[tech-3d] Stereo Base Calculation With a $20.00 Handheld

["Michael K. Davis" <zilch0@xxxxxxxxxxxx>]

Here's an easy way to calculate the General Solution for stereo base (that
which John Bercovitz and others use), with a $20.00 calculator (the HP 30S
Scientific) that prompts you for each variable, allowing you to overwrite
only those you wish to change, before calculating the stereo base in
millimeters.  Using this document to program the HP 30S, you can get base
figures in less than fifteen seconds after turning it on.  The Hewlett
Packard 30S is a thin, light, two-line calculator that fits the bill
perfectly.  

If you oppose the General Solution approach to calculating base, you might
want to skip down to the section I call "Comments Regarding the Percent of
MAOFD Desired."  As old as the debate is, I believe I have shed some new
light on the subject of the 1/30 Rule vs. the General Solution and you'll
even be pleased to find I make a case for using the 1/15 Rule…

For those who want to get right to how the inexpensive HP 30S can be used
to calculate base according to the General Solution, check out the
simplicity of the actual keystrokes used in the field:

KEYSTROKES TO CALCULATE STEREO BASE:

Note:  In this document, <- and -> refer to the arrows on the HP 30S cursor
control pad.

1)   Power ON
2)   VRCL <-  ENTER      Recalls the Equation from memory
3)   ENTER               You'll be prompted for C, Percent of MAOFD Desired
(i.e. 80) 
4)   ENTER               You'll be prompted for A, Near Point in feet (i.e.
14)
5)   ENTER               You'll be prompted for B, Far Point in feet (i.e.
25000)
6)   ENTER               Stereo base in millimeters will be calculated and
displayed!

With each prompt you can supply a new value or you can just press ENTER to
accept the last value used and go on to the next prompt.   It's easy, fast
and accurate!

THE FORMULA:

Here's the General Solution (useful for all but macro work), with the
formula for maximizing camera focus embedded.  See:
http://home.mira.net/~kiewavly/bases.html :

             af*an      1         1
   b0 = d * (-----) * ( -  -  --------- )
             af-an      f        an*af
                              2*(-----)
                                 an+af

Here's the same formula as it looks once stored in the HP 30S EQN variable:

    .026C*((304.8AB/(B-A))*(1/65-1/(609.6AB/(A+B))))

  where:

    .026C is d (deviation) - more on the constant and the variable C, below...
     
    304.8 is a constant to convert from feet to millimeters
    
    AB is af*an (or Far*Near) 
    
    B-A is af-an (or Far-Near)

    1/65 is 1/f (or 1 over the Camera Focal length - more on this below...)

    609.6 is the above conversion factor (304.8) multiplied by 2

    AB is an*af (or Near*Far) 

    A+B is an+af (or Near+Far)

OK, here's the fun part - programming the HP 30S…

KEYSTROKES TO INITIALIZE THE VARIABLES:

1)   Power ON
2)   14 STO ENTER         Writes to variable A, to begin with 14 ft. as a
Near Point
3)   25000 STO -> ENTER   Writes to variable B, to begin with 25000 ft. as
a Far Point
4)   100 STO -> -> ENTER  Writes to variable C, to begin with 100 as the
Percent of MAOFD Desired
5)   CL
6)   2nd FIX     
7)   use <- or -> as necessary to underscore the number 1, then ENTER


KEYSTROKES TO LOAD THE FORMULA INTO THE CALCULATOR:
 
Notes:  To move the cursor left or right, use the arrow pad.
            It's safest to tap an arrow, moving one character position at a
time, 
            rather than holding it down.
        To delete everything to the right of cursor position, use 'CL'
        To delete only the character at the cursor position, use 'DEL'
        To insert a character at the cursor position, use '2nd INS'
        To cancel INS mode, issue '2nd INS' again

After pressing CL to clear the display, enter these keystrokes - begin with
".026", not  "1)    .026" :


1)   .026                This is a hard coded constant WHICH MUST BE
CUSTOMIZED TO SUIT
                         YOUR EQUIPMENT.  It MUST be recalculated using the
following
                         equation:

                            Constant = Viewer FL/3000 

                            i.e.  78/3000 = 0.0260   for a 78mm Viewer
Focal Length

                              or  47/3000 = 0.0157   for a 47mm Viewer
Focal Length


                         This constant adjusts the on-film deviation to
accommodate any
                         mismatch between Viewer Focal Length and Camera
Focal Length.


