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George Huxtable wrote:
>
> Well, all I was thinking of, really, was that if a zoom lens was being
> used, that has a whole range of possible expansion factors, and presumably,
> a somewhat different lens-distortion for each one, so the "lot of work" was
> just to measure calibration curves at all these zoom settings. But I'm
> aware that it's difficult, if not impossible, to return to exactly the same
> known zoom factor as before, in the absence of a precisely marked
> scale-of-zoom. So I expect that the only way to use a zoom lens for such a
> purpose in practice is probably to use it at either maximum or minimum
> zoom, not in-between, and hope it returns to exactly the same value when
> doing so.

You can get pretty close by trial and error. Use the markings on the
lens for the initial setting, then take a shot. My camera doesn't show
the focal length in real time, but the value is recorded for each photo
and can be viewed immediately after. The display sensitivity seems to be
2 mm, i.e., I can get it to read 48 mm and 50 mm, but not 49 mm. A
definitely perceptible rotation of the zoom ring is possible without
changing the readout. If the desired setting is one of the numbers
marked on the zoom ring, I believe you could be repeatable within 1 mm
simply by observing this ring.

> Presumably Greg's picture was taken from on land. I suggest that observing
> from the unsteady footing of a small craft would be a very different
> matter, even at 1/500 sec.

Many amateur grade cameras have a system of angular rate sensors and
servos to remove most of the unsteadiness of the photographer's hands.
The implementation varies among different manufacturers. For instance,
Olympus puts the image sensor on a moveable x-y carriage, while Nikon
uses a moving element inside the lens. At any rate, these systems
typically let you use a shutter speed four to eight times slower than
would be practical otherwise.

The old 35 mm rule of thumb for the slowest safe handheld shutter speed
was to take the reciprocal of the focal length. So with a 200 mm lens
you would try to shoot at 1/250 or faster.

> "I could however imagine that there exist somewhere professional
programs
> which would help to do also this sort of calibration and analysis of the
> measurements."
>
> I can only agree. I'm pretty sure that he and I must be retracing
> well-trodden ground, if only we knew where to look.

Imatest sells the "studio" version of their program for $100.
http://www.imatest.com/solutions

According to their documentation, the basic distortion correction
equation is an odd order 3rd degree polynomial. If Ru is the undistorted
radius and Rd the distorted radius of a circle centered on the
intersection of the optical axis with the image plane, Ru = Rd + k *
Rd^3. The sign of k determines whether the distortion is barrel or
pincushion.

http://www.imatest.com/docs/iqfactors.html#distortion

It's common nowadays for software (in the camera or on a computer) to
apply distortion corrections based on the known characteristics of the lens:
http://www.dpreview.com/articles/distortion/

In the case of my camera, manual correction "by eye" can be applied
after the photo is taken, or you can select auto correction. For the
latter option you must have a modern lens with the electrical contacts
that allow the camera to identify the lens, and, I suspect, the zoom
focal length.


> Of course, it's a poor test of the correctness of the tangent model, simply
> because it's restricted to such a small  range of angles from the optic
> axis, so the resulting distortion is so low. All we can really say is that
> it doesn't discredit that model by showing any disagreement with
> observation.

I see nothing wrong with George's desire to demonstrate it
experimentally, but the mathematical basis for tangent formula can be
found in books. For instance, Smart ("Textbook on Spherical Astronomy")
derives a formula for the distance between the point where the optical
axis intersects the photographic plate, and the image of a star on the
plate. It equals the telescope focal length, times the tangent of the
angle between the optical axis and the star.

Green ("Spherical Astronomy") says, "It is seen that the process of
imaging stars on to the photographic plate is similar to a central
projection of the stars on to the tangent plane to the celestial sphere
at A." (The point he calls A is the intersection of the optical axis
with the celestial sphere.)

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