First edition version

Bercovitz Formulae
for stereo base.

3D base controversy further confused 
by John Wattie

  Version: 12/09/24


  1. Stereo Base Computation in stereoscopy. The Bercovitz formula.
  2. Davis Modification when L<2N
  3. Wattie modification to ensure correct stereo roundness
  4. Di Marzio formula
  5. The easier infinity formula
  6. Pierre Meindre Formula
  7. Bercovitz formula spreadsheets
  8. Choice of linear parallax: Carlsson anaglyph experiment

  9. Wattie magnification formula. Stereo base, not focal length of the lens, determines magnification. A stereoscopic photography paradox, when stereoscopic infinity is in the picture.

  10. 1/30 rule for stereo pairs: mathematics of the 2 meter stereo window.
  11. Stereo base computation for close-up photography. Edition 2 version
  12. Stereo Base Computation for telephoto lenses. Edition 2 example

  13. The Di Marzio Equations (Memorial page).
  14.  The "PePax" principle of McKay and why some people say it is wrong because it is not orthostereoscopic - but it was never intended to be!
  15. Computation of stereo parallax and its use in photogrammetry.
  16. Using your eyes as a range-finder with the help of a vernier caliper.
  17. Depth of field in stereo photography: why digital cameras are better.
  18. Practical limitations of stereo base, in keystone-free macro-photography.

  19. Experiment: effect of diffraction in digital cameras.
  20. Experiment: lack of effect of pixel numbers in macro photography.

Calculating the stereo base

This has become more critical now twinned (or cha-cha) digital cameras have replaced 35mm film cameras. Just what do you do about zoom lenses and tiny CCD chips?

The secret is to enter 35mm equivalent focal length into the formulae, not the actual focal length, which often you do not know anyway.

For digital cameras: MAOMD = Maximum Allowable On Media Deviation. The media (CCD Chip etc) is hidden inside your tiny digital marvel and you usually have no idea how big it is. MAOMD becomes a big problem. So forget it: use 35mm equivalent measurements. The images from all cameras are enlarged to a consistent size for viewing and so the actual on media measurements do not matter. only the ratio of media size to lens focal length is important. In other words, we are only interested in the camera field of view. (There is always an exception: depth of focus, as we will compute later).

Edition 2: I now use MAD: Maximum Acceptable Deviation


Bercovitz formula

{ John  Bercovitz} and others have developed a formula which includes both the parallax and depth of field factors to give a maximum permissible stereoscopic effect.

  1. The Bercovitz full formula is thought to be the best description yet achieved for defining the parameters involved in stereoscopic photography.
  2. A simpler Bercovitz formula applies if the most distant object is at infinity.

1) Full Bercovitz formula

B = P(mn/(m-n)) (1/F - (m+n)/2mn)

B = Stereo Base (distance between the camera optical axes)
P = Parallax aimed for, in mm on the film
m = Largest distance from the camera lens
n = Nearest distance from the camera lens
F = Focal length of the lens

m-n = T = the front to back thickness of the subject.
With a bit of luck it is also the depth of field of the image, but does not have to be, especially at macro and telephoto distances where a big depth of field is hard to achieve.

The formula is not for orthoscopic viewing.
     orthoscopic viewing.

When you focus the camera it is best to focus between the nearest and furthest object and then stop down until both are in focus. This is using the hyperfocal distance, as marked on the lens barrel, to get an evenly sharp picture.
This sensible focusing manoeuvre is included in the formula, to allow for magnification of stereoscopic parallax resulting from the lens being racked forward from its infinity setting.

(Linear parallax = MAOFD: Maximum allowable on film deviation)

2) Simpler Bercovitz formula when Longest distance is infinity

B = P (n/F - 1/2)

This is sometimes quoted as B = PN/F, but the simple version only works if the lens is focussed on infinity. Most people focus closer than infinity, usually on the nearest object, using digital auto-focus. Experts focus onto the hyperfocal distance. Some Canon cameras do this hyperfocal computation directly, after you have focussed on the closest object and then on the most distant. The camera then adjusts the f number and the focus, so everything is in focus at once. The version of the formula which includes 1/2 allows for focussing on the hyperfocal distance.


