FOAIE DE CALCULE

TRNSFORMAREA UNGHIURILOR

φ1 λ1 φ2 λ2 Δλ φ2-φ1
40.416 -68.482 38.178 -9.465 59.017 -2.238
din gms in grd 40.69333 -68.80333 38.29667 -9.77500 59.028 -2.397
din grd. in rad. 0.71023 -1.20084 0.66840 -0.17061 1.03024 -0.04183
din rad in grad
din grd in gms 40.416
din gms in zecimi de gms
din zecimi de gms in gms
din gms in gms
din gms in Mm 2441.6 -4128.2 2297.8 -586.5 3541.7 -143.8
din gms in Mm poz 2441.6 4128.2 2297.8 586.5 3541.7 143.8
POZITIV 59.0

FUNCTII TRIGONOMETRICE

φ1 λ1 φ1 λ2 Δλ φ2-φ1
NR./FCT. 40.693 -68.803 38.297 -9.775 59.028 -2.397
unghiuri in RAD 0.71023 -1.20084 0.66840 -0.17061 1.03024 -0.04183
SIN 0.65201 -0.93234 0.61973 -0.16978 0.85742 -0.04182
COS 0.75821 0.36157 0.78481 0.98548 0.51461 0.99913
TG 0.85993 -2.57860 0.78966 -0.17228 1.66615 -0.04185
CTG 1.16288 -0.38781 1.26637 -5.80448 0.60019 -23.89250
COSEC 1.53372 -1.07256 1.61360 -5.88999 1.16629 -23.91342
sin + 0.65201 0.93234 0.61973 0.16978 0.85742 0.04182
lgsin 9.81425 9.96958 9.79220 9.22989 9.93319 8.62136
cos + 0.75821 0.36157 0.78481 0.98548 0.51461 0.99913
lgcos 9.87979 9.55819 9.89477 9.99365 9.71148 9.99962
tg + 0.85993 2.57860 0.78966 0.17228 1.66615 0.04185
lgtg 9.93446 0.41138 9.89744 9.23624 0.22171 8.62174
ctg + 1.16288 0.38781 1.26637 5.80448 0.60019 23.89250
lgctg 0.06554 9.58862 0.10256 0.76376 9.77829 1.37826
cosec + 1.53372 1.07256 1.61360 5.88999 1.16629 23.91342
lgcosec 0.18575 0.03042 0.20780 0.77011 0.06681 1.37864

cosM= 0.71030 lgtgD= 1.28055 ctgD1= 0.30696 ctgD2=

cosM + = 0.71030 lgtgD-tf= 1.28055 tgD1= 3.25777 tgD2=
lgcosM= 9.85144 tgD= 19.07895 D1 in rad= 1.27297 D2 in rad=
M in rad.= 0.78088 D in rad.= 1.51843 D1 in grad= 72.93568 D2 in grad=
M in grade= 44.74107 D in grade= 86.99966 D1 in dms= 72.561 D2 in dms=
M in gms= 44.445 D in gms= 87.000 D1 in dzm= 72.9 D2 in dzm=
M in Mm= 2684.5 D in gzm= 87.0 D1'+' dzm= 72.9 D2'+' dzm=
cosD= 0.05234 360-D1= 287.1 360-D2=
secD= 19.10514 D1+180= 252.9 D2+180=
secD + = 19.10514 180-D1= 107.1 180-D2=
lgsecD= 1.28115 D1= 72.9 D2=
λ1 in rad.= -1.201 λ2 in rad.=
 lgctgΔλv1= 10.32717 lgctgΔλv2= 10.17399 Δλv1 in rad.= 0.440 λv2 in rad.=
lgctgΔλv1-tf.= 0.32717 lgctgΔλv2-tf.= 0.17399 λv1 in rad.= -0.761 λv2 in rad.=
ctgΔλv1= 2.12410 ctgΔλv2= 1.49278 λv1 in grd.= -43.593 λv2 in grd.=
tgΔλv1= 0.47079 tgΔλv2= 0.66989 λv1 in gms= -43.356 λv2 in gms=
Δλv1 in rad.= 0.44001 Δλv2 in rad.= 0.59023 λv1 in grd= -43.593 λv2 in grd=
Δλv1 in grade= 25.21050 λv2 in grade= 33.81783 λv1 in gms= -43.356 λv2 in gms=
Δλv1 gms= 25.126 Δλv2 gms= 33.491
Δλv1 gms= 25.126 Δλv2 gms= -33.491
1.Calculul distantei ortodromice (M)
VERIFICARE 59.028 lga= 19.60646 lgb=
59.017 lga-tf.= 9.60646 lgb-tf.=
a= 0.40407 b=

