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7/23/2019 calcul ortodroma.xlsx
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FOAIE DE CALCULE
TRNSFORMAREA UNGHIURILOR
φ1 λ1 φ2 λ2 Δλ φ2-φ1
-6.080 155.400 41.440 -124.120 80.080 47.520
d! "#$ ! "%d -6.1&&&& 155.66667 41.7&&&& -124.20000 80.1&& 47.867
d! "%d. ! %'d. -0.10705 2.716(0 0.728&8 -2.16770 1.&(85( 0.8&54&d! %'d ! "%'d
d! "%d ! "#$ -6.080
d! "#$ ! )*+# d* "#$
d! )*+# d* "#$ ! "#$
d! "#$ ! "#$
d! "#$ ! M# -&68.0 (&40.0 2504.0 -7452.0 4808.0 2872.0
d! "#$ ! M# ,) &68.0 (&40.0 2504.0 7452.0 4808.0 2872.0
O/ITI 80.1
FUNCTII TRIGONOMETRICE
φ1 λ1 φ1 λ2 Δλ φ2-φ1
NR.FCT. -6.1&& 155.667 41.7&& -124.200 80.1&& 47.867
!"3% ! RAD -0.10705 2.716(0 0.728&8 -2.16770 1.&(85( 0.8&54&
SIN -0.10684 0.41204 0.66566 -0.82708 0.(8521 0.7415(
COS 0.((428 -0.(1116 0.74625 -0.56208 0.171&6 0.67086
TG -0.10746 -0.45222 0.8(201 1.47146 5.74(4( 1.1054&
CTG -(.&05(( -2.211&2 1.12106 0.67(60 0.17&(& 0.(046&
COSEC -(.&5(57 2.426(2 1.50226 -1.20(07 1.01501 1.&4846
$! 0.10684 0.41204 0.66566 0.82708 0.(8521 0.7415(
"$! (.02874 (.614(4 (.82&26 (.(1755 (.((&5& (.87016
+$ 0.((428 0.(1116 0.74625 0.56208 0.171&6 0.67086
"+$ (.((751 (.(5(60 (.87288 (.74(80 (.2&&(0 (.8266&" 0.10746 0.45222 0.8(201 1.47146 5.74(4( 1.1054&
"" (.0&124 (.655&5 (.(50&7 0.16775 0.75(6& 0.04&5&
+" (.&05(( 2.211&2 1.12106 0.67(60 0.17&(& 0.(046&
"+" 0.(6876 0.&4465 0.04(6& (.8&225 (.240&7 (.(5647
+$*+ (.&5(57 2.426(2 1.50226 1.20(07 1.01501 1.&4846
"+$*+ 0.(7126 0.&8506 0.17674 0.08245 0.00647 0.12(84
+$M 0.05602 ""D 0.18(05 +"D1 0.(1880 +"D2
+$M 0.05602 ""D- 0.18(05 "D1 1.088&7 "D2
"+$M 8.748&5 "D 1.54542 D1 ! %'d 0.8276( D2 ! %'d
M ! %'d. 1.51475 D ! %'d. 0.((648 D1 ! "%'d 47.42&0( 2 ! "%'d
M ! "%'d*86.7885& D ! "%'d*57.0(40( D1 ! d#$ 47.254 2 ! d#$M ! "#$ 86.47& D ! "#$ 57.056 D1 ! d)# 47.4 2 ! d)#
M ! M# 5207.& D ! ")# 57.1 D199 d)# 47.4 299 d)#
+$D 0.54&26 &60-D1 &12.6 &60-D2
$*+D 1.84074 D1180 227.4 D2180
$*+D 1.84074 180-D1 1&2.6 180-D2
"$*+D 0.264(( D1 47.4 D2
λ1 ! %'d. 2.717 2 ! %'d.
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"+"Δλ:1 (.06552 "+"Δλ:2 10.52841 λ:1 ! %'d. -1.455 2 ! %'d.
