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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

Navigational Algorithms

http://www.geocities.com/andresruizgonzalez

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