                         For the curious, this is actually derived from the
equation:

                         Constant = (Camera FL/30)*(Viewer FL/Camera
FL)*(1/100)   

                            i.e. (65/30) * (78/65) * (1/100) = 0.026

                         Explanation:
 
                         65/30   for a 65mm camera FL, this is the MAOFD in
mm 

                         78/65   the ratio of viewer focal length to camera
focal length

                         1/100   this converts a % value to a decimal number 
                                 i.e. 80% to 0.80 

                         The Camera FL's cancel each other out, reducing
the equation to:
                             
                         Constant = Viewer FL/3000 
           
                         
2)   VRCL -> -> ENTER    Writes the variable name C to the formula - % of
MAOFD Desired
                         This variable encourages you to use a deviation
that is some
                         percentage of the Maximum Allowable On-Film
Deviation.

3)   X((304.8            The constant 304.8 comes from 12*25.4, to convert
feet to mm

4)   VRCL ENTER          Writes the variable name A to the formula - Near
Point in feet

5)   VRCL -> ENTER       Writes the variable name B to the formula - Far
Point in feet
6)   /                   The key for divide (I can't type the symbol.)

7)   (

8)   VRCL -> ENTER       Writes the variable name B to the formula

9)   -                   The key for subtraction

10)  VRCL ENTER          Writes the variable name A to the formula

11)  ))X(1/65-1/(609.6   The constant 65 is the Camera Lens Focal Length in
mm 
                         THIS MUST BE CUSTOMIZED TO SUIT YOUR EQUIPMENT!
(i.e. If your
                         camera has a 50mm lens, change the constant 65 to
50!) 
                         The constant 609.6 is 2*304.8 - the 2 is part of
calculating
                         the best focus distance - leave this as is.

12)  VRCL ENTER          Writes the variable name A to the formula

13)  VRCL -> ENTER       Writes the variable name B to the formula

14)  /(                  Divide, then close parenthesis

15)  VRCL ENTER          Writes the variable name A to the formula

16)  +                   The key for addition

17)  VRCL -> ENTER       Writes the variable name B to the formula

18)  ))))

19)  STO  <- ENTER       Writes the formula to the variable named EQN and
runs it

20)  CL



KEYSTROKES TO CONFIRM CORRECT ENTRY OF THE FORMULA IN THE VARIABLE EQN:
 
1)   VRCL <- ENTER       Recalls the variable EQN to the first line of the
display

2)   <-                  Use <- repeatedly to scroll back to the beginning...

3)   ->                  Then -> to scroll forward, comparing the formula
to this:

                         .026C*((304.8AB/(B-A))*(1/65-1/(609.6AB/(A+B)))
                         ----                      --

If you modified the underlined constants for Viewer FL or Camera
FL, check for those values instead.   


4)   ENTER               You'll be prompted for C, % of MAOFD - leave it at
100 for now.

5)   ENTER               You'll be prompted for A, Near in ft. - leave it
at 14 for now.

6)   ENTER               You'll be prompted for B, Far in ft.- leave it at
25000 for now.

7)   ENTER               A base value will be displayed, in millimeters.
If you modified
                         the constants in the formula above, as instructed,
the
                         calculated base value should be within a few mm of
the value had
                         with this formula:

                               Base = (Near/30) * 308.4 * (Viewer FL/Camera
FL)

                         i.e.  Base =   (14/30) * 308.4 * (78/65) = 172.7mm 
                                 versus a General Solution value of 169.5mm

		  This formula employs the 1/30 Rule, discussed below.			

See the top of this document for everyday usage instructions (Keystrokes to
Calculate Stereo Base).

Note:  Downstream, if you ever find that you don't receive the expected
prompts, something has become corrupted in memory.  Use these keystrokes to
clear all memory before reprogramming the calculator:

    2nd RESET -> ENTER

It's a good idea to keep a printed copy of this text (and spare batteries)
with the calculator.  That's why I wrote this document in the first place -
for ME to use when I've forgotten everything since the last time I used it!)


