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    Direct methods for finding position
    From: Herbert Prinz
    Date: 2002 Apr 12, 17:42 +0000

    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.
    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.
    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.
    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.
    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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