P3D Re: The "PerPax principle" Challenge, PePax principle is incomplete

[abram klooswyk <abram.klooswyk@xxxxxx>]

First, McKay called it "PePax", not PerPax. Second, this posting is
not an entry for the challenge. :-)

PePax and related subjects is one of the topics on which I did some
stereohistorical research, to be found back in Koo Ferwerda 's "The
World of 3-D" page 247, in the chapter which also discusses
deformations. The subject of geometric deformations IS technical. 
Maybe McKay didn't know, maybe PePax was another instance of
rediscovery. But the principle goes back to Claudet (also inventor of
the stereowindow) and von Helmholtz (also inventor of the
Telestereoscope). However, PePax theory as formulated by McKay is
incomplete. 

There are four geometrical stereo deformations.
1. Base deformation (Hyper and Hypo).
2. Distance deformation (e.g. long camera focal distance, sitting
   far from projection screen.
3. Convergence deformation (also called Frustum deformation). This
   does NOT deal with optical convergence (convergence of camera 
   optical axes), but occurs when the viewing convergence of the eyes 
   is substantially different from photographic convergence. The 
   latter is defined as the angle between the stereocamera's 
   "viewing lines" to an object.
4. Oblique deformation: Parallelogram effect when sitting at the
   side of the projection screen. (This last deformation has hardly
   practical significance).

It should be emphasized that geometrical deformation does not
necessarily mean perceptible deformation. Geometrically, the
stereoslides made with the Super Duplex (base 30 mm) should show
gigantism, with everything two times enlarged. This is not the case,
enlargement is hardly noticeable, only a somewhat diminished depth
effect compared with the normal base.

- Geometrically a large base gives hypers in which all 3 dimensions
  are smaller,   a large cube becomes a small cube, but stays a cube.
- Geometrically in stereo a long focal distances of camera lenses
  enlarge width and height but not the third dimension, so a small
  cube becomes like a box for 10 computer diskettes.
- The combination of these two (PePax) can result in normal height and
  width, but still the depth is squeezed. PePax theory neglects this,
  but now the Deformation of the Third kind comes in.
 
When you look at some stereo window at 7' your eyes converge about 2
degrees.
Now take hypers with 4 or more degrees photographic convergence, say
60 cm base for a distance of 9 meter (1 in 15, not 1 in 30). You
mount these as usually "to the window", so they are viewed with again
the standard 2 degrees convergence at the window. In fact the images
are pulled apart till they fit in the standard.
Then you have added the third deformation. Theoretically a cube
becomes a frustum of a four sided pyramid, but in practice only
stretch is visible. And the diskettes box is stretched to a cube more
or less original in size. You will not notice the "less".

Mckay and many writers before and after him didn't notice the third
deformation, but many have used it with great effect, it is the
background of what I call "telestereoscopic closeups". 
Paul Wing used them for his bird pictures, an Peter Wysnevski too.
Among the most impressive applications are Allan Griffin's action
shots of surfers and rodeo performers.  

Another example of the third deformation:
With a Macro Realist you can take pictures with a base of 16 mm at a
distance of about 80 mm. In projection often stretch is visible,
especially when flowers know to be circular appear as oval. This is
the third deformation. The base is 1 in 5, a photographic convergence
much larger than the viewing convergence possible in conventional
viewers (let alone projection).
To avoid the stretch effect you would need a special viewer and
special mounting. In mounting the separation of the chips should be
much less than normal, perhaps by cutting out a central strip of 5P
mounts, and you need a viewer with an interocular (distance between
centre of viewer lenses, or whatever seems clear) substantially more
than the slides's separations, and with strong focusing possibility
so that your virtual viewing distance is at about 30 cm. 
(In England someone actually has made a system for 'near-point
stereoscopy'. He used *bended* normal no-glass 5P-mounts and put 
them in a special viewer with converging optical axes.)

Still another well known effect of the third deformation is seen in
aerial stereos as used for cartography. If you happen to live in a
more or less civilised country, you probably can buy such pictures of
your own living area (if you don't live is such a country, ask the
CIA or the KGB).
When viewed in the way we are used to, these pictures show
lilliputism combined with considerable stretch, the latter from the
third deformation. This is no wonder as the photographic convergence
in aerial survey photography is excessive.

But this is not all, the strangest effect for math adapts is that
the deformation which works so strong in the examples given, is not
visible at all when convergence differences are small (again: this is
not about optical convergence).  Some stereo projectionists
superimpose infinity points on the screen. Others try to have them
6.5 cm apart, and still others superimpose the window on the screen, 
which makes the separation of infinity points on large screens more 
than 6.5 cm. 

Pat Whitehouse used this last technique, she explained that the
subject matter of her slides was not at infinity, and that she
changed slides so quickly that it did no harm. She was right of
course, and even with longer viewing times slight viewing divergence
is not a big deal. 

But these different ways of projection have great effects in
geometrical theory. The seize of a mountain NEAR the screen is a few
feet, it must differ from a mountain seen at infinity. But it doesn't.
Showing the different ways of projection to a naive audience, asking
them to close their eyes while on-screen changes are made, makes
clear that perception is not different.
These facts are disastrous for math-only theories. For example in
the Spottiswoode book everything depends on formula's, there is
hardly any reference to the perceptibility of computed space.

I have tried to 'explain' this perceptional behaviour of the
geometrical convergence deformation (the Third one) with postulation
of a Convergence Threshold.
Differences between photographic and viewing convergence of less
than two degrees are generally not perceptible. Differences of more
than 2 or 3 degrees can become significant for perception, and larger
differences mostly have a remarkable effect.

You will notice that the seize of this threshold is very large
compared to the average natural stereo acuity, which is less than 
one minute of arch (down to 2 seconds with some persons under lab
conditions), but even large compared to photographic stereo acuity
which is probably in the order of 3 to 5 minutes of arch.

Kurt Goedel proved the incompleteness of number theory. I hope I
have convinced you that PePax theory is incomplete too.

Abram Klooswyk


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