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Re: Greg's article on hav-Doniol in "Ocean Navigator"
From: Hanno Ix
Date: 2015 Jun 15, 06:38 -0700

Christophe,

you are certainly invited to join this discussion!

First, I commend you on the form and clarity of your hav/4 table if you allow me this

change of name. Very neat!

As far as your approach goes, I cannot judge the practicality of it - personally I would

be perfectly happy with it. See, I have proposed a similar approach before, not quite as

neat, and I ran into a wall. Greg Rudzinski has made his index card (RIC) as a

measurement of practicality, and my approach did not really fit. The reason was

keeping track of the signs in sums with more than 2 components. The sign rules

have been discussed in length again and again - even for sums with only

2 components. Personally I am totally comfortable with the standard algebraic

sign rules but your CelNav friend might not be.

Also, the pain of doing multiplication is very different from person to person as
I have observed. Greg and Peter fly through it, I labor and make many mistakes.

A "scientific" approach would count the number of elementary steps i.e. of elementary

additions / multiplication of just 2 digits in the process and assigning a probability of
correctness to them - for the sake of discussion here say 99 % for a single addition,
95% for a single multiplication and 90% for a sign decision - and finally arrive at an
estimated overall probability of correctness by multiplying all in to a single product P.
Since P is a product it falls very fast with the number of the components that go into it

And your approach and my own prior approach have a lot of such elementary operations!

You could also assign times for individual operation and ad them up for comparison.

I fear your proposal, as well as my own prior proposal, would not fare very well either way.

But if you are interested in pursuing it further I encourage you to make such

measurement based comparisons and see what you get. I'd be interested in your

results.

On Mon, Jun 15, 2015 at 12:56 AM, Christophe C. <"NoReply_ChristopheC."@fer3.com> wrote:

Congratulations again to the article – and thanks for the many interesting discussions on that topic around the past few months, they’ve kept my brain busy!
And wondering: couldn’t we also remove the last multiplication? Well, it turns out that the formula can be rewritten as (use signed values):
hav(ZD) = ½ hav(L-d) - ½ hav(L+d) + ¼ hav(L+d+LHA) + ¼ hav(L+d-LHA) + ¼ hav(L-d+LHA) + ¼ hav(L-d-LHA)
This is actually very easy to compute with a table of ¼ haversines (“eighth-versine”?? I‘ll call it “qhav” for now). See attached 2-page table and form. No multiplication anymore, a few sums, one substraction, one division by 4 (can be done "visually"): the computation takes me approx. 5 mins, and slightly less if the AP is such that L-D (or L+d) and LHA are whole numbers of degrees.
Regarding precision, with the table rounded to 5 decimals the error on ZD is below 1nm (it does go up to a few nm for 2 points opposite each other near the poles).
Thanks to everyone in the group – I’m learning a lot and enjoying it.
Christophe

Attached File:

(quart-haversine-form.pdf: Open and save or View online)

Attached File:

(quart-haversine-table.pdf: Open and save or View online)

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