Example of Great Circle Sailing Calculation

Vessel depart from Abo Lat 60°27’N Longitude 022°16’E to Amarracao  Lat 02°53’S Longitude 041°40’W.Find initial course, final course, distance  and vertex position make during voyage?

Lat  60°27’N                Long   022°16’E

Lat  02°53’ S                Long   041°40’W

                                   -------------------------

                                  d’long A to B 63°56’ W  LHA=P

                                  d’long B to A 63°56’  E  LHA= 296°04’ P=63°56’

PA = 90° - 60°27’

      = 29°33’

PB = 90° + 02°53’

      = 92°53’

Dist AB

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cos AB = cos PA cos PB + sin PA sin PB cos APB

cos AB = cos 29°33’ cos 92°53’ + sin 29°33’ sin 92°53’ cos 63°56’

AB       = 80°03.38’

AB       = 4803.38 NM

Initial course

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cos A = cos PB - cos PA cos AB / (sin PA sin AB)

cos A = cos  92°53’ - cos 29°33’ cos 80°03.38’ / (sin 29°33’ sin 80°03.38’ )

A  =  N  114°22.8’W

A  =  S  65°37.2’W

A  =  245°37.2’ T

Final course

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cos B = cos PA - cos PB cos AB / (sin PB sin AB)

cos B = cos  29°33’ - cos 92°53’ cos 80°03.38’/ (sin  92°53’ sin 80°03.38’)

B  =   N  26°43.75’ E

Final  = S  26°43.75’ W

         = 206°43.75’ T

PV1

Sin PV1 = cos © PA cos © A

Sin PV1 = sin PA sin A

Sin PV1 = sin 29°33’ sin 65°37.2’

PV1 = 26˚41.56’

Vertex lat V1 = 63˚18.44’N

P

Sin © PA = tan © P tan © A

Cos PA = 1 / tan P tan A

Tan P = 1/ tan A cos PA

Tan P = 1/ tan 65°37.2’ cos 29°33’

P = 27˚31.07’ E

Long A    022˚16.0’E

P            27˚31.07’ E

Long v1   049˚47.07’E

Vertex position  63˚18.44’N   049˚47.07’E