|
|
|
|
|||||
|
||||||
Subject: Direct methods for finding position
From: Herbert Prinz (hprinz@XXX.XXX)
Date: Fri Apr 12 2002 - 13:42:06 EDT
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".
|