Tz World Map for Azimuth Calculations

The 27/28 Mz "LONG PATH" TECHNICAL NOTICE (English only)


REMINDER

     The real meaning of the word "Tz" is for hours "Time Zone",
     then there is 24 Tz (for 24 hours) on the horizontal axis.

    The Earth is a 360 degree sphere, then one Tz is also 15 degree,
    and as the Earth circumference is 40.000 Km then one Tz is also
    about 1666 Km at the equator level as weel as for each meridian.

   Then there is also 12 Tz on the vertical axis !!! The Tz marking is
    very interesting for Ham radio using Time or Distance as Angle.

  To trace Tz on your own world map symply trace lignes at each 15 degree
  and mark them 1 to 24 in horizontal axis, and 1 to 11 in vertical axis.

Formulas are :

   Arc AB  = ArcCos ((Cos TzA x Cos TzB) + (Sin TzA x Sin TzB x Cos TZ))
 
   Angle A = ArcCos ((Cos TzB - (Cos AB x Cos TzA)) / (Sin AB x Sin TzA))

   Angle B = ArcCos ((Cos TzA - (Cos AB x Cos TzB)) / (Sin AB x Sin TzB))

Usable directly on a good scientific calculator able to use parenthesis.
(We have compute examples with "Casio" type Fx-180, Fx-8500G and Fx-9750G)


Example No 1 : Mexico with Mauritius
Mexico    is about LATITUDE 4.5 Tz  then  TzA =  4.5 x 15 =  67.5
Mauritius is about LATITUDE 7.5 Tz  then  TzB =  7.5 x 15 = 112.5 
Time Zone is about         10.5 Tz  then  TZ  = 10.5 x 15 = 157.5  

Arc AB  = ArcCos ((Cos TzA x Cos TzB) + (Sin TzA x Sin TzB x Cos  TZ))
------             C  67.5   C 112.5     S  67.5   S 112.5   C 157.5
        = 159 degree length

        (then the distance is 159 x 1666 / 15 = about 17.600 Km)

Angle A = ArcCos ((Cos TzB - (Cos AB x Cos TzA)) / (Sin AB x Sin TzA))
-------              112.5    C  159   C  67.5      S  159   S  67.5
        = 95 degree (direct Azimuth toward East)

Angle B = ArcCos ((Cos TzA - (Cos AB x Cos TzB)) / (Sin AB x Sin TzB))
-------            C  67.5    C  159   C 112.5      S  159   S 112.5
        = 85 degree

        (for West Azimuth you must compute 360 - 85 = 275 degree)  

Example No 2 : France with New Caledonia
France        is about LATITUDE  3 Tz then  TzA =  3 x 15 =  45.0
New Caledonia is about LATITUDE  7 Tz then  TzB =  7 x 15 = 105.0
Time Zone     is about          11 Tz then   TZ = 11 x 15 = 165.0

Arc AB  = ArcCos ((Cos TzA x Cos TzB) + (Sin TzA x Sin TzB x Cos  TZ))
------                  45       105          45       105       165
        = 147 degree length

        (then the distance is 147 x 1666 / 15 = about 16.300 Km)

Angle A = ArcCos ((Cos TzB - (Cos AB x Cos TzA)) / (Sin AB x Sin TzA))
-------                105       147        45         147        45
        = 30 degree (direct Azimuth`toward East)

Angle B = ArcCos ((Cos TzA - (Cos AB x Cos TzB)) / (Sin AB x Sin TzB))
-------                 45       147       105         147       105
        = 20 degree

        (for West Azimuth you must compute 360 - 20 = 340 degree)  

Example No 3 : Brasil with France
Brasil    is about TzA = 6.5 Tz x 15 = 97.5
France    is about TzB = 3   Tz x 15 = 45
Time Zone is about TZ  = 4   Tz x 15 = 60

Arc AB  = ArcCos ((Cos TzA x Cos TzB) + (Sin TzA x Sin TzB x Cos  TZ))
------                97.5        45        97.5        45        60
        = 75 degree length

        (then the distance is 75 x 1666 / 15 = about 8.000 Km)

Angle A = ArcCos ((Cos TzB - (Cos AB x Cos TzA)) / (Sin AB x Sin TzA))
-------                 45        75      97.5          75      97.5
        = 40 degree (direct Azimuth toward East)

