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Re: AP terminology
From: Peter Hakel
Date: 2009 Nov 13, 15:16 -0800
Since LOPs are one-dimensional objects, you need precisely one parameter to characterize them.  It is this parametrization that amounts to "calculating the LOP directly" (answer to Geoffrey Kolbe's question).  Sumner used longitude as the parameter.  The parameter doesn't have to be longitude and in the case of LOP=meridian, it is in fact unsuitable.

The LOP is what it is, based on the Ho and GP, independent of any AP.  As I said in an earlier post, you need to have a coordinate system to do anything practical.  The AP is the origin of such a convenient local coordinate system.  We have some freedom in choosing where to place this AP=origin.  The computed intercept distance then tells us how far the chosen AP is from the LOP, and the computed azimuth tells us the orientation/direction.  That is what the solution of the celestial triangle AP-GP-Pole is about; it gives the relative position of the (independently existing and fixed) LOP and our arbitrarily chosen AP.  In practice St. Hilaire proceeds in reverse; we choose the AP first, solve the triangle, and then we construct the LOP at the appropriate distance from the AP and orientation with respect to true North.

On a Mercator plotting chart LOPs look like straight lines (for not too high Hs's).  A straight line lying in a plane can be defined by two points, or by one point and one direction.  Sumner uses the former; St. Hilaire uses the latter.  St. Hilaire is less work because one detail regarding the direction part is automatic; it is the angle between the LOP and the azimuth line toward the GP, which is always 90 degrees.  We can therefore plot the LOP at the right angle to the azimuth line (at the intercept distance) because Mercator mapping used in plotting charts is conformal, i.e. it preserves angles.

Peter Hakel

From: John Karl <jhkarl---.net>
To: NavList <navlist@fer3.com>
Sent: Fri, November 13, 2009 12:33:27 PM
Subject: [NavList 10635] Re: AP terminology, WAS: 2-Body Fix -- take three

No one has addressed my question of why the St Hilaire method
calculates an altitude at a location our ship is NOT at, when we've
just measured the altitude where our ship IS at.  (For politically
correct reasons, I'm not using the name of this location.)

Now lets go back to Sumner's 1837 calculation, where he picked three
different longitudes and calculated three points on the circular LOP.
This calculation is exact, and the equation for each point is the same
as the one of the two necessary in the St Hilaire method (thus each
Sumner point is half the work of a St Hilaire reduction).  And he
could calculate as many exact points as he wished.

So I'll put my question yet another way:  Why is the St Hilaire method
superior to Sumner's and consequently the only one used today??

I claim that the answer to this question has been made confusing
because of the conventional name (names?) used for the location of the
St Hilaire altitude calculation.  As evidence of this confusion I note
that some authors write that we need to assume some point because the
distance between the GP and the LOP is too great to plot, that there's
insufficient information to plot the LOP, or that iterations are
required to get exact points on the LOP.  The Sumner calculation
demonstrates that none of this is correct.

JK

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