PePax Principle

  1. Stereo Base Computation in stereoscopy. The Bercovitz formula.
  2. Bercovitz formula spreadsheets
  3. Choice of linear parallax: Carlsson anaglyph experiment
  4. 1/30 rule for stereo pairs: mathematics of the 2 meter stereo window.
  5. Stereo base computation for close-up photography
  6. Stereo Base Computation for telephoto lenses.
  7. The Di Marzio Equations (Memorial page).
  8.  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!
  9. Computation of stereo parallax and its use in photogrammetry.
  10. Using your eyes as a range-finder with the help of a vernier caliper.
  11. Depth of field in stereo photography: why digital cameras are better.
  12. Practical limitations of stereo base, in keystone-free macro-photography.
  13. Experiment: effect of diffraction in digital cameras.
  14. Experiment: lack of effect of pixel numbers in macro photography.

NB - the mathematical symbols used for PePax are not the same as in the rest of the stereo mathematics section. This is a very old section and will/might be updated shortly...

Herbert C McKay wrote a book on stereoscopic photography. His word "parastereoscopy" referred to photographs not following the rules of orthostereoscopy. He claimed a parastereoscopic image could not be distinguished from orthostereoscopy as long as his "PePax principle" was followed. (PePax was a word he invented, of dubious etymology, probably perspective / parallax.)      orthostereoscopy   etymology

PePax states the stereo base and the focal length of the taking lens must remain in proportion. 

PePaxif you double the focal length from standard,
the stereo base must also double.

  • Magnification depends on the focal length, and so it doubles. In other words it seems we are half as far from the object.

  • Stereo depth depends on the stereo  base, so it doubles too. 

  • This is the desired result: the object looks half as far away and so it should have twice the stereo depth. It looks normal.

  • It is just as if the stereograph was taken from half the distance and with the standard lens instead of a telephoto.

McKay recognised  there was different perspective in telephoto parastereoscopy, and he stated it in an interesting way (telephoto lenses pack more matter into the same space...) However, the people he tried PePax out on were unable to tell the difference between true orthostereoscopy and parastereoscopy.

The modern use of this effect is taking stereo with two cameras on a bar, each equipped with a zoom lens (e.g. Sony cameras used with the Lanc Shepherd controller). If you do not change the stereo base, then zooming the lenses in to telephoto reduces 3D parallax. Often the stereo base is too big, because the camers will not sit close together on the bar. Do not worry, just operate them telephoto and move back a bit. The problem will be when you want to take a wide angle view from near-by.

Note that PePax is answering a different question to the Bercovitz formula. 

There is only one problem with the PePax principle, according to some people writing on the internet, it is wrong!

Telephoto lens versus standard lens.

the stereo base, B, is constant 
the same two objects, m and n are imaged, 
from exactly the same place
only the focal length changes from F1 to F2

The effect on stereo parallax is to change from p1 to p2

By similar triangles:
p2 / p1    =    F2 / F1

But this is not the conditions of the PePax principle, where the magnification is fixed. McKay wants to make a distant object the same size and stereo depth as if it were photographed from nearby with a standard stereo camera using short focal length lenses.


Consider a stereo pair taken with everything the same, except the focal length of the lenses (F) is changed. 

Let us assume F1 is the standard focal length for the camera format and F2 is a telephoto lens.

MacKay was correct:
The magnification of the image is given by the ratio of focal lengths:

magnification = F2/F1

MacKay was "wrong":
The parallax angle between distant object m and nearer object n is constant (say we set it up as 2 degrees to suit the standard lens).

BUT the linear parallax on the film is magnified, just the same as everything else in the picture is enlarged by F2/F1.

SO the telephoto lens (F2) has a bigger parallax on the image. It has already magnified the stereo depth.

The same thing happens if you enlarge just part of the pictures and mount them as a stereo pair. The parallax is also enlarged. If you have followed the Bercovitz formula to achieve maximum stereo base and then enlarge the prints, you will make the new stereo pair difficult or impossible to mount and fuse.

Doing nothing but change from short focus to  telephoto lens gives hyperstereoscopy. 

Those who prefer orthostereoscopy will have to reduce the stereo base (B) when shooting telephoto.

But reducing stereo Base as focal length increases is exactly opposite to the PePax principle isn't it? 

Yes it is, but the arrangement in the diagram is not PePax. Read on!


The PePax principle quotes the special case where the magnification is the same despite changing from a standard to a telephoto lens. 

Dropping the magnification in telephotography (with a fixed focus lens) can only be done by moving further away from the subject. 

To get the same magnification changing from F1 to F2
    n must increase in proportion to:

 F2 / F1

Now consider the parallax between front and distant objects. If you doodle on the back of an envelope, using similar triangles and the above diagram, the  parallax as you get further away from the subject is found to be:

  1. Stereo parallax (p) = FBd / mn

  2. Simpler than the general Bercovitz formula, because it is for pin-hole cameras, while the general formula includes a depth of field correction for lenses.

The surprising consequence of this parallax equation is:

  1. magnification of the front object (F/n) decreases linearly with distance but
  2. parallax between the front and back object decreases by the square of the "average" distance.
    (actually the square of the geometric mean distance between front and back object, or by 1 / [m n] ).

So PePax is right: you must do two things to regain "proper" stereo depth when photographing a distant object: 

  1. Increase the magnification (change to a telephoto lens: F is bigger) That also recovers part of the stereo depth which was lost by moving away.
  2. Increase the stereo base in proportion to F. That recovers the rest of the stereo depth.

Then the stereo parallax is back to "normal".

In other words we must use a quadratic (square) function for the stereo depth but a linear function for the magnification. 

Stereo depth perception decreases by the square of the distance: a stereoscopic inverse square law.

Just as McKay said - only he put it differently and confused everybody ever since!

PePax is still not orthostereoscopy, as the attractive purple column to the right explains.

The reason perspective decreases with telephotography comes from the change in magnification with distance.

 Magnification on the film changes with focal length in a linear fashion: 
   by (F / n) for the near object and 
   by (F/ m) for the distant object.

For a subject with fixed depth, d:
   (d = m-n) 
the further away you get, m and n get closer to being the same
   (m / n approaches 1).

So the difference in magnification between the near object (n) and the distant object (m)
   (F / n minus F / m) 
drops as we move away.

We say the perspective has decreased in a telephoto picture, because we are losing the size cue of distant objects looking smaller than closer objects.

Variation in stereo base and focal length is shown very nicely by Tony Alderson {here}

Derivation of the parallax formula

(according to non-mathematician John Wattie)

q is the real relative parallax between 
distant object m and 
near object
when a stereo camera of focal length F  
has a lens separation of B (both lenses focused on infinity, or "pin-hole camera properties".) 
p is the parallax recorded on the film, in other words the photographic image of q.

(In "stereo-plotter talk", q is the measured relative parallax on  the two photographs corrected for photographic magnification).


q / p   =   n / F              ---(1)

q / (m-n)   =   B / m     ---(2)


q   =   pn / F                 ---(1)

q   =  B(m-n) / m         ---(2)

OR since q = q, combine (1) and (2):

pn/f   =   B(m-n) / m    ---(3)

CROSS MULTIPLY equation (3):

pnm  =  Bf(m-n)


p  =  BF(m-n) / nm

IF d = depth of the object,

d = m-n


p = BdF / nm

nm is the square of the geometric mean distance from the camera to n and m because
 geometric mean = [square  root] (nm)

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