Corrections to Bercovitz formula to avoid excessive stereoscopic depth in close-ups.

  1. Davis modification: an arbitrary adjustment
  2. Wattie modification: for correct stereoscopic perceived depth (roundness)

1) Davis Modification

The Bercovitz formula requires such a large stereo base for very shallow close-up objects that the resulting stereo depth magnification can distort and become disturbing.
This is avoided in practice by only using the uncorrected Bercovitz formula when the most distant plane (L) is more than twice as far away as the near plane (N).

Once L is less than 2N, the Bercovitz formula is adjusted by making

m = 2n

This stops the stereo base getting absurdly big. LINK

The Davis correction is built-in to the
Di Marzio formulae.

2) Wattie Modification to produce correct roundness (from edition 2 of this web site)

  1. Compute the stereo base needed for correct stereoscopic roundness first, as this gives a realistic stereoscopic shape.
        Correct roundness must be set up separately for different viewing distances (V).
        For example, computer viewing at 1 meter needs a bigger stereo base than cinema viewing.

    (Base) = (Nearest object distance) * (Interocular distance) / (Viewing distance)

    B = n i / V

  2. Then compute base using the Bercovitz formula, to make sure the parallax using correct roundness has not become too big.
  3. Finally use the smaller of these two computations for stereo base.


The Wattie correction will ensure correct roundness, if possible, but default to maximum acceptable deviation (MAD) if necessary.

Comfortable viewing takes precedence over realistic stereo depth. "Flat stereo" trumps true shape.

The Wattie correction is especially useful in macro stereo photography, since the Davis correction can still cause excessive deviation in very high magnification macro stereo.

Di Marzio equation

Frank Di Marzio presents the Bercovitz equation as:

B = P (mn/(m-n)) (1/F - 1/S)

S = The distance focused on for maximum depth of field
S = 2mn/(m+n)

Di Marzio proceeds to simplify until:

B = H/60

H = the hyperfocal distance that has the near and far points in focus.

(I (Wattie) pointed out that H varies with the diameter of the circle of confusion which is chosen, making the formula not quite so universal as Frank hoped. Worse than that, hyperfocal distance and depth of field depends on the size of the image recording CCD, as discussed later. Also, modern digital cameras usually do not have the hyperfocal distances marked on the lenses - which are usually zoom lenses and usually do not give the focal length either. I am sorry Frank died before all this could be discussed with him and I only had time to demonstrate to him that depth of field was based on image magnification in the camera (reproduction ratio).

Meindre formula

Pierre Meindre provides a calculator suitable for a PDA or computer running html based on a simplification of the Bercovitz formula. This simplification excludes the S factor of Di Marzio and can be called a "pin-hole formula," since it does not involve focusing the camera. He warns of this by saying it will only work when the closest distance is quite a lot more than the focal length, which excludes close-ups. (Actually in practice the Meindre formula works "near enough" down into the low magnification close-up range, but certainly not in the macro range.)

You should use Meindre's formula for standard stereoscopic photography, since you can work it out on a pocket calculator.

Base = (P/F) * (mn/(m-n))



Stereo Base when the longest distance is infinity

By re-arranging the Bercovitz infinity formula, it is possible to compute how close the nearest object (N) "should be" when changing the stereo base and using a telephoto lens.  
stereo base

n = ( BF/P ) + F/2

The graph shown here was prepared by the author for 

  • 35mm format, 
  • where P = 1.2mm:
  • and infinity is in the picture
  1. 35mm wide angle lens,  
  2. 135mm short telephoto 
  3.  300mm telephoto lens.

Graph showing how stereo base varies with different focal lengths

Telephoto hyperstereoscopy with reduced subject depth

As in close-up stereoscopy, the stereo base can be increased if you do NOT include photographic infinity. The size of this correction can be very significant. It is well worth knowing in forest photography, where infinity is obscured by the trees.  This means stereoscopic infinity can be replaced by the deepest plane in the picture.

For example, using the full Bercovitz formula:

B (stereo base) = 120cm

If infinity is included in the picture, then B falls to 40cm

Photographic "infinity" is not mathematical infinity. It is a rather silly term for distance from the camera beyond which everything is in focus when the lens is set at its focal length from the film. Beyond photographic "infinity," things still get smaller as the distance is increased further still.