0.4040725
 FUNCTII LOGARITMICE
Δφc= -185.6
Δφc + = 185.6
Δλ lgΔφc= 2.268657
59.017
59.028
1.03024

59.0

M D1 D2

1.27297 -1.17729
0.95598 -0.92357
0.7102951
3.25777 -2.40874

0.95598 0.92357
9.98045 9.96547
0.71030
9.85144
3.25777 2.40874
0.51292 0.38179

-0.41515 lgcosφv1= 19.86024 lgcosφv2= 19.86024

-2.40874 lgcosφv1-tf= 9.86024 lgcosφv2-t 9.86024
-1.17729 cosφv1= 0.72483 cosφv2= 0.72483
-67.45399 φv1 rad= 0.76001 φv2 rad= 0.76001
-67.272 φv1 grad= 43.54523 φv2 grad= 43.54523
-67.5 φv1 dms= 43.327 φv2 dms= 43.327
67.5 φv1 dms= 43.327 φv2 dms= -43.327
292.5
247.5
112.5
112.5
-0.171
 -0.590
-0.761 Δλv1 in grd= 25.2105
-43.593 Δλv2 in grd= 33.81783
-43.356 Δλ in grd.= 59.02833
136.407 118.0567
136.244 118.034

dromice (M) dist.lox.m 3.Calculul drumului initial D1 4.Calculul drumului final D2

