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Re: Computaion of table of offsets.
From: Andrés Ruiz
Date: 2007 Oct 30, 15:31 +0100

Gary, Consider the following figure and the equation of the offset:

R = (21600/2PI) / TAN(H)  [nm]

D [nm]

theta = ASIN(D/R)

Offset = R*(1.0-COS(theta)) [nm]

For a Circle of equal altitude: CoP

Centre: GP(Lat, Lon) = ( Declination, GHA)

Radius: zd = 90º-Ho

This circle is small circle on the surface of The Earth.

Using the Mercator projection, a straight line on a Mercator chart is a rhumb line or Loxodromic, but a CoP is curve depend on Ho.

To see this fact, on the Mercator map the CoPs of a fictitious body of altitude H = 0, 10, 20,…, 80, 90º are drawn:

So the radius of the CoP on a Mercator chart has no sense for 0º <= Ho <= 45.xxº

And R is only the radius of the CoP for hight altitudes.

I have a book that Wolfgang Köberer recommended me, is very technical, all mathematics, a good book: "La navigation astronomique" by Philippe Bourbon published by the Institut océanographique (Paris 2000), and formula for the offset at Bowditch is only a simplification of a most general one.

I generate a table of offsets using Bowditch equations and the result is OK for calcute the offset:

 H/D 0 5 10 15 20 25 30 35 40 45 H 0 nan nan nan nan nan nan nan nan nan nan 0 1 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 2 3 0 0 0 0 0 0 0 0 0 0 3 4 0 0 0 0 0 0 0 0 0 0 4 5 0 0 0 0 0 0 0 0 0 0 5 6 0 0 0 0 0 0 0 0 0 0 6 7 0 0 0 0 0 0 0 0 0 0 7 8 0 0 0 0 0 0 0 0 0 0 8 9 0 0 0 0 0 0 0 0 0 0 9 10 0 0 0 0 0 0 0 0 0 0.1 10 11 0 0 0 0 0 0 0 0 0 0.1 11 12 0 0 0 0 0 0 0 0 0 0.1 12 13 0 0 0 0 0 0 0 0 0.1 0.1 13 14 0 0 0 0 0 0 0 0 0.1 0.1 14 15 0 0 0 0 0 0 0 0 0.1 0.1 15 16 0 0 0 0 0 0 0 0.1 0.1 0.1 16 17 0 0 0 0 0 0 0 0.1 0.1 0.1 17 18 0 0 0 0 0 0 0 0.1 0.1 0.1 18 19 0 0 0 0 0 0 0 0.1 0.1 0.1 19 20 0 0 0 0 0 0 0 0.1 0.1 0.1 20 21 0 0 0 0 0 0 0.1 0.1 0.1 0.1 21 22 0 0 0 0 0 0 0.1 0.1 0.1 0.1 22 23 0 0 0 0 0 0 0.1 0.1 0.1 0.1 23 24 0 0 0 0 0 0 0.1 0.1 0.1 0.1 24 25 0 0 0 0 0 0 0.1 0.1 0.1 0.1 25 26 0 0 0 0 0 0 0.1 0.1 0.1 0.1 26 27 0 0 0 0 0 0 0.1 0.1 0.1 0.2 27 28 0 0 0 0 0 0 0.1 0.1 0.1 0.2 28 29 0 0 0 0 0 0.1 0.1 0.1 0.1 0.2 29 30 0 0 0 0 0 0.1 0.1 0.1 0.1 0.2 30 31 0 0 0 0 0 0.1 0.1 0.1 0.1 0.2 31 32 0 0 0 0 0 0.1 0.1 0.1 0.1 0.2 32 33 0 0 0 0 0 0.1 0.1 0.1 0.2 0.2 33 