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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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