COMMENTS REGARDING THE PERCENT OF MAOFD DESIRED: 

The on-film deviation, d, is too often used as a constant when calculating
the General Solution.  Some people assume this should always be equal to
the camera FL divided by 30 (i.e. 1.2mm for a 36mm lens), but that is only
the value one should specify if you are CHOOSING to produce the Maximum
Allowable On-Film Deviation, or MAOFD.  John Bercovitz and others have
preceded me in communicating that one should select a DESIRED on-film
deviation - something more comfortable to view, for example - a value less
than 1.2mm for a 36mm lens.  Using the formula I have documented above for
the HP 30S calculator, the first variable you are prompted for is the
Percent of MAOFD Desired.  Please feel free to select a value other than
100%.  

Note that 100% will produce base values very close to those had with the
1/30 Rule (Near/30) when the Near:Far ratio is at least 1:50.  Below 1:50,
as the Near:Far ratio falls, the 1/30 Rule produces inconsistent results,
with the stereo effect becoming increasingly 'flatter', as if the user had
purposely chosen to have increasingly smaller on-film deviation.  Is that
the intent of everyone who uses the 1/30 Rule with Near:Far ratios less
than 1:50?  (I can't resist the sarcasm - I want the point to hit home.) 

Have a look at this table of base values contrasting the General Solution
at 100% MAOFD against the 1/30 and 1/15 Rules (1/15 is often used for
Near:Far ratios of 1:2 and lower.)  This table was calculated for a 65mm
lens and matching viewer.  Please make an effort to interpret this table
yourself, before reading further, comparing 100 % MAOFD to the 1/30 or 1/15
Rules.: 


Near      Far      N:F Ratio   100% MAOFD     1/30     1/15    Difference

15ft    25,000ft    1:1667       151mm       152mm              +0.66 %

15ft    1500ft      1:100        153mm       152mm              -0.65 %

15ft    750ft       1:50         154mm       152mm              -1.30 %

15ft    375ft       1:25         158mm       152mm              -3.80 %

15ft    150ft       1:10         168mm       152mm              -9.52 %

15ft    75ft        1:5          189mm       152mm             -19.58 %

15ft    52ft        1:3.5        212mm       152mm             -28.30 %

15ft    37.5ft      1:2.5        252mm       152mm             -39.68 %

15ft    31.5ft      1:2.1        288mm       152mm             -47.22 %

15ft    30ft        1:2          302mm                 305mm    +0.99 %

15ft    20ft        1:1.33       602mm                 305mm   -49.34 %

At Near:Far ratios greater than 1:50, the linear 1/30 Rule manages to
adequately mimic a fairly straight portion of the 100% MAOFD curve, but at
ratios less than 1:50 the 1/30 Rule can no longer hug the MAOFD curve.  At
ratios just above 1:2, the bases given by the 1/30 Rule are much shorter
than they should be, as seen in the Difference column of the table.   If
your goal is to produce stereo images with a consistent on-film deviation,
you simply can't be satisfied with even the 10% reduction in stereo effect
at 1:10, much less the 20% loss suffered at 1:5, or the 47% at 1:2.1, had
when using the 1/30 Rule.  We can not expect a single straight line to
approximate the MAOFD curve all the way down to a 1:2 Near:Far ratio.  We
could use multiple straight-line segments to approximate the curve, but it
would take about 6 more 1/n Rules, between 1/30 and 1/15 to deliver
reasonably consistent deviation on-film.  The slope of the MAOFD curve
changes too rapidly through this range to use even four rules, much less a
single rule.