34 0 0 0 0 0 0.1 0.1 0.1 0.2 0.2 34 35 0 0 0 0 0 0.1 0.1 0.1 0.2 0.2 35 36 0 0 0 0 0 0.1 0.1 0.1 0.2 0.2 36 37 0 0 0 0 0 0.1 0.1 0.1 0.2 0.2 37 38 0 0 0 0 0 0.1 0.1 0.1 0.2 0.2 38 39 0 0 0 0 0 0.1 0.1 0.1 0.2 0.2 39 40 0 0 0 0 0 0.1 0.1 0.1 0.2 0.2 40 41 0 0 0 0 0.1 0.1 0.1 0.2 0.2 0.3 41 42 0 0 0 0 0.1 0.1 0.1 0.2 0.2 0.3 42 43 0 0 0 0 0.1 0.1 0.1 0.2 0.2 0.3 43 44 0 0 0 0 0.1 0.1 0.1 0.2 0.2 0.3 44 45 0 0 0 0 0.1 0.1 0.1 0.2 0.2 0.3 45 46 0 0 0 0 0.1 0.1 0.1 0.2 0.2 0.3 46 47 0 0 0 0 0.1 0.1 0.1 0.2 0.2 0.3 47 48 0 0 0 0 0.1 0.1 0.1 0.2 0.3 0.3 48 49 0 0 0 0 0.1 0.1 0.2 0.2 0.3 0.3 49 50 0 0 0 0 0.1 0.1 0.2 0.2 0.3 0.4 50 51 0 0 0 0 0.1 0.1 0.2 0.2 0.3 0.4 51 52 0 0 0 0 0.1 0.1 0.2 0.2 0.3 0.4 52 53 0 0 0 0 0.1 0.1 0.2 0.2 0.3 0.4 53 54 0 0 0 0 0.1 0.1 0.2 0.2 0.3 0.4 54 55 0 0 0 0 0.1 0.1 0.2 0.3 0.3 0.4 55 56 0 0 0 0 0.1 0.1 0.2 0.3 0.3 0.4 56 57 0 0 0 0.1 0.1 0.1 0.2 0.3 0.4 0.5 57 58 0 0 0 0.1 0.1 0.1 0.2 0.3 0.4 0.5 58 59 0 0 0 0.1 0.1 0.2 0.2 0.3 0.4 0.5 59 60 0 0 0 0.1 0.1 0.2 0.2 0.3 0.4 0.5 60 61 0 0 0 0.1 0.1 0.2 0.2 0.3 0.4 0.5 61 62 0 0 0 0.1 0.1 0.2 0.2 0.3 0.4 0.6 62 63 0 0 0 0.1 0.1 0.2 0.3 0.3 0.5 0.6 63 64 0 0 0 0.1 0.1 0.2 0.3 0.4 0.5 0.6 64 65 0 0 0 0.1 0.1 0.2 0.3 0.4 0.5 0.6 65 66 0 0 0 0.1 0.1 0.2 0.3 0.4 0.5 0.7 66 67 0 0 0 0.1 0.1 0.2 0.3 0.4 0.5 0.7 67 68 0 0 0 0.1 0.1 0.2 0.3 0.4 0.6 0.7 68 69 0 0 0 0.1 0.2 0.2 0.3 0.5 0.6 0.8 69 70 0 0 0 0.1 0.2 0.2 0.4 0.5 0.6 0.8 70 71 0 0 0 0.1 0.2 0.3 0.4 0.5 0.7 0.9 71 72 0 0 0 0.1 0.2 0.3 0.4 0.5 0.7 0.9 72 73 0 0 0 0.1 0.2 0.3 0.4 0.6 0.8 1 73 74 0 0 0.1 0.1 0.2 0.3 0.5 0.6 0.8 1 74 75 0 0 0.1 0.1 0.2 0.3 0.5 0.7 0.9 1.1 75 76 0 0 0.1 0.1 0.2 0.4 0.5 0.7 0.9 1.2 76 77 0 0 0.1 0.1 0.3 0.4 0.6 0.8 1 1.3 77 78 0 0 0.1 0.2 0.3 0.4 0.6 0.8 1.1 1.4 78 79 0 0 0.1 0.2 0.3 0.5 0.7 0.9 1.2 1.5 79 80 0 0 0.1 0.2 0.3 0.5 0.7 1 1.3 1.7 80 81 0 0 0.1 0.2 0.4 0.6 0.8 1.1 1.5 1.9 81 82 0 0 0.1 0.2 0.4 0.6 0.9 1.3 1.7 2.1 82 83 0 0 0.1 0.3 0.5 0.7 1.1 1.5 1.9 2.4 83 84 0 0 0.1 0.3 0.6 0.9 1.2 1.7 2.2 2.8 84 85 0 0 0.2 0.4 0.7 1 1.5 2 2.7 3.4 85 86 0 0.1 0.2 0.5 0.8 1.3 1.9 2.6 3.4 4.2 86 87 0 0.1 0.3 0.6 1.1 1.7 2.5 3.4 4.5 5.7 87 88 0 0.1 0.4 0.9 1.7 2.6 3.8 5.2 6.9 8.8 88 89 0 0.2 0.8 1.9 3.4 5.5 8 11.3 15.3 20.3 89

Best regards,

Andrés Ruiz

http://www.geocities.com/andresruizgonzalez

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