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A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: MP through linear fit of randomly timed observations
From: Peter Hakel
Date: 2012 Jan 28, 14:36 -0800
From: Jaap vd Heide <Jaap.vdHeide@xs4all.nl>
To: NavList@fer3.com
Sent: Thursday, January 26, 2012 2:49 PM
Subject: [NavList] Re: MP through linear fit of randomly timed observations
From: Peter Hakel
Date: 2012 Jan 28, 14:36 -0800
Jaap,
Your Eq. 12 correctly expresses the fact that a secant line across the graph of a quadratic function at points x1 and x2 has the same slope as the tangent line to the same graph at the midpoint between x1 and x2. The derivation preceding that is also correct, except I would not use differentials "dx" and "dy" in Eqs. 6-8, since you are really expressing finite differences here; the notation "Delta_x" and "Delta_y" is customary in this context. In fact, the development in Eqs. 6-8 is a commonly used proof that the derivative of Eq. 4 is indeed Eq. 5.
Yes, you can get the moment of transit "t" in this way and you will also get the curvature "alpha" of the altitude curve. However, by working with the slope (i.e. rate of change of altitude formula) the procedure automatically loses track of the "beta" (Eq. 4) i.e. "H_transit" (Eq. 3), which is really the main prize in getting the latitude in the end. (Indeed, there is no "beta" in your Eqs. 6-12). To get that, you have to back to the altitudes themselves, which is a point you make at the end of the first paragraph.
If you have three observations, then, as you say, picking either one of the three will get you the same answer: the unique parabola running through all three points. But what if you have more than three observations, which are not lying exactly on the one and the same parabola? That will certainly be the case with real life observations which are affected by errors.
Statistical methods are needed for cases like these, which are overdetermined and whose data are not 100% internally consistent. At this point you might do a least-squares parabolic fit to the data, which will give you both the "alpha" and the "beta". This leads me to believe that your procedure would provide a useful consistency check on the "alpha" and "t" but would not really yield the ultimate goal: "beta" = "H_transit" and the ex-meridian latitude. To do so, you would have to make a decision on which of the N>3 altitudes is the one to fix "beta" - a decision, which is difficult to make in a mathematically rigorous way.
Peter Hakel
Your Eq. 12 correctly expresses the fact that a secant line across the graph of a quadratic function at points x1 and x2 has the same slope as the tangent line to the same graph at the midpoint between x1 and x2. The derivation preceding that is also correct, except I would not use differentials "dx" and "dy" in Eqs. 6-8, since you are really expressing finite differences here; the notation "Delta_x" and "Delta_y" is customary in this context. In fact, the development in Eqs. 6-8 is a commonly used proof that the derivative of Eq. 4 is indeed Eq. 5.
Yes, you can get the moment of transit "t" in this way and you will also get the curvature "alpha" of the altitude curve. However, by working with the slope (i.e. rate of change of altitude formula) the procedure automatically loses track of the "beta" (Eq. 4) i.e. "H_transit" (Eq. 3), which is really the main prize in getting the latitude in the end. (Indeed, there is no "beta" in your Eqs. 6-12). To get that, you have to back to the altitudes themselves, which is a point you make at the end of the first paragraph.
If you have three observations, then, as you say, picking either one of the three will get you the same answer: the unique parabola running through all three points. But what if you have more than three observations, which are not lying exactly on the one and the same parabola? That will certainly be the case with real life observations which are affected by errors.
Statistical methods are needed for cases like these, which are overdetermined and whose data are not 100% internally consistent. At this point you might do a least-squares parabolic fit to the data, which will give you both the "alpha" and the "beta". This leads me to believe that your procedure would provide a useful consistency check on the "alpha" and "t" but would not really yield the ultimate goal: "beta" = "H_transit" and the ex-meridian latitude. To do so, you would have to make a decision on which of the N>3 altitudes is the one to fix "beta" - a decision, which is difficult to make in a mathematically rigorous way.
Peter Hakel
From: Jaap vd Heide <Jaap.vdHeide@xs4all.nl>
To: NavList@fer3.com
Sent: Thursday, January 26, 2012 2:49 PM
Subject: [NavList] Re: MP through linear fit of randomly timed observations
You know what,
I have that itchy feeling I missed something somewhere. So here is the (unfinished) setup of my reflections on the change of altitude formula.
The "trick" is in equations (4) - (12).
Did I go wrong? If so, where and why?
Or did I really stumble upon a possibility for fitting observations of the path of a heavenly object near M.P./transit without rogue computing power?
Shoot.
Jaap
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