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Herbert,

Many thanks for this very thorough tutorial. I want to diagram out your
"cornerstone" method, it does seem to be a fundamental extension of the
basic navigational triangle. Along with other key selections from this
list, this goes in the permanent file for future reference.

Just to be sure I am not totally confused let me confirm that in your
first statement of the indirect method,
D = declination
L = local hour angle
T = latitude

I think you have indirectly answered my question about the Nautical
Almanac's procedures for obtaining "Position from intercept and azimuth
by calculation" (pp. 282-283).  As the procedure is iterative, it is
indirect.  I would be curious what the math of this procedure is doing.
The inputs are the calculated azimuths and intercepts from two or more
sights and the two outputs (after much addition, multiplication, and
squaring of sines and cosines) are a change in latitude and a change in
longitude that when applied to your original AP give an improved
estimate of position. I can replicate the math in a spreadsheet; I can't
begin to understand how it works. My best guess is that it is a form of
best fit algorithm for the data from the sights.

Again, many thanks for this tutorial.

Regards,
Arthur Pearson

-----Original Message-----
From: Navigation Mailing List
[mailto:NAVIGATION-L@LISTSERV.WEBKAHUNA.COM] On Behalf Of Herbert Prinz
Sent: Friday, April 12, 2002 1:42 PM
To: NAVIGATION-L@LISTSERV.WEBKAHUNA.COM
Subject: Direct methods for finding position

Arthur,

Did you ask "What is a direct method in general?" (as opposed to an
indirect
method), or "What, specifically, are the direct methods for finding
position
at sea from two altitudes and GMT?" After answering the first question
very
briefly, I shall give only an overview over the various types of methods
available. Finally I outline one old method that I consider a
cornerstone
of  spherical astronomy.

Direct versus indirect. Consider the formula

H = arcsin(sin D * sin L + cos D * cos L * cos T)

With this you find the altitude of a star directly from its declination
and
local hour angle and your latitude.

What if you know H, D, L and want to find T? Shuffle terms around and
get

T = arccos((sin H - sin D * sin L) / (cos D * cos L))

Now you have a direct method of finding T from H,D and L.

What if you know H, D, T and want to find L? Then things become messy. I
leave it to you to try to get L to the left and everything else to the
right
side of the equation sign. It's rather tedious. If you succeed, you are
rewarded with a direct method for finding L.

But you may as well decide that it's not worth the trouble. As an
alternative method, you can guess L, insert it into the above formula
and
see whether it gives the right H. If so, you are done; if not, you can
try
another value for L. You would not want to try wildly arbitrary values
for L
ad nauseam. A method that gives you guidance as to what value to start
with
and how to proceed, if the result does not quite satisfy you, is called
an
indirect method. Other names are "trial and error", "heuristic",
"iterative", "regula falsi", "brute force", all with slightly different
meanings, but with that same basic idea.

Now, on to the direct methods for finding position at sea from two
altitudes
and GMT. I think it makes sense to distinguish geometric, algebraic,
trigonometric methods and hybrids thereof.

Geometric.

Take a globe and draw two circles of equal altitude on it. They must
intersect each other (normally twice) unless the "sight" was actually a
halluzination. Read off the co-ordinates of that intersection of the two
which makes more sense to you - und you have a direct geometrical
method.
Don't laugh, it has been done.

Algebraic.

The globe, being a sphere, can be represented in 3 dimensional space by
an
algebraic equation

x^2 + y^2 + z^2 = 1

A circle of equal altitude is the intersection of this sphere with a
plane.

a*x + b*y +c*z = d

We find the cooefficients a,b,c very easily, because we know the
orientation
of the plane in space: it is perpendicular to the line from the origin
(the
center of the sphere) to the star. d is then chosen so as to make the
radius
of the circle the right size.The two planes containing the two circles
of
equal altitude intersect each other in a line. This line cuts through
the
surface of the globe in exactly two points. One of them is your
position.

Hybrids.

This would be algebraic equations containing trigonometric terms. For
instance, the system of two equations in two variables, L and T

 sin H1 = sin D1 * sin L + cos D1 * cos L * cos (T1 - T)
 sin H2 = sin D2 * sin L + cos D2 * cos L * cos (T2 - T)

where T is your sidereal time and T1, T2 are the SHAs of two stars, can
be
solved for L and T. Likewise, if you interpret T as your longitude and
T1,
T2 as GHAs of two stars. Expand the right sides and eliminate one
variable.
Then express all terms in the remaining variable by means of one and the
same trig. function. It takes a lot of pencil, paper, and most of all,
eraser, to do so.

Last, not least, a purely trigonometric method. It is the most beautiful
and
oldest method, based on technology from the 15th century. All we need is
the
cosine theorem for spherical triangles. Where is the rub? Well, the
theorem
has to be applied 5 times!

Consider two stars A and B. Let P be the celestial pole and Z be the
zenith
of the observer. In the following AB means the side of the triangle from
point A to point B, APB means the angle at P enclosed by sides PA and
PB,
etc. The symbol => means that by applying the cosine theorem we can get
the
right side from the left.

APB, AP, BP => AB
BAP, AB, AP => BAP
AB, AZ, BZ => BAZ
PAZ = BAP +/- BAZ, the sign depends on the actual configuration of
observer
and stars.
PAZ, AP, AZ => PZ. Hurrah, we found our co-latitude.
AZ, AP, PZ => APZ. This is the local hour angle of star A, from which
LST
(local sidereal time)

If we have GMT we get longitude from LST. So this procedure is a jack of
all
trades: Latitude from combined altitude, local time, position from GMT.
You
could even use it to find your ecliptic co-ordinates, should you ever
get
lost in the solar system.

Regards,

Herbert

Arthur Pearson wrote:

> Gentlemen,
>
> Also, Herbert states below that "When starting from a wildly wrong DR
> position, St. Hilaire will get you the right fix, albeit only after 2
or
> 3 iterations. That's not surprising, because direct methods will not
> even need a DR."  What are the "direct methods".

   

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