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    How good is St. Hilaire?
    From: Richard Reed
    Date: 2010 Feb 25, 17:38 -0800

    I'm new to the calculations discussed here, so forgive me if this has been done to death. Also, sorry for the lack of pictures, but I'll make the effort to get set up to do them if necessary.

    After reading a few descriptions of how two altitude-intercept lines of position (LOP) are plotted, I finally realised that acccurate (to observation limits) circles of equal altitude were being calculated, and their intersection estimated by plotting tangent lines near what must be the intersection, the point from which both sights were done.

    Knowing only this, I was intrigued as to how accurately this process found the intersection of the circles, and what effect bad dead reckoning (DR) had on the process. I did see the threads about John Karl's exact mathematical method, and have sent away for his book, but I was still curious about the plotting method.

    I decided to create an easier plane geometrical example representing altitude circles and come up with expressions for errors introduced by a representation of a bad DR. I drew two circles of equal radius that intersected at 90 degrees. I chose an 'error DR' on the inside bisector between the exact LOP's at the circle intersections. I then drew the two symmetrical 'error LOP's' on the azimuths through the error DR location.

    I could see that if I 'slid' the DR point on the bisector inside and away from the circle intersection, the error LOP's intersected further away on the other side of the circle intersection as the errors of the azimuths through the DR increased and the error LOP's slid down the circles. When the error azimuths reached 45 degrees of error, the error LOP's were parallel, or infinite error from the intersection.

    This symmetrical setup made the geometry a little easier and I got expressions for the LOP intersection error and the 'DR error' (its distance from the circle intersection), both in terms of the 'error angle' of the azimuths through the DR point. I typed these into my 15-year-old version of Mathematica and found the following:

    - even when the 'azimuth error angle' was 27.5 degrees, the ratio of intersection error to DR error was less than one.
    - at an azimuth error angle of 13 degrees, the error ratio is down to .25, and decreases nearly linearly as the error angle to zero for the correct azimuths.

    To put this into perspective, for middling altitudes, resulting in circle radii of 2500 nm or more, the 13 degree azimuth error represents a DR error of about 500 nm or more, so the error ratio most often encountered is nearer .05 or less. Even if DR is 100 nm off, the plotted intersection error will be less than 5 nm, and even less if the sights are of bodies with GP's more than 2500 nm away.

    Even better, if the distance between DR and the LOP intersection is less than 100 nm and the GP distance is 2500 nm or greater, the first result will be within 5 nm and one more iteration starting from the LOP intersection will theoretically get within 0.0125 nm, much smaller than a drawn line.

    Even though my example lacks generality, I'm convinced of the robust practicality of St. Hilaire -- no surprise, but I just had to take it apart ;-)

    Richard

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