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> Bill wrote:
> "How the heck do I correct for  parallax error for two points at varying
> distance? "

> Frank responded
> I don't think anyone  bothers with this. It might be an interesting
> mathematical puzzle, but I don't  think it would ever matter in practice.

It was interesting.  And led to a better understanding of constraints in
sextant design. The first pieces of the puzzle for me were the distances
between the silvered surfaces of the index and horizon mirrors when
parallel, which I can measure to .001" (probably not close enough for the
geometry/trig solution I propose).  The second, the precise angle of the
horizon mirror to the line-of-sight through the scope--which is proving
tougher to determine accurately.  Any nifty tricks for finding the angle
accurately?

My current thinking: since the line through the sight tube/scope and horizon
glass to a point establishes a "base line", distance of the point viewed
through the scope and horizon glass doesn't matter a bit; only the distance
of the point from the index mirror matters for one-point corrections; or
distance from the horizon mirror to the elevated point for two-point
corrections.  No need to combine the distances/parallax corrections.

I am also currently of the opinion that if one knows the distance between
the silvered surfaces of the index and horizon mirrors when they are
parallel, and the angle of the horizon glass to the axis of the sight tube,
one has all needed to solve for the parallax correction angle.  First solve
for the distance between the scope line-of-sight through the horizon glass
to the mirror surface on the index mirror axis. Then solve for a line from,
and perpendicular to, the line-of-site to the index mirror axis.  Then you
have everything you need to solve for one or two point parallax error. The
latter, noting the precision required, is probably not doable in a
living-room lab for establishing error along the arc.

One could calculate these corrections for various distances rigorously and
build a table, or use the small angle formula you suggested (with less than
.0007 percent error between it and rigorous calculations--less than
operator/sextant measurement errors). Here is where it gets fun more me than
why Twinkie- and potato(e)-chip packages expand.  With this table one could
use the sextant as a rangefinder.

Easier than calculating all the distances, calculate the correction for one
distance, and use that as a reference angle.  "Angles *ARE* Ratios" is the
magic bullet. For small angles, there seems to be an approximately
inverse-linear relationship between angle and distance.  Since the greatest
parallax error one might encounter in reality is well below 1d, this should
work. Double the reference angle, half the distance.  One third the
reference angle, 3X the distance, and so on.

Really simple--so probably best--motor over to a local football field.
Position yourself 100 yards from the goalpost upright, bring it into
alignment at your eye level (sextant held horizontally), take the difference
between 0 and the angle-off-the-arc measured, and correct for IE. That is a
100-yard reference angle for your sextant that you can put on an index card
and place in your sextant case. Divide the reference angle by the observed
angle (corrected for IE) and multiply by the distance used to establish the
reference angle--and you have distance to the object.  Better than a
Datascope which requires height of object.

I like this concept as Bowditch tables 15 and 16 require height above the
water, and this can be problematic as many charts give (light) height above
the base. Using the sextant as a rangefinder not only eliminates the need to
know the height (and apply dip), but also corrects for parallax errors.
This should work for distances approaching .5 nm, where sextant/observer
errors approach the correction angle.

What have I missed?

I would not be bold enough to call it a peer review, but if any of the list
are interested in grading a student's test I would be happy to send PDFs,
gifs or jpegs of the diagrams and solutions on request.  The 300 dpi PDFs
total 50KB, the GIFs 100KB, and JPEGs 300KB.

> And:
> "and a three-arm  protractor is precise to 0.1'."
>
> But is it that accurate? Let's  suppose the arms are 50 cm long. Placing the
> end points at 0.1 arcminute accuracy would be equivalent to placing the tips
> of the arms with a linear  accuracy of 0.015 millimeters while the width of a
> line on a chart is probably around 0.5 mm in most cases--that's over 30
> times larger. I wouldn't count on angles laid out with a three-arm protractor
> to be any better than 0.1 degree  accuracy. Then again, maybe I've missed
> something.

No, I don't feel you've missed anything. Think you are right on the money. I
believe missing the obvious belongs in my bailiwick.  At .5 nm the parallax
error could be 0.2' to 0.3', so not significant in your example.  And how
much distortion/error is already built into the chart?  Do think I need to
switch to a Pentel with 0.3 mm leads.
>
> As an aside,  here's a little mantra: Angles *ARE* Ratios.
> An angle of one arcsecond  is a ratio of 1:206,265. An angle of one arcminute
> is a ratio of 1:3438. And an  angle of one degree is a ratio of 1:57.3.
> Memorize any one of these and you  never need trig for small angle
> calculations.

Thanks.  If I recall, arc divided by radius yields sine, tangent and angle
(in radians) in small angles. Your factor of 1:57.3 simply converts from
rads to degrees. Yes?

Thank you,

Bill

   

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