29.48604 lgtgD= 11.28055 lgm= 19.84403 lgn= 19.59254 lgm= 19.89604
9.48604 lgtgD-tf.= 1.28055 lgm-tf.= 9.84403 lgn-tf.= 9.59254 lgm-tf.= 9.89604
0.30622 19.07895 m= 0.69829 n= 0.39133 m= 0.78711
 LATITUDINI CRESCANDE
φc=7915.70447lg{tg(45+φ/2)[(1-esinφ/1+esinφ)la put(e/2)]}
φ1 φ2
φ= 40.693 38.297
φ in rad.= 0.71023 0.66840
7915.70447 7915.70447
e= 0.0818191986 0.0818191986
45+φ/2= 1.14051 1.11960
tg(45+φ/2)= 2.17883 2.06385
1-esinφ= 0.94665 0.94929
1+esinφ= 1.05335 1.05071
(1-esinφ/1+esinφ)= 0.89871 0.90348
(1-esinφ/1+esinφ)la put.e/2= 0.99564 0.99586
tg(45+φ/2)(1-esinφ/1+esinφ)la put(e/2)= 2.16933 2.05530
lgtg(45+φ/2)(1-esinφ/1+esinφ)la put(e/2)= 0.33633 0.31287
φc= 2662.25471 2476.62083
0.25471
φc1 φc2
ul drumului final D2
lgn= 19.57049 19.86024
lgn-tf.= 9.57049 9.86024
n= 0.37196
 REZOLVARE PROBLEME NAVIGATIE ORTODROMICA
PUNCT DE PLECARE φ1= 40 41.6 N λ1= 068 48.2 W Boston
PUNCT DE SOSIRE φ2= 38 17.8 N λ2= 009 46.5 W Lisabona
1.Calculul distantei ortodromice (M) cosM=sinφ1sinφ2+cosφ1cosφ2cosΔλ
φ2= 38 17.8 lgsinφ1= 9.81425 lgcosφ1= 9.87979
-φ1= 40 41.6 lgsinφ2= 9.79220 lgcosφ2= 9.89477
Δφ= -02 23.8 lg a= 9.60646 lgcosΔλ= 9.71148
Δφ= 143.8 Mm a= 0.40407 lg b= 9.48604
λ2= -009 46.5 b= 0.30622
-λ1= -068 48.2 a= 0.40407
Δλ= 059 01.7 +b= 0.30622 M= 44 44.5
Δλ= 3541.7 Mm cosM= 0.71030 M= 2684.5 Mm
lgcos M= 9.85144
2.Calculul distantei loxodromice (m) tgD=Δλ/Δφc m=ΔφsecD
φc2= 2476.62 lgΔλ= 3.54921 lgΔφ= 2.15776
-φc1= 2662.25 +cologΔφc= 7.73134 lgsecD= 1.28115
Δφc= -185.63 lgtgD= 1.28055 lgm= 3.43891
lgΔφc= 2.26866 D= 87 00.0 m= 2747.3 Mm
cologΔφc= 7.73134 D= 87 0 m= 2747.3
Diferenta dintre distanta loxodromica si distanta ortodromica C=m-M -M= 2684.5
C= 62.9 Mm
3.Calculul drumului initial D1 4.Calculul drumului final D2
ctgD1=tgφ2cosφ1cosecΔλ-sinφ1ctgΔλ ctgD2=-tgφ1cosφ2cosecΔλ+sinφ2ctgΔλ
lgtgφ2= 9.89744 lgsinφ1= 9.81425 lgtgφ1= 9.93446 lgsinφ2= 9.79220
lgcosφ1= 9.87979 lgctgΔλ= 9.778287 lgcosφ2= 9.89477 lgctgΔλ= 9.77829
lgcosecΔλ= 0.06681 lgn= 9.59254 lgcosecΔλ= 0.06681 lgn= 9.57049
lgm= 9.84403 n= 0.39133 lgm= 9.89604 n= 0.37196
m= 0.69829 m= 0.78711

m= 0.69829 D1= 72 9 m= 0.78711

+n= 0.39133 D2= 112 5 +n= 0.37196
ctgD1= 0.30696 D1= 72 56.1 D2= 67 27.2 tgD2= -0.41515
5.Calculul coordonatelor vertexului
a.Latitudinea vertexului
cosφv1=cosφ1sinD1 cosφv2=cosφ2sinD2
lgcosφ1= 9.87979 lgcosφ2= 9.89477
lgsinD1= 9.98045 lgsinD2= 9.96547
lgcosφv1= 9.86024 lgcosφv2= 9.86024
φv1= 43 32.7 φv2= -43 32.7
b.Longitudinea vertexului
ctgΔλv1=sinφ1tgD1 ctgΔλv2=sinφ2tgD2 Verificare
lgsinφ1= 9.81425 lgsinφ2= 9.79220 Δλv1= 025 12.6
lgtgD1= 0.51292 lgtgD2= 0.38179 Δλv2= 033 49.1
lgctgΔλv1= 0.32717 lgctgΔλv2= 0.17399 Δλ= 059 01.7
Δλv1= 025 12.6 Δλv2= -033 49.1 = 118 03.4

λ1= -068 48.2 Δλv1= 025 12.6

+Δλv1= 025 12.6 +Δλv2= 033 49.1
λv1= -043 35.6 λv2= 136 24.4 Δλ= 059 01.7
6.Calculul coordonatelor punctelor inter tgφz=tgφvcosΔλz
Z Z1 Z2 Z3 Z4 Z5 Z6
λz -60 -50 -40 -30 -20 -10
Δλz=λv-λz 16.244 6.244 -3.356 -13.356 -23.356 -33.356
lgtgφv= 9.97794 9.97794 9.97794 9.97794 9.97794 9.97794
lgcosΔλz= 9.98194 9.99728 9.99915 9.98766 9.96209 9.92064
lgtgφz= 9.95988 9.97522 9.97708 9.96560 9.94003 9.89858
φz= 42.214 43.220 43.293 42.440 41.034 38.222