I suppose there are some people using the 1/30 Rule who are fully aware of
the reduced on-film deviation it delivers below the 1:50 Near:Far ratio.
There is nothing wrong with making this choice knowledgeably, since they
might be intentionally seeking a flatter look as the Near:Far ratio
shrinks.  That's their choice, but I wonder about the users of 1/30 who
make the choice to bump the on-film deviation back up to 100% of MAOFD when
they hit a Near:Far ratio of 1:2 -- where they abruptly switch to the 1/15
Rule.  Just above 1:2, at 1:2.1, the 1/30 Rule forces 47% less on-film
deviation than 100% of MAOFD, but by shifting to the 1/15 Rule at a
Near:Far ratio of 1:2, this two-rule approach suddenly doubles the on-film
deviation, from 152mm to 305mm, for example,  to rejoin the 100% MAOFD
curve again.  Some might argue that I'm splitting hairs, but with only two
rules, at some point you have to switch from one to the other.  If not
precisely at 1:2, nor at 1:2.1, then where?  

Let's look at how the 1/15 Rule handles Near:Far ratios below 1:2.  The
slope is changing so rapidly here (i.e. the curve is so curvy), that the
linear 1/15 Rule can be nothing more than tangent to the curve at a single
point.  It rapidly pulls away from the curve as the Near:Far ratio
continues to decrease, below 1:2.   Again, we would need more straight-line
segments to adequately approximate this section of the curve, but ONLY IF
it's our choice to maintain a consistent on-film deviation.  In my opinion,
maintaining a consistent on-film deviation at Near:Far ratios below 1:2 is
NOT desirable.  (Surprise!)  Doing so is one of the reasons that some of
the people who have tried the General Solution, using 100% of MAOFD as a
value for d, have incorrectly concluded that the entire formula is invalid,
when really they just failed to treat d as a variable, reducing its value
to suit their tastes.  

It is my choice to use the General Solution ONLY for Near:Far ratios
greater than 1:2, not at 100% of MAOFD, but rather at 80% of MAOFD.  I find
this delivers a much more natural look, allowing the stereo effect to be
transparent in the mind of my audience.  Too much base is the stereography
equivalent of a garish clown face painted on black velvet.  I don't want
the medium itself to take center stage.  I may adjust this later, but so
far, having shot only 26 rolls of twin-camera 120 at this writing, I am
quite pleased with the look of 80% MAOFD.  (Yes, I'm new to this sport, but
quite obsessed!)

For Near:Far ratios less than 1:2, I have chosen to employ the 1/15 Rule
(modified to reduce the deviation to 80% at 1:2, instead of 100%, and to
accommodate the mismatch of my viewer focal length to my camera focal
length.)  For a 65mm camera FL and 78mm viewer FL, the rule becomes 1/15.8
for 80% of MAOFD.  Using a linear rule for Near:Far ratios less than 1:2 is
actually a good idea, in my opinion.  It automatically tapers the on-film
deviation as the Near:Far ratio shrinks toward 1:1 and that's a lot easier
than accomplishing the same thing by bumping the Percent of MAOFD variable
down incrementally.   I have chosen to do this because I believe it is more
appealing than sticking with a high, fixed percentage of MAOFD when the
subject depth is so shallow.  

If you choose to do this, you really should modify the 1/15 Rule to
accommodate the ratio of Viewer FL to Camera FL and the Percent of MAOFD
Desired that you use for ratios above 1:2.   To determine the value of the
denominator for this 1/n Rule, just use the calculator, entering your
chosen Percent of MAOFD Desired (i.e. 80), then something like 20 ft. and
40 ft. for the Near and Far distances (to get a ratio of 1:2).  Take the
base that's calculated, convert it from mm to feet, then divide the Near of
20 ft. by that base in feet.   For example, if you calculated a base of
387mm, divide it by 308.4 to get the base in feet.  That's 1.25 ft..  Then
divide the Near of 20 ft. by the calculated base of 1.25 ft. and you'll get
the factor 16.0, for this example.  Thus 1/16 would be the rule to use for
Near:Far ratios below 1:2, not 1/15.

In summary, let me say that I'm convinced there is no right or wrong in how
you go about determining stereo base as long as the results you get are
pleasing to your eye and hopefully to your audience.  If the method(s) you
choose allow you to accurately pre-visualize the results you desire,
terrific - you must be doing something right.  If you get surprised from
time to time, by too much or too little stereo effect, you might want to
rethink your approach and consider that there are benefits to borrowing
techniques from more than one school of thought.

I vigorously encourage critical review of this article, not just for my
edification, but for the people I hope it will serve.  

Mike




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