# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Graphically solving haversine calculation
From: Jaap vd Heide
Date: 2015 Nov 12, 00:11 -0800

Francis,

I will get back on it in more detail tonight (CET), but what I did is basically constructing the haversine using a unit circle like you can do for the sine and cosine (see https://en.wikipedia.org/wiki/Trigonometry under "extending the definitions"). You might recognise that when cos(a) ranges from 1 (at a=0) to -1 (at a=180), 1-cos(a) will range from 0 (at a=0) to 2 (at a=180).
Dividing it all by 2, so (1-cos(a))/2 then ranges from 0 (at a=0) to 1 (at a=180). (1-cos(a))/2 = haversine(a) by definition.

The idea of solving the haversine form of the spherical law of cosines I got from Erik de Man's nautical pages. He has a worksheet for graphically reducing the sight with the sine/cosine form of the equation.

My first mentioning of graphically reducing the sight as I propose was in a post on condensed tables. Not a whole lot more than referring to the model I had put on Geogebra.

I don't think I can do a better job than John Karl did in "Celestial navigation in the GPS age" in showing it all comes down to applying the spherical law of cosines to the navigational triangle.

Regards,

Jaap

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