φv= 43.327
λv= -043.356
N E
S W
Punct de plecare φ1= 40.416 λ1= -68.482
Punct de sosire φ2= 38.178 λ2= -9.465
1.Calculul distantei ortodromice (M) cosM=sinφ1sinφ2+cosφ1cosφ2cosΔλ
φ2= 38.18 lgsinφ1= 9.81425 lgcosφ1=
-'φ1= 40.42 lgsinφ2= 9.79220 lgcosφ2=
Δφ= -2.24 lg a= 9.60646 lgcosΔλ=
Δφ= 143.8 Mm a= 0.40407 lg b=
b=
λ2= -9.47 1822.6089 a= 0.40407
λ1= -68.48 1822.6089 +'b= 0.30622
Δλ= 59.02 cosM= 0.71030 M= 44.445
Δλ= 3541.7 Mm lgcos M= 9.85144 M= 2684.5
2.Calculul distantei loxodromice (m) tgD=Δλ/Δφc m=ΔφsecD
φc2= 2476.6 lgΔλ= 3.54921 -80.2 lgΔφ= 2.15776
-φc1= 2662.3 +cologΔφc= 7.73134 lgsecD= 1.28115
Δφc= -185.6 lgtgD= 1.28055 lgm= 3.43891
lgΔφc= 2.26866 D= 87.000 m= 2747.3
cologΔφc= 7.73134 D= 87.0 m=
Diferenta dintre distanta loxodromica si distanta ortodromica C=m-M M=
C=
3.Calculul drumului initial D1 4.Calculul drumului final D2
ctgD1=tgφ2cosφ1cosecΔλ-sinφ1ctgΔλ ctgD2=-tgφ1cosφ2cosecΔλ+sinφ2ctgΔλ
lgtgφ2= 9.89744 lgsinφ1= 9.81425 lgtgφ1= 9.93446 lgsinφ2=
lgcosφ1= 9.87979 lgctgΔλ= 9.77829 lgcosφ2= 9.89477 lgctgΔλ=
lgcosecΔλ= 0.06681 lgn= 9.59254 lgcosecΔλ= 0.06681 lgn=
lgm= 9.84403 n= 0.39133 lgm= 9.89604 n=
m= 0.69829 m= 0.78711

m= 0.69829 m= 0.78711
n= 0.39133 D1= 72 56.1 n= 0.37196
ctgD1= 0.30696 D1= 72.9 ctgD2= -0.41515 D2=
5.Calculul coordonatelor vertexului D2= -67
a.Latitudinea vertexului
cosφv1=cosφ1sinD1 cosφv2=cosφ2sinD2
lgcosφ1= 9.87979 lgcosφ2= 9.89477
lgsinD1= 9.98045 lgsinD2= 9.96547
lgcosφv1= 9.86024 lgcosφv2= 9.86024
φv1= 43.327 φv2= -43.327
b.Longitudinea vertexului
ctgΔλv1=sinφ1tgD1 ctgΔλv2=sinφ2tgD2 Verificare
lgsinφ1= 9.81425 lgsinφ2= 9.79220 Δλv1= 25.126
lgtgD1= 0.51292 lgtgD2= 0.38179 Δλv2= 33.491
lgctgΔλv1= 0.32717 lgctgΔλv2= 0.17399 Δλ= 59.017
Δλv1= 25.126 Δλv2= -33.491 = 118.034

λ1= -68.482 -9.465 Δλv1= 25.126

+Δλv1= 25.126 -33.491 +Δλv2= 33.491
λv1= -43.356 λv2= 136.244 Δλ= 59.017
6.Calculul punctelor intermediare tgφz=tgφvcosΔλz