"+"Δλ:1-. (.06552 +"Δλ:2-. 0.52841 λ:1 ! %'d. 1.262 2 ! %'d.
+"Δλ:1 0.11628 +"Δλ:2 &.&7606 λ:1 ! "%d. 72.2(( 2 ! "%d.
"Δλ:1 8.5((61 "Δλ:2 0.2(620 λ:1 ! "#$ 72.180 2 ! "#$
Δλ:1 ! %'d. 1.4550& λ:2 ! %'d. 0.287(7 λ:1 ! "%d 72.2(( :2 ! "%d
Δλ:1 ! "%'d* 8&.&6718 2 ! "%'d* 16.4((48 λ:1 ! "#$ 72.180 2 ! "#$
Δλ:1 "#$ 8&.220 Δλ:2 "#$ 16.&00Δλ:1 "#$ -8&.220 Δλ:2 "#$ 16.&00
1.Calculul distantei ortodromic
ERIFICARE ((.867 "' 18.85200 ";
((.520 "'-. 8.85200 ";-.
' 0.07112 ;
0.071121
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FUNCTII LOGARITMICE
Δφ+ &111.1
Δφ+ &111.1
Δλ "Δφ+ &.4(2(1(
-27(.520
-27(.867
-4.8845(
27(.5
M D1 D2
0.8276( 1.&7612
0.7&6&7 0.(8111
0.056021
1.088&7 5.07171
0.7&6&7 0.(8111
(.86710 (.((172
0.05602
8.748&51.088&7 5.07171
0.0&678 0.70515
0.1(717 "+$φ:1 1(.8646 "+$φ:2 1(.8646
5.07171 "+$φ:1- (.86460 "+$φ:2- (.86460
1.&7612 +$φ:1 0.7&215 +$φ:2 0.7&215
78.845(4 φ:1 %'d 0.74(&2 φ:2 %'d 0.74(&2
78.508 φ:1 "%'d 42.(&266 :2 "%'d 42.(&26678.8 φ:1 d#$ 42.560 :2 d#$ 42.560
78.8 φ:1 d#$ -42.560 :2 d#$ 42.560
281.2
258.8
101.2
78.8
-2.168
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0.288
-1.880 Δλ:1 ! "%d 8&.&6718
-107.701 Δλ:2 ! "%d 16.4((48
-107.420 Δλ ! "%d. 80.1&&&&
-107.701 180
-107.420 180.000
(M) d$.<.# 3.Calculul drumului initial D1 4.Calculul drumul
2(.1042( ""D 10.18(05 "# 1(.(54&5 "! 18.26(11 "# 18.(105(
(.1042( ""D-. 0.18(05 "#-. (.(54&5 "!-. 8.26(11 "#-. 8.(105(
0.12714 1.54542 # 0.(0022 ! 0.01858 # 0.081&(
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LATITUDINI CRESCANDE
φ+7(15.70447"=">45φ2?@>1-*$!φ1*$!φ?' ,>*2?B
φ1 φ2
φ -6.1&& 41.7&&
φ ! %'d. -0.10705 0.728&8
7(15.70447 7(15.70447
* 45φ20.7&187 1.14(5(
">45φ2? 0.8(8&0 2.2&204
1-*$!φ 1.00874 0.(4554
1*$!φ 0.((126 1.05446
>1-*$!φ1*$!φ? 1.01764 0.8(670
>1-*$!φ1*$!φ?' ,.*21.00072 0.((555
">45φ2?>1-*$!φ1*$!φ?' ,>*2? 0.8(8(4 2.22211
"">45φ2?>1-*$!φ1*$!φ?' ,>*2? -0.04627 0.&4677
φ+ -&66.245(5 2744.8(2&1
0.75405
φ+1 φ+2
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i fnal D2
"! 1(.06&6& 1(.86460
"!-. (.06&6& (.86460
! 0.11578