Angle B = ArcCos ((Cos TzA - (Cos AB x Cos TzB)) / (Sin AB x Sin TzB))
-------               97.5        75        45          75        45
        = 120 degree

        (for West Azimuth you must compute 360 - 120 = 240 degree)  

Example No 4 : New Caledonia with Mexico
New Caledonia  is about TzA = 7   Tz x 15 = 105
Mexico         is about TzB = 4.5 Tz x 15 = 67.5
Time Zone      is about TZ  = 7   Tz x 15 = 105
 (you compute 24 Tz - 17 Tz = 7 regarding the Tz world map)

Arc AB  = ArcCos ((Cos TzA x Cos TzB) + (Sin TzA x Sin TzB x Cos  TZ))
------                 105      67.5         105      67.5       105
        = 110 degree length

        (then the distance is 110 x 1666 / 15 = about 12.000 Km)

Angle A = ArcCos ((Cos TzB - (Cos AB x Cos TzA)) / (Sin AB x Sin TzA))
-------               67.5       110       105         110       105
        = 70 degree (direct Azimuth toward East)

Angle B = ArcCos ((Cos TzA - (Cos AB x Cos TzB)) / (Sin AB x Sin TzB))
-------                105       110       67.5        110       67.5
        = 100 degree

        (for West Azimuth you must compute 360 - 100 = 260 degree)  


Example No 5 : France with Indien Ocean
France       is TzA = 3   Tz x 15 = 45
Indian Ocean is TzB = 7.5 Tz x 15 = 112.5
Time Zone    is  TZ = 4   Tz x 15 = 60 

Arc AB  = ArcCos ((Cos TzA x Cos TzB) + (Sin TzA x Sin TzB x Cos  TZ))
------                  45     112.5          45     112.5        60
        = 87 degree length

        (then the distance is 87 x 1666 / 15 = about 9.600 Km)

Angle A = ArcCos ((Cos TzB - (Cos AB x Cos TzA)) / (Sin AB x Sin TzA))
-------              112.5        87        45          87        45
        = 125 degree (direct Azimuth toward East)

Angle B = ArcCos ((Cos TzA - (Cos AB x Cos TzB)) / (Sin AB x Sin TzB))
-------                 45        87     112.5          87     112.5
        = 40 degree

        (for West Azimuth you must compute 360 - 40 = 320 degree)  

Example No 6 : France with SE Australia
France       is TzA = 3   Tz x 15 = 45
SE Australia is TzB = 8.5 Tz x 15 = 127.5
Time Zone    is  TZ = 8.5 Tz x 15 = 127.5

Arc AB  = ArcCos ((Cos TzA x Cos TzB) + (Sin TzA x Sin TzB x Cos  TZ))
------                  45     127.5          45     127.5     127.5
        = 140 degree length

        (then the distance is 140 x 1666 / 15 = about 15.500 Km)

Angle B = ArcCos ((Cos TzB - (Cos AB x Cos TzA)) / (Sin AB x Sin TzA))
-------              127.5       140        45         140        45
        = 100 degree (direct Azimuth toward East)

Angle B = ArcCos ((Cos TzA - (Cos AB x Cos TzB)) / (Sin AB x Sin TzB))
-------                 45       140     127.5         140     127.5
        = 60 degree

        (for West Azimuth you must compute 360 - 60 = 300 degree)  

Example No 7 : Brazil with Mauritius
Brazil    is TzA = 6.5 Tz x 15 =  97.5
Mauritius is TzB = 7.5 Tz x 15 = 112.5
Time Zone is  TZ = 8   Tz x 15 = 120

Arc AB  = ArcCos ((Cos TzA x Cos TzB) + (Sin TzA x Sin TzB x Cos  TZ))
------                92.5     112.5        92.5     112.5       120
        = 116 degree length

        (then the distance is 116 x 1666 / 15 = about 13.000 Km)

Angle A = ArcCos ((Cos TzB - (Cos AB x Cos TzA)) / (Sin AB x Sin TzA))
-------              112.5       116      92.5         116      92.5
        = 115 degree (direct Azimuth toward East)

Angle B = ArcCos ((Cos TzA - (Cos AB x Cos TzB)) / (Sin AB x Sin TzB))
-------               92.5       114     112.5         114     112.5
        = 105 degree

        (for West Azimuth you must compute 360 - 105 = 255 degree)  

The 27/28 Mz "LONG PATH" TECHNICAL NOTICE (English only)

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