Z -60 -50 -40 -30 -20 -10

Δλz=λv-λz 16.244 6.244 -3.356 -13.356 -23.356 -33.356
lgtgφv= 9.97794 9.97794 9.97794 9.97794 9.97794 9.97794
lgcosΔλz= 9.98194 9.99728 9.99915 9.98766 9.96209 9.92064
lgtgφz= 9.95988 9.97522 9.97708 9.96560 9.94003 9.89858
φz= 42.214 43.220 43.293 42.440 41.034 38.222
42.214 43.220 43.293 42.440 41.034 38.222
φv= 43.327
λv= -43.356
Δλz-poz 16.244 6.244 3.356 13.356 23.356 33.356
φ1cosφ2cosΔλ
9.87979
9.89477
9.71148
9.48604
0.30622

Mm
m=ΔφsecD

2747.32
2684.46
62.9

+sinφ2ctgΔλ
9.79220
9.77829
9.57049
0.37196

112.5
27.2

0.000

1.000
9.997
0.000
9.997
 0 0

L1 -68.482 -68.482
Z1,Δλ<180 -68.482 -58.482
Z1,Δλ>180 -78.482 -88.482
Z1' -68.482 -58.482 -48.482 -38.482 -28.482 -18.482 -8.482 1.518
Z2 -60 -50 -40 -30 -20 -10 0 10
Z3 -60 -50 -40 -30 -20 -10 0 10
Z4,Δλ<180 -60 -50 -40 -30 -20 -10
Z5,Δλ>180 -60 -50 -40 -30 -20 -10 0 10
Z6 -60 -50 -40 -30 -20 -10
Z in rad. -1.047198 -0.872665 -0.698132 -0.523599 -0.349066 -0.174533
λv in gms= -43.356 -43.356 -43.356 -43.356 -43.356 -43.356
λv in rad.= -0.760838 -0.760838 -0.760838 -0.760838 -0.760838 -0.760838
Δλz in rad. 0.286359 0.111826 -0.062707 -0.23724 -0.411773 -0.586306
Δλz in grd. 16.407 6.407 -3.593 -13.593 -23.593 -33.593
Δλz in gms. 16.244 6.244 -3.356 -13.356 -23.356 -33.356
φv in gms.= 43.327
φv in rad.= 0.76001 0.76001 0.76001 0.76001 0.76001 0.76001
φv in rad.+ = 0.76001 0.76001 0.76001 0.76001 0.76001 0.76001
tgφv= 0.95047 0.95047 0.95047 0.95047 0.95047 0.95047
lgtgφv= -0.02206 -0.02206 -0.02206 -0.02206 -0.02206 -0.02206
lgtgφv + = 9.97794 9.97794 9.97794 9.97794 9.97794 9.97794
cosΔλz= 0.95928 0.99375 0.99803 0.97199 0.91641 0.83299
cosΔλz + = 0.95928 0.99375 0.99803 0.97199 0.91641 0.83299
lgcosΔλz= -0.01806 -0.00272 -0.00085 -0.01234 -0.03791 -0.07936
lgcosΔλz + = 9.98194 9.99728 9.99915 9.98766 9.96209 9.92064
lgtgφz= 19.95988 19.97522 19.97708 19.96560 19.94003 19.89858
lgtgφz tf.= 9.95988 9.97522 9.97708 9.96560 9.94003 9.89858
0.91176 0.94453 0.94860 0.92384 0.87102 0.79173 #VALUE! #VALUE!
0.73928 0.75688 0.75903 0.74583 0.71657 0.66968 #VALUE! #VALUE!
42.35737 43.36600 43.48895 42.73311 41.05649 38.36969 #VALUE! #VALUE!
42.214 43.220 43.293 42.440 41.034 38.222
 11.518 21.518 31.518 41.518 51.518 61.518 71.518 81.518 91.518
20 30 40 50 60 70 80 90 100
20 30 40 50 60 70 80 90 100

20 30 40 50 60 70 80 90 100

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!
 101.518 111.518
110 120
110 120

110 120

#VALUE! #VALUE!
#VALUE! #VALUE!
#VALUE! #VALUE!