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RE/OLARE ROLEME NAIGATIE ORTODROMICA
UNCT DE LECARE φ1 06 08.0 S λ1 155 40.0 E ','%'$
UNCT DE SOSIRE φ2 41 44.0 N λ2 124 12.0 3'#'1.Calculul distantei ortodromice (M) cosM=sinφ1sinφ2+cosφ1cosφ2cosΔλ
φ2 41 44.0 "$!φ1 (.02874 "+$φ1 (.((751
-φ1 -06 08.0 "$!φ2 (.82&26 "+$φ2 (.87288
Δφ 47 52.0 " ' 8.85200 "+$Δλ (.2&&(0Δφ 2872 M# ' 0.07112 " ; (.1042(
λ2 -124 12.0 ; 0.12714
-λ1 155 40.0 ' -0.07112
Δλ 080 08.0 ; 0.12714 M 86 47.&
Δλ 4808 M# +$M 0.05602 M 5207.& M#
Δλ -27(.52 "+$ M 8.748&5
2.Calculul distantei loxodromice (m) tD=Δλ!Δφc m=ΔφsecD
φ+2 2744.8( "Δλ &.681(6 "Δφ &.45818
-φ+1 -&66.25 +"Δφ+ 6.50708 lgsecD= 0.264((
Δφ+ &111.14 lgtgD= 0.18(05 lgm= &.72&18
"Δφ+ &.4(2(2 D= 57 05.6 m= 5286.6 M#
"Δφ+ 6.50708 D= 57 1 m= 5286.6D*%*!' d!%* d$'!' <d%#+' $ d$'!' %d%#+' C=m-M -M= 5207.&
C= 7(.& M#
3.Calculul drumului initial D1 4.Calculul drumului final D2
+"D1"φ2+$φ1+$*+Δλ-$!φ1+"Δλ +"D2-"φ1+$φ2+$*+Δλ$!φ2+"Δλ
""φ2 (.(50&7 "$!φ1 (.02874 ""φ1 (.0&124 "$!φ2 (.82&26
"+$φ1 (.((751 "+"Δλ (.240&7 "+$φ2 (.87288 "+"Δλ (.240&7
+$*+Δλ 0.00647 "! 8.26(11 "+$*+Δλ 0.00647 "! (.06&6&
"# (.(54&5 ! 0.01858 "# 8.(105( ! 0.11578
# 0.(0022 # 0.081&(
# 0.(0022 D1 47 4 # -0.081&(
! -0.01858 D2 78 8 ! 0.11578+"D1 0.(1880 D1 47 25.4 D2 78 50.8 "D2 0.1(717
".Calculul coordonatelor #ertexului
'.L'd!*' :*%*<
+$φ:1+$φ1$!D1 +$φ:2+$φ2$!D2
"+$φ1 (.((751 "+$φ2 (.87288
"$!D1 (.86710 "$!D2 (.((172
"+$φ:1 (.86460 "+$φ:2 (.86460
φ:1 -42 56.0 φ:2 42 56.0
;.L!"d!*' :*%*<
+"Δλ:1$!φ1"D1 +"Δλ:2$!φ2"D2 *%+'%*
"$!φ1 (.02874 "$!φ2 (.82&26 Δλ:1 08& 22.0
""D1 0.0&678 ""D2 0.70515 Δλ:2 016 &0.0+"Δλ:1 (.06552 "+"Δλ:2 0.52841 Δλ 080 08.0
Δλ:1 -08& 22.0 Δλ:2 016 &0.0 180 00.0
λ1 155 40.0 Δλ:1 08& 22.0
Δλ:1 -08& 22.0 Δλ:2 016 &0.0
λ:1 072 18.0 λ:2 -107 42.0 Δλ 0(( 52.0
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6.Calculul coordonatelor punctelor interm"φ)"φ:+$Δλ)
/ $1 $2 $3 $4 $" $% $& $'
λ) 160 170 180 -170 -160 -150 -140 -1&0Δλ)λ:-λ) -87.420 -(7.420 -107.420 -117.420 -127.420 -1&7.420 -147.420 -157.420
""φ: (.(686& (.(686& (.(686& (.(686& (.(686& (.(686& (.(686& (.(686&
