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Here's my answers to several questions from the recent 2-body post.
I'm starting a new post since these are unrelated to the 2-body
question (although they've been somewhat addressed in much early
posts).

A.   DIRECT CALCULATION:  By direct calculation I mean computing the
Lats and Lons of points on the celestial LOP from some an equation, or
equations.  These points trace out the LOP in Lats & Lons, just as one
would make an x-y plot of Sin A versus the angle A, by evaluating Sin
A and plotting the result versus A.  For a CN example, we can find LHA
from the single equation Cos LHA = (Sin Ho - Sin Lat Sin dec)/ Cos Lat
Cos dec, getting the longitude of a LOP point that has latitude Lat
(from Lon = LHA - GHA).  This is an example of a direct calculation of
the Lats & Lons of the LOP's coordinates.  Also tables could be easily
constructed to give the LHA in terms of entries of Dec, Lat, and Ho.
(The navigator then takes LHA - GHA to get longitude.)

B.  PLOTTING VARIABLES:  I think (not sure) that the discussion on the
List about parameters is what I would call plotting variables.  In the
above, the plotting variable is the latitude.  Sumner used the
longitude as a plotting variable to plot three points on his LOP.
Another plotting variable is the true bearing from the GP to the LOP
point (that is, the angle at the GP between the coH side and the coDec
side of the nav triangle).  This is convenient in computer
applications.  (These bearing-angle equations are discussed in
Exercise 1.22 in my book.)  In high altitude sights using a drafting
compass with its point on the GP, its distance equal to the observed
coH, and striking an arc on a chart, we are mechanically using the
bearing angle as the plotting variable.

C.  ST. HILAIRE:  The St. Hilaire method uses two equations to plot a
point on the LOP, the altitude and azimuths equations.  The point at
the end of the intercept, at the computed azimuth, is exactly on the
LOP, just as in the above examples.  (In all practical applications
the difference between the rhumb line and great-circle distances is
negligible. Moreover, the great circle intercept distance is indeed
exact, just as the Lats & Lons are from the direct method.  So chart
projections are irrelevant in this whole discussion.)  If desired,
many other APs can be used to compute many other points exactly on the
LOP.  Thus St. Hilaire can be used to trace out the LOP -- exactly.

D.  ST. HILAIRE & DIRECT:  Both methods compute points on the LOP
exactly.  There are no assumptions, no estimates, no iterations.
These computations might be done with calculators, computers,
logarithms, Bygrave slide rules, or tabulated results (which are the
results of someone else doing the same math).  The means of
calculating has no bearing on this topic.  And forget whole degrees --
they have nothing to do with this discussion.  (Also BTW, they are
sometimes used in log calculations and calculator applications for
convenience, the same reason they're used in tables.)  Both methods
provide straight line approximations to the circular LOP.  In the
direct calculation, two points on the LOP are connected with a cord,
giving a approximation similar to St. Hilaire's, and in some cases
even better.

E.  ST. HILAIRE vs. DIRECT:  At first it seems that St. Hilaire is
inferior to direct calculation for two reasons:  It requires two trig
equations for each LOP point, and it requires plotting because it
doesn't give Lats & Lons directly.  But St. Hilaire has a trump card:
IT'S ROBUST (i.e., it's not fussy, works the same in all cases, has no
gotcha's).  But the direct calculation isn't robust.  For example, in
calculating a Lon corresponding to a selected Lat, a nearly east-west
LOP can yield a Lon that's too far away.  Worse yet, the LOP might not
even intersect the parallel of latitude.  Likewise, specifying Lons
and calculating Lats gives the same problem with nearly north-south
LOPs.

The essential point is that the St. Hilaire method completely avoids
these problems by using a completely different approach to specifying
which section of the LOP we wish to plot.  It specifies a point by
both Lats & Lons to identify our area of interest.  It says that we
want the section of the LOP that is closest to this point.  This point
is a reference point, a locator point, usually abbreviated
"AP" (doesn't "AP" stand for A locator Point?).  There is no
assumption, no estimation, only a decision.  We need to decide where
we want to plot the LOP.  We must decide this before plotting any
celestial LOP by any method (unless we wish to plot the whole
circle).  Once we have the point on the LOP closest to our area of
interest, a straight line drawn perpendicular to the azimuth to the GP
gives an approximation suitable for all but very high altitude cases.
This is similar to the direct-calculation straight-line approximation.

F.  TERMINOLOGY:  Some list members have pointed out that our
terminology may depend on our background: how, where, and when we
learned celestial.  Yes, this is true of many words, but from what
I've read by members, we have pretty good agreement on DR and EP.
However, I was surprised to see a member write that the term "LOP"
means a straight line.  Not in my book.  There are many LOPs that are
not straight:  a range LOP from a lighthouse's height, the double
horizontal-angles LOP from two objects, a LORAN LOP, and a bathometric
contour, just to name a few.  An LOP is simply any line of position,
no matter what shape.

As some member's know by now, "AP" is nomenclature that really irks
me, for two reasons:  First we don't assume anything when we pick an
AP, we just decide where we want to plot a section of the LOP.  We
select it -- we don't assume anything.  It's like deciding to plot Sin
A between 45 and 63 degrees.  We're deciding, not assuming.  The 2002
edition of Bowditch also lists CHOSEN POSITION for the term, which
matches well with what I'm saying.  Second, if you think this is
putting too fine a point on terminology, let me say that every CN book
I've seen either doesn't explain why we're using an AP, or if they
attempt to explain it, they either have it wrong, or give the wrong
implication.  I believe the term "assumed position" contributes to
this confusion.

G.  Someone asked where, or how, I learned CN.  Well, when I was about
8 or 9-yrs old in Flint MI, I purchased a new 1945 edition of Bowditch
since we were all sailors in my family.  While it was hard going for
me, I think I learned a lot from it, but with no sextant and no
horizon, wasn't able to practice sights.  Then in 1960, while a
physics undergrad at M.I.T., I stumbled upon a sextant in a Chelsea
pawn shop.  I bought this Hezzanith for $18, went home and pulled
Bowditch off the self, and started taking sights looking at the sky
and ocean from near my home, then in Marblehead.  In those days I used
logs from Bowditch, and later, tables from H.O. 211.  Many years
later, when I was invited to teach CN aboard the S/V Denis Sullivan on
a Milwaukee-Montreal leg, I become interested in teaching this stuff.
So I wrote a couple of books, a course manual, a PowerPoint
presentation, and now teach on land or sea whenever I get the chance.

JK


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