Image size set by the stereo base

"Frozen magnification."

An equation derived by John Wattie, where the camera is focussed on the hyperfocal distance and not on infinity shows:

Focal length drops out of the Bercovitz infinity formula and
agnification in stereoscopic photography depends on the
Base and the linear

M = 1/(B/P - 1/2)

The consequences of this are discussed here

An even simpler version is provided by Dr T (George Themelis) using a "pin-hole" derivation where the lens does not have to be focussed, but it works pretty well in ordinary photography:

M = P/B


Choice of Linear Parallax (MAOFD)
(Maximum Acceptable On Film deviation)

The Bercovitz formula allows any linear parallax (MAOFD) to be set with precision.

Just what Parallax to choose depends on how the stereo pair will be viewed and the stereoscopic experience of the audience. 3D photographers often under-estimate the difficulty beginners have to diverge their eyes and adjust focus simultaneously. The MAOFD suitable for neophytes is far smaller than the experienced photographer can handle himself.

Lake Wanaka and Mt Iron in 3D

This is hyperstereoscopy, which some people have difficulty fusing when seen at full size. The deviation is 1/10 rather than the recommended 1/30.
However infinity separation on a 336mm monitor (horizontal width of a 17 inch monitor) is only 33mm, which is much less than your interocular distance. People experienced with 3D viewing should have no difficulty.

Click the image of Lake Wanaka to see it bigger on Flickr, then choose all images sizes.
This is a popular image on Flickr, which implies that most people have no trouble with the deviation.

Impression of depth

There is controversy about the impression of depth. Some say the depth impression should be realistic. Others say viewing must be comfortable, but that unreal depth is perfectly acceptable, even encouraged, as in hyperstereoscopy of the sun. Parallax and perceived stereo depth should be in harmony: the sun should look round like a ball and not flat or conical.

The maker of stereo pairs for computer viewing cannot control the screen size or where the audience will sit and so "impression of depth" becomes a hazy concept. When stereo is projected to an audience, only one person at a time can sit in the best possible seat in the house for seeing ortho stereo.

The formulae compute a base to give consistent stereo parallax.

1/30 rule

The 1/30 "rule" comes from the first treatise on stereo photography by Brewster who expressed it as an angular parallax of 2 degrees. The two degree rule only works for standard focal length lenses, which makes it useless for modern zoom lenses.

Nowadays the 1/30 rule is best expressed in terms of the final stereo image as:
"linear parallax at infinity measuring 1/30 of the width of a landscape format, 4:3, picture."

4:3 is the format of your computer screen (unless you are using a "letter-box" shape). This definition avoids big problems which arise with telephoto lenses, which were impossible on antique stereoscopic cameras, and fits in with the ISU definition.

(60mm infinity separation on your computer screen is equivalent to a "1/5 rule," which represents the theoretical limit for anaglyphs, but would be hopeless for a parallel stereo pair.)

Ramped stereo

Beginners more easily handle stereo parallax when it increases continuously into the distance ("ramped stereo").

Neophytes do not like close objects separated by a gap from distant objects ("stepped stereo"). They notice double vision, which is less obvious if their eyes glide smoothly into the depth. This may also explain why people often prefer a 3D picture taken from under an over-hang (tree branches meeting above a path for example.) This give another "stereo ramp" into the distance.

It is wise to start a stereoscopic projection show with gentle stereo (small, ramped parallax) and introduce bigger parallax as the show progresses, so the audience has time to adapt.

Projection stereo

In the ideal situation the audience should sit close to the screen so that it covers 40 degrees of their field of view. This is about the same as a person sitting at a computer and about the same as recorded by a 50mm  lens on 35mm film. Only one person can sit dead centre and see 40 degrees and he is said to be in the "ortho-stereo seat"

In practice in a hall, people all want to sit well back, far behind the ortho-stereo seat. This has the fortunate side effect that the stereo angular parallax is reduced. Even stereo incompetents can fuse the 3D. In this situation people even enjoy deep stereo, which at a computer monitor only stereo experts can fuse.

Danish anaglyph experiments on computer monitors: what should P be set to?