"+$Δλ) 8.60&&( (.1270( (.482(& (.667&1 (.78642 (.86(02 (.(26(( (.(6624
""φ) 8.57202 (.0(572 (.45157 (.6&5(5 (.75505 (.8&765 (.8(56& (.(&487φ) 2.'3 &.%3 1".4&% 23.232 2*.3'2 34.31* 3'.1' 4.432
φ: -42.560
λ: 072.180
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N E
S
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!+ d* ,*+'%* φ1 -6.080 λ1 155.400
!+ d* $$%* φ2 41.440 λ2 -124.120
1.Calculul distantei ortodromice (M) cosM=sinφ1sinφ2+cosφ1cosφ2
φ2 41.44 "$!φ1 (.02874 "+$φ1
-9φ1 -6.08 "$!φ2 (.82&26 "+$φ2
Δφ 47.52 " ' 8.85200 "+$ΔλΔφ 2872.0 M# ' 0.07112 " ;
;
λ2 -124.12 82&.87(4 ' -0.07112
λ1 155.40 82&.87(4 9; 0.12714
Δλ 80.08 -27(.52 +$M 0.05602 M 86.47&
Δλ 4808.0 M# "+$ M 8.748&5 M 5207.&
2.Calculul distantei loxodromice (m) tD=Δλ!Δφc m=Δφsec
φ+2 2744.( "Δλ &.681(6 -80.2 "Δφ &.45818
-φ+1 -&66.2 +"Δφ+ 6.50708 "$*+D 0.264((
Δφ+ &111.1 ""D 0.18(05 "# &.72&18
"Δφ+ &.4(2(2 D 57.056 # 5286.6
+"Δφ+6.50708 D 57.1 #D*%*!' d!%* d$'!' <d%#+' $ d$'!' %d%#+' C#-M M
C
3.Calculul drumului initial D1 4.Calculul drumului fnal D2
+"D1"φ2+$φ1+$*+Δλ-$!φ1+"Δλ +"D2-"φ1+$φ2+$*+Δλ$!φ2+
""φ2 (.(50&7 "$!φ1 (.02874 ""φ1 (.0&124 "$!φ2
"+$φ1 (.((751 "+"Δλ (.240&7 "+$φ2 (.87288 "+"Δλ
"+$*+Δλ 0.00647 "! 8.26(11 "+$*+Δλ 0.00647 "!
"# (.(54&5 ! 0.01858 "# 8.(105( !
# 0.(0022 # 0.081&(
# 0.(0022 # -0.081&(
! -0.01858 D1 47 25.4 ! 0.11578+"D1 0.(1880 D1 47.4 +"D2 0.1(717 D2
".Calculul coordonatelor #ertexului D2 78
'.L'd!*' :*%*<
+$φ:1+$φ1$!D1 +$φ:2+$φ2$!D2
"+$φ1 (.((751 "+$φ2 (.87288
"$!D1 (.86710 "$!D2 (.((172
"+$φ:1 (.86460 "+$φ:2 (.86460
φ:1 -42.560 φ:2 42.560
;.L!"d!*' :*%*<
+"Δλ:1$!φ1"D1 +"Δλ:2$!φ2"D2 *%+'%*
"$!φ1 (.02874 "$!φ2 (.82&26 Δλ:1 8&.220
""D1 0.0&678 ""D2 0.70515 Δλ:2 16.&00"+"Δλ:1 (.06552 "+"Δλ:2 0.52841 Δλ 80.080
Δλ:1 -8&.220 Δλ:2 16.&00 180.000
λ1 155.400 -124.120 Δλ:1 8&.220
Δλ:1 -8&.220 16.&00 Δλ:2 16.&00
λ:1 72.180 λ:2 -107.420 Δλ ((.520
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%.Calculul unctelor intermediare "φ)"φ:+$Δλ)
/ 160 170 180 -170 -160 -150 -140
Δλ)λ:-λ) -87.420 -(7.420 -107.420 -117.420 -127.420 -1&7.420 -147.420
""φ: (.(686& (.(686& (.(686& (.(686& (.(686& (.(686& (.(686&
