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Re: Review of the Longhand Haversine Sight Reduction method
From: Hanno Ix
Date: 2015 Jun 22, 10:28 -0700

Andres,

recently you showed us an example of doing multiplications by

the Napier bones approach. In return I'll show you here an example of the

Butterfly / Vedic method with detailed explanations.

This one has the visual advantage of lining up the columns vertically - nice but not essential. The real advantage is the possibility of stopping when the number of digits of the product is sufficient! I know of no other method that could do that.

The idea behind Vedic - a mystical name for something so dry - is collecting specifically those digits of the factors that contribute to a specific digit of the product. It turns out, those can be found in certain graphical patterns as described. They are easy to remember. I am sure I can spare you the math since you will understand when you see the details.

I only show finding the 4 leading digits of the product which is what I need for CelNav. There is a way, not described here, to continue the patterns so as to yield the full result. The www has examples of the full procedure.

Stopping a multiplication will cause a truncation, not a rounding, of the product. That truncation causes a systematic positive error, and I have made a simulation that shows that error's mean as being very close to 6 units of the 5th digit to the right. That's what one would expect. It can be included in the scheme as a carry 6, shown in red, into the 4th column. Also according to this simulation, most truncation errors are 6 +-1.5. Good enough for me.

This method reduced my multiplication error rate drastically to the point of becoming actually useful. Before I just could never trust my results. The orderly record keeping of even the smallest details make the process easy to check. Others don't need all that, they barely make any errors. I wish I did!

On Sun, Jun 21, 2015 at 10:41 AM, Andrés Ruiz <NoReply_AndresRuiz@fer3.com> wrote:

picking up the thread of what was said in:
http://fer3.com/arc/m2.aspx/HavDoniol-checker-Andr%C3%A9sRuiz-jun-2015-g31782

Altitude

By computation (Hanno) -> test OK

ZD = Zenith Distance

Hc = (90 - ZD) Calculated Altitude

hv (ZD) = n+(1-(n+m))*a

Same Name

n = hv (|B| - |dec|)

m = hv (|B| + |dec|)

Contrary Name

n = hv (|B| + |dec|)

m = hv (|B| - |dec|)

a = hv(t)

Azimuth

By computation (Bergman) -> test OK

hv (Z) = (a-n)/(1-(n+m))

Same Name

a = hv (90º-|dec|)

Contrary Name

a = hv (90º+|dec|)

m = hv (B+Hc)

n = hv (B-Hc)

N Latitude:

If LHA> 180◦, Zn = Z

if LHA< 180◦, Zn = 360◦ − Z

S Latitude:

If LHA> 180◦, Zn = 180◦ − Z

if LHA< 180◦, Zn = 180◦ + Z

Z by Hanno azimuth diagram -> test NOOK
An ambiguity in the value of azimuth may arise: see the attached spreadsheets.
Z is E/W as the name of the meridian angle.
But the N/S name is not determined.
RECAP: good enought if Z is mesured to avoid th ambiguity.
Sign criteria with azimuth diagram

If( LHA > 180º ) |t| = 360º-LHA

If( |t| > 90º ) |t| = 180º-|t|
enter with t in the diagram

Meridian angle

if (0 <= LHA && LHA <= 180)

t = -LHA [W]

else if (180 < LHA && LHA <= 360)

t = 360º - LHA [E]

hv(x) = hv(360º-x)

hv(x) = hv(-x)

-> hv(LHA) = hv(t)

Azimuth

sin LHA * cos dec = sin Z * cos Hc = C

Z = asin(sin(LHA) * cos(Dec) / cos(Hc))

0 <= LHA <= 360º

-180º <= t <= 180º (the diagram has values for t between 0 and 90º).

sin(t) = - sin(LHA)

cos(x) = cos(-x)

-90º <= Hc <= 90º -> sg(cos Hc) = +

-90º <= Dec <= 90º -> sg(cos Dec) = +

sg(C) = sg(sin Z) = sg(sin LHA) = - sg(sin t)

sin(x) = -sin(-x) = sin(180º-x) = -sin(180º+x)

Z -> Zn

0 <= Z <= 90º is E/W as the meridian angle is.

Sun: morning E, afternoon W

References

Doniol Hc http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-Rudzinski-jun-2014-g27982

http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-HannoIx-jun-2014-g27989

http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-Rudzinski-jun-2014-g27991

Hanno Hc http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-HannoIx-nov-2014-g29121

Bergman Z http://fer3.com/arc/m2.aspx/Longhand-Sight-Reduction-Bergman-nov-2014-g29441
Greg Rudzinski article http://fer3.com/arc/m2.aspx/Gregs-article-havDoniol-Ocean-Navigator-FrankReed-jun-2015-g31625

Doniol, NavList : Hanno Ix, Lars Bergman, Greg Rudzinski
Other diagrams for Z:
Rust and Weir

--
Andrés Ruiz
Navigational Algorithms
http://sites.google.com/site/navigationalalgorithms/

Attached File: 131824.hv-sr.nn1.xlsx (no preview available)

Attached File: 131824.hv-sr.nn2.xlsx (no preview available)

Attached File: 131824.hv-sr.nn3.xlsx (no preview available)

Attached File: 131824.hv-sr.nn4.xlsx (no preview available)

Attached File: 131824.hv-sr.ns.xlsx (no preview available)

Attached File: 131824.hv-sr.sn.xlsx (no preview available)

Attached File: 131824.hv-sr.ss.xlsx (no preview available)

Attached File: 131824.longhand-haversine-sight-reduction.docx (no preview available)

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Altitude

By computation (Hanno) -> test OK

Azimuth

By computation (Bergman) -> test OK

Sign criteria with azimuth diagram

Meridian angle

Azimuth

References

Other diagrams for Z:

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