equations

The derivation of the Bercovitz/Spicer equation is shown above.
Here are the common equations which were used to calculate the
additional values which are equally applicable to flat photography.

    2 * af * an   This gives the correct focusing distance to make
a = -----------   objects at af and an equally sharp (or equally
      af + an     fuzzy, depending on your personal philosophy).

             d*(af - an)    This gives the angular fuzziness due to
rg = Arctan ------------    geometry which will result from using a 
             2 * af * an    linear aperture of diameter d.

      1.92        This gives the angular fuzziness due to diffraction 
rd = --------     which will result from using a linear aperture of 
       d          diameter d.

d is the linear diameter of the aperture and is obtained from the 
f/number and the focal length