Experiments by Louis Carlsson on anaglyphs shows Danish "Vikings" (his term), who are not trained in stereoscopic vision, can handle a far point separation of 18mm on a computer screen at the usual viewing distance.
The near point separation (in front of the screen surface) that the general public can handle is 5mm.
According to Carlsson, anaglyph stereoscopic space for untrained people should be set in the range of :
-5mm to +18mm. Total separation 23mm.

18mm on a 17 inch (336mm horizontal) computer monitor is equivalent to a 35mm camera parallax setting (MAOFD) when taking the stereo pair of 1.8mm. So enter a maximum of 1.8mm into the above spreadsheets when using 35mm format for anaglyphs. (Effectively a 1/20, rather than the hallowed 1/30 rule.)

For digital cameras, enter 35mm equivalent focal lengths but not the tiny true focal length and still set Parallax to 1.8mm.
If objects are desired to project in front of the window ("poke your eyes out stereo" or "pop-out"), increase the total parallax setting to 2.4mm.

StereoPhotoMaker program (SPM) automatically adjusts maximum deviation, during post-processing, to 1/25, which is equivalent to P = 1.4mm. You can over-ride this if you wish by adjusting the separation of the two images (using the horizontal arrow keys). Often you do have to over-ride SPM automatic setting if the stereo base during photography has been "too big", in order to get the stereo window just where you want it. You do this at the expense of getting a wide infinity deviation, which neophytes may not be able to tolerate.

Anaglyphs set up for minimum ghosts, or red enhancement in the Mirachrome system, should have a 35mm equivalent parallax setting of 0.6mm or even less.

People having difficulty fusing computer 3D should just move further away from the computer screen.

1.8mm MAOFD is for anaglyphs and does NOT apply to stereoscopic slides or Holmes cards, where the 3D infinity separation is set by the viewing system and picture mounting technique required for two separate images. For slides and Holmes cards you must stick with MAOFD of 1.2mm maximum.

Exhaustive review of parallax / deviation / MAOFD etc, all meaning much the same thing.


Hyper stereo example: Big baseline

red/cyan goggles for viewing the next anaglyph

Red/Cyan anaglyph, hyper-stereoscopic 3D panorama of Lake Wanaka

This small anaglyph is easily fused.

The parallax deviation at the deepest mountain is only 3mm on a 17 inch monitor set at 1024x768 pixels.

1024 pixel anaglyph, still far smaller than the original.(Cha-cha hyper stereo photographed by John Wattie)

The 2x bigger anaglyph panorama should still be easy for just about everybody.
Deepest stereoscopic linear parallax is only 6mm on a 17 inch computer monitor (336mm horizontal) running at 1024x768 pixels.
With the monitor resolution set on 800 x 600 pixels, infinity parallax is nearer 8mm, still easy for most people.

Note how the size of the computer monitor and the setting of the video card make a big difference. This shows how the maker of computer anaglyphs or stereo pairs has no control over how the audience will see the result. If you have difficulty, sit further away from the computer monitor.

Distances from The Geographic Atlas of New Zealand:

Map: portion of Lake Wanaka from Geographic Atlas of New Zealand

Nearest Distance: 4.1km
Longest Distance: 16.6km
Focal Length: 7mm Sony V3 = 34mm in 35mm equivalent.
(i.e. 7mm actual focal length on the Sony has the same angle of view as 34mm focal length on a 35mm camera, like the digital Canon 5D).
Base: approximately 100meters (paced out)
Linear Parallax: 0.62mm (1/56 of 35mm): a conservative stereo base which will allow considerable enlargement of the anaglyph and still be viewable by neophytes..

Bercovitz formula spread-sheets

DOWNLOAD: Spreadsheet 1.2
(Compute stereo base: full Bercovitz formula: from macro to hyper stereo: Davis correction included: measure N from maps or pacing: camera screen magnification method also included.)

DOWNLOAD: Spreadsheet 2.3
(Complex: Stereo base and Magnification: Compute image size on digital camera view screen. Bercovitz formula with Wattie modification allows stereo base to be assessed directly from the digital camera view screen. Explanation here)

Spreadsheet 1:
Longest and shortest distance, any parallax, any focal length.

