version 05.03.08

It so happens people are not much worried about "sensible" 3D. If they have spent good money on a stereograph, they expect to see good depth. Devotion to rigid mathematics is not the point  the aim is to make a picture people enjoy. Too many stereo pairs on the market or internet are actually flat, because the base was not big enough. People do not judge distances by stereoscopy alone. Looking at a photographic stereo pair they cannot readily tell if binocular stereoscopic information is "wrong". They have to think hard about information from stereo parallax, angle of view and perspective before realising these three are not working together as usual.
In the real world, where all your senses are working, inconsistent stereo and perspective information causes remarkable confusion.
You can prove this for yourself. Get a "Mini Wheatstone viewer" or better still a ScreenScope and walk around looking at the real world through it. The mirrors effectively increase the distance between your eyes. Stereo depth is increased in proportion to the raised stereo base, which is amusing, but you can still function. At first you will not reach far enough to grab things but quickly you compensate.
Hyperstereoscopic viewing of the real world makes things seem too small. Look at your feet through the ScreenScope and they look miniature. And yet they occupy the same space on the retina of your eyes as they usually do. Hyperstereoscopy says your feet are closer than normal and so they should be bigger. The feet are not bigger and so the brain assumes they are too small in preference to thinking the stereo depth is wrong. Everybody I have tried this on so far ends up thinking their feet have shrunk rather than the stereoscopic distance is wrong.
When the stereoscopic information is powerful, it overcomes perspective information and even common sense. But when stereo information is weak, perspective and common sense take over. This is proved with the face mask illusion. A translucent plastic mould of a face is hung up against the window and viewed from the concave side. Although concave, the backlit mould actually looks like a real face and seems to have a convex surface. The nose pokes out at you, despite your knowing it is really depressed into the mask. Even motion parallax does not solve the illusion, for if you move sideways, the face seems to turn and follow you. Stereoscopic vision finally gives the right answer only when you come really close to the mould and realise it is shaped like a basin. Perspective works at all distances but stereo vision is only useful nearby. This is "the inverse square law of stereoscopy" working in real life.
Driving a car while looking through a ScreenScope would be disastrous. You are not likely to try it in real life, but consider driving a car by remote control while looking at a stereoscopic display. If the two TV cameras which are replacing your eyes are not set up for true stereo viewing (orthostereoscopy) the remotely controlled car is going to crash! In machine vision orthostereoscopy is important (e.g. robotics or surgery by remote control). Even in machine vision the computer can be programmed to allow for rigs which are not orthostereo. Just as our brains also make allowances, but only if given time to learn the new sensations. It is better to set up remote stereo viewing systems properly in the first place.
For remote stereo control you do NOT use the Bercovitz formula and you do NOT apply the PePax principle, since neither of them give you orthostereoscopy. They do give you nicely satisfying stereo art, but art has nothing to do with reality. A plastic surgeon basing his practice on the art of Picasso would soon be in serious trouble!
The photogrammetry basic equation can be derived from the parallax formula by solving for d (the depth of the object). In this way a pair of photographs taken looking straight down from an aircraft flying at H meters above ground level can be used to measure the height or depth (D) of objects.
Parallax formula:
P = BDF / NL
The product NL (Nearest times Largest distances) can be replaced with H^{2.} to give an approximate formula which is still very good.
H is the aircraft height above ground and H is much bigger than the depth of most objects being measured. This means m and n are both virtually the same as H.
Approximate Photogrammetry formula:
D = P H^{2 }/ B F
D = depth or height of the object (meters)
P = relative parallax (mm) measured on the stereo pair (e.g. with a parallax bar)
H = Height (meters) of the aircraft above ground
B = Base length (meters) (how far the aircraft flew between pictures)
F = focal length of the camera lens (mm)
In practice a far more complex computation is done by computer to allow for the aircraft not flying straight and level between pictures, photographic distortion, lens aberrations, base stations not being at sea level etc...
(You may not like F and p being measured in millimetres when all the other terms are in meters. However the ratio of p/F in the formula means the millimetres cancel out.)
Now let us see if the approximate formula derived here is consistent with the precise photogrammetry formula used when working with a parallax bar: The precise formula is: D = P H / (Bm + p)Notice that the square of the aircraft height is not in the formula. We see H but not H^{2}, which worried me since it seemed to deny "the inverse square law of stereoscopy." However, the height squared is actually lurking inside the equation as we will now see. Bm is the Base length in mm as measured on the photographs, which avoids having to calculate the true base by correcting for plate magnification. But if we do the magnification correction: Bm = BF/H Now making the approximation that Bm is a lot bigger than P, so [ Bm + P ] is near enough to Bm: D = [aprox] P H / Bm D = [aprox] (P H) ( H / BF ) D = [aprox] P H^{2} / BF Despite first impressions, the practical photogrammetry formula does contain "the inverse square law of stereoscopy". 
You can imagine how useful the Bercovitz formula was while taking 300mm telephoto {stereoscopic photographs in the Amazon}. I had to work fast from a canoe in a flooded jungle and did not have a "laser reflector rangefinder" to measure the distances and no spreadsheet on a computer to calculate the stereo base. So I just took the pictures anyway, which is what you should do too.
Place your vernier caliper at arm's length and measure how far it is from your eyes, you only need to do this once.
Look at a very remote object with both eyes. Your eye axes should be parallel under these conditions.
The caliper is now out of focus and seen double. You will have to work with it like that.
Adjust the caliper until the two measuring limbs fuse on the distant object. (Take care not to look at the caliper and be sure the distant object does not look double, or you are converging your eyes too close).
The measurement is your interocular distance (I).
Do the same as for I, but looking at the near object this time.
Your eyes will be converged and the caliper will measure less than your interocular distance.
(By the geometry of similar triangles):
Example:
I = 68mm Interocular distance
L = 500mm Arm's length from your eyes
E = 60mm Converged optical axes at arm's lengthN = 500 / (1  60/68)
= 4,250mm or 4.25meters
This gives you the correct stereo base from the caliper measurement.
(This example is the version for 35mm photography in which "infinity" is included in the pictures):
This way you can use a vernier caliper to compute the stereo base for hyperstereoscopy. It means carrying a caliper and a calculator.
Or carry a spreadsheet on a PDA, preloaded with this formula.
Or work it all out beforehand; to produce a small piece of paper giving stereo base for your own lenses directly from the caliper measurement).