Hi Hanno Ix
It would be helpful if I read all of your questions before shooting off a response!
The limitations of the Bygrave are primarily a computational problem related to the trigonometric functions. If you look closely at the multiplicity of scales being produced by the list, you will note that the numbers
are bunched and separated as a function of the log distance of the trig function. Hence around zero degrees, the log(cosine) bearly changes, resulting in an in-ability to resolve the angle on the scale. The scales on the bygrave are are also truncated due
to the long length required for angles above 89 degrees. Again, you will see the scales produced with this truncation. When you play with the equations, you will rapidly see the limitations.
The commander will choose his navigational objects wisely. You want to get the lines of position to cross at steep angles. Depending only on meridian shots and prime meridian shots was a method used long ago. Modern
methods advocate using LOPs independent of these special cases. You shouldn't avoid these lines, rather, they are just another line without special emphasis.
Consider the rapid fire sun observation recently on the list posed by Frank. Observation after observation of the sun is taken. Multiple LOPs are created. These lines cross at shallow angles, yet with enough lines,
we can resolve both longitude and latitude to a reasonable margin of safety. Each of these LOPs can be solved with a Bygrave but the cocked hat created by them will require a least squares fit (for accuracy). Nothing special about these LOPs.
Just be careful when you know you are treading into the problem areas!!
Best Regards
Brad
From: NavList@fer3.com [NavList@fer3.com] On Behalf Of Brad Morris [bmorris@tactronics.com]
Sent: Saturday, July 25, 2009 1:41 PM
To: NavList@fer3.com
Subject: [NavList 9223] Re: The Bygrave
Hi Hanno Ix
modest angular errors at merdian crossings cause minor latitude errors. The primary purpose of a merdian shot is to get a special line of position which is exactly your latitude.
angular errors at 90 degree hour angles cause errors in longitude. This is known as a prime meridian shot and gives you a special line of position which is exactly your longitude.
Bear in mind that these two special lines can special just for the outcome. The Bygrave will give you lines of position (when properly plotted) for any angle in between, hence these errors are not so terrible. For
the purist mathematician, the erroneous result is troubling, but for practical navigation, not so much!
Best Regards
Brad
From: NavList@fer3.com [NavList@fer3.com] On Behalf Of Hanno Ix [hannoix@sbcglobal.net]
Sent: Friday, July 24, 2009 6:54 PM
To: NavList@fer3.com
Subject: [NavList 9216] Re: The Bygrave
Thank you for your response, Brad!
However, if you permit, answers to questions 1 and 2 seem not quite complete.
In my Engineering past it has bcome my experience that computational issues typically point to ill-posed problems. Do computational issues in our application here similarly indicate we may be using a disadvantageous constellation for the problem we are trying
to solve? I am thinking of cases like shooting "stars", all very close to each other, for position, for instance.
Are there alternative solutions known from the manuals of the original Bygrave or the German MHR - provided they still exist - for special cases like the ones I mentioned? What was the poor Commander to do in such instances? I assume he would try to shoot a
different constellation rather than mess with computational issues in the margins, no?
Regards again
H
--- On Fri, 7/24/09, Brad Morris <bmorris@tactronics.com> wrote:
From: Brad Morris <bmorris@tactronics.com>
Subject: [NavList 9213] Re: The Bygrave
To: "NavList@fer3.com" <NavList@fer3.com>
Date: Friday, July 24, 2009, 8:53 AM
Hi Hanno Ix
See my answers interspaced below
Best Regards
Brad
From: NavList@fer3.com [mailto:NavList@fer3.com]
On Behalf Of hannoix@sbcglobal.net
Sent: Friday, July 24, 2009 11:37 AM
To: NavList@fer3.com
Subject: [NavList 9212] Re: The Bygrave
May I please ask some questions?
1. I am not a sailor. So, what are the useful ranges of a Bygrave? In other terms, what are the critical trigonometrical/nautical Regardssituations where the knowledge of angles >89deg or angles <1deg is important?
Suppose your celestial body is the sun. If reducing for a meridian crossing (sun on your line of longitude), then the hour angle T is 0 (less than 1 degree). As a practical matter,
the Bygrave is quite compressed under 5 degrees for the LogCosine scale, so reductions when the sun is close to your meridian suffer from accuracy issues.
If reducing when the sun is directly east or west of you, when the hour angle is 90 degrees (sunrise, sunset) similar issues result.
2. Are there practical approximations by polynomials etc. for such extreme cases?
Just use the standard azimuth and altitude equations that can be directly reduced using a calculator.
3. More generally: Are we aiming at historical replicas of the Bygrave, MHR, etc. or are we seeking at a modern implemetation of the idea? Or both?
Depends upon who you talk to. Due to cost, we probably won’t be looking at historical replicas.
4: Similarly: Are we interested in the Bygrave just because it is so interesting - which it is! - or are we looking for the very simplest means to do accurate sight reduction? For the latter, there might be attractive alternatives.
HO229 offers accuracies to 0.1 arc-minutes, about 10-20X the accuracy of the Bygrave. It is heavier, larger and requires tabular lookup instead of slide rule manipulation.
Others will have different preferences for tabular reduction or for computer aided reduction. The beauty of the Bygrave is the small size, small weight and rapidity of reduction
Regards</table
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