Below is an example of Spreadsheet v 1.0 (improved versions now available).
It is possible to calculate distances from a map or from simple pacing, and computation for those is provided. A laser range-finder, as used for shooting or golf is ideal (but costs a heap - only for obsessive neurotics!)
The Excel spreadsheet works in a PDA running Windows (such as an iPaq), allowing use in the field.
It works very well with paired Sony V3 digital cameras, but will work with any camera.

If you do not know the 35mm equivalent focal length of your zoomed lenses, that is a problem and your rig is not really suitable for variable zoom, variable base, precision, stereo photography using this spreadsheet. Do not despair, it can still be done using Wattie's method of digital screen magnification.

Spread sheet version 1.3, Berkovitz formula, Wattie   Nearest distance from a fixed base but varying focal length

Fortunately Sony V3 cameras (frequently used for digital stereo since they can be synchronised accurately) quote the focal length as a multiplying factor on the screen. Wide angle on a Sony V3 is equivalent to 34mm. Telephoto zoom times 4 is equivalent to 136mm. ("Equivalent" means giving the same angle of view as a 35mm camera would do).

1.2mm parallax on the Sony V3 screen, which is 50mm wide, becomes
1.2 x 50/36 = 1.7mm
Still use 1.2mm in the formula.
Use 1.7mm if you are adjusting stereo base by directly measuring parallax on the camera screen. The formula operates with 35mm equivalent, not with actual size.

Remember that cropping the image during post processing and then enlarging it again is the same as using digital telephoto. So if you will crop the vertical height of the image to half and then bring up to full size in soft-ware, double the focal length setting in the above formulae.

Cropping horizontally is NOT a problem since it does not change the magnification of the final image.

Cropping vertically and then magnifying the picture to fill the screen is a problem. Magnification of the image is the same as using a telephoto lens. There is a fundamental difference between vertical cropping and resizing versus horizontal cropping for window corrections. The horizontal cropped picture cannot be magnified - if you try the vertical dimension will be too big for the screen.

See here for confusion about cropping.

Reduced depth impression with telephoto stereo:

Notice how the stereo base is inversely proportional to the focal length: double the focal length and the required stereo base halves.

Or, double the focal length and keep the base constant: the nearest allowable distance doubles.

The mathematics of telephoto depth is analysed in the PePax section

How to overcome reduced telephoto depth impression

To overcome the flattening of depth when using a telephoto lens, you must increase the stereo base by T/S, where:
T is the telephoto focal length and
S is the standard focal length.

Base increase = T/S

For example:

You want to take a head and shoulders stereoscopic portrait with a telephoto lens, rather than the standard lens.
(This is good photographic practice as professionals have used a longer focal length portrait lenses since one and a half centuries ago, so can it work in stereo?)

50mm Standard lens with a 65 mm stereo Base provides an orthostereoscopic view on 35mm format.
(Because 65mm is an adult interocular distance and you plan to view the stereo picture in a standard stereoscope with 50mm focal length lenses, to match the standard camera lens.)

Now use a 100mm Telephoto lens (twice standard), what Base is needed for an orthostereoscopic view?
(Edition 2: this is not orthostereoscopic but it is orthoplastic.)

Base increase = T/S = 100/50 = 2
Base for "orthostereoscopic depth" = 2 x 70mm = 140mm

Note that this is a constant magnification situation, since you want the portrait to be a standard head and shoulders view. (PePax requires a constant magnification).

There are three problems:

  1. The depth impression is only correct for the portrait,
  2. Objects further away are too big (wrong perspective) and show decreasing depth.
  3. Infinity cannot be included in the picture, or stereoscopic disparity will be too big.

You will probably conclude this is a fair compromise, since:

  1. you want to have a close background for the portrait anyway and
  2. the reduced perspective is just fine for overcoming familiar photographic artefacts, like the nose seeming to project too far forwards on standard focal length portraits.

The Base increase = T/S formula works well for stereoscopic wild-life photography, where telephoto lenses are essential to avoid disturbing the animals. Just remember, it only works if the background is close behind the animal. You cannot make this work for a Stag on a Crag, with a distant mountain behind him. The Bercovitz formula would never permit that!

Example using a 600mm lens

Bercovitz curves are available in graph form { here }