"+$Δλ) 8.60&&( (.1270( (.482(& (.667&1 (.78642 (.86(02 (.(26((
""φ) 8.57202 (.0(572 (.45157 (.6&5(5 (.75505 (.8&765 (.8(56&φ) 2.08& 7.06& 15.476 2&.2&2 2(.&82 &4.&1( &8.108
-2.08& 7.06& 15.476 2&.2&2 2(.&82 &4.&1( &8.108
φ: -42.560
λ: 72.180
Δλ)-,) 87.420 (7.420 107.420 117.420 127.420 1&7.420 147.420
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cosΔλ
(.((751
(.87288
(.2&&(0(.1042(
0.12714
M#
D
5286.5(5207.&1
7(.&
"Δλ
(.82&26
(.240&7
(.06&6&
0.11578
78.8
50.8
1.000
0.000
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-1&0
-157.420
(.(686&
(.(6624
(.(&48740.4&2
40.4&2
157.420
(.127
0.000
(.127
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0 0
L1 155.4 155.4
/1ΔλJ180 145.4 1&5.4
/1ΔλK180 165.4 175.4
/19 165.4 175.4 185.4 1(5.4 205.4 215.4 225.4 2&5.4
/2 160 170 180 1(0 200 210 220 2&0
/& 160 170 180 -170 -160 -150 -140 -1&0
/4ΔλJ180 160 170 180
/5ΔλK180 160 170 180 -170 -160 -150 -140 -1&0
/6 160 170 180 -170 -160 -150 -140 -1&0
/ ! %'d. 2.7(2527 2.(6706 &.1415(& &.&16126 &.4(065( &.6651(1 &.8&(724 4.014257
λ: ! "#$ 72.180 72.180 72.180 72.180 72.180 72.180 72.180 72.180λ: ! %'d. 1.261864 1.261864 1.261864 1.261864 1.261864 1.261864 1.261864 1.261864
Δλ) ! %'d. -1.5&066 -1.7052 -1.87(7& -2.05426 -2.2287( -2.40&&& -2.57786 -2.752&(
Δλ) ! "%d. -87.701 -(7.701 -107.701 -117.701 -127.701 -1&7.701 -147.701 -157.701
Δλ) ! "#$. -87.420 -(7.420 -107.420 -117.420 -127.420 -1&7.420 -147.420 -157.420
φ: ! "#$. -42.560
φ: ! %'d. 0.74(&2 0.74(&2 0.74(&2 0.74(&2 0.74(&2 0.74(&2 0.74(&2 0.74(&2
φ: ! %'d. 0.74(&2 0.74(&2 0.74(&2 0.74(&2 0.74(&2 0.74(&2 0.74(&2 0.74(&2
"φ: 0.(&0&2 0.(&0&2 0.(&0&2 0.(&0&2 0.(&0&2 0.(&0&2 0.(&0&2 0.(&0&2
""φ: -0.0&1&7 -0.0&1&7 -0.0&1&7 -0.0&1&7 -0.0&1&7 -0.0&1&7 -0.0&1&7 -0.0&1&7
""φ: (.(686& (.(686& (.(686& (.(686& (.(686& (.(686& (.(686& (.(686&
+$Δλ) 0.04012 -0.1&400 -0.&0404 -0.46485 -0.6115& -0.7&(64 -0.84527 -0.(2521
+$Δλ) 0.04012 0.1&400 0.&0404 0.46485 0.6115& 0.7&(64 0.84527 0.(2521"+$Δλ) -1.&(661 -0.872(1 -0.51707 -0.&&26( -0.21&58 -0.1&0(8 -0.07&01 -0.0&&76
"+$Δλ) 8.60&&( (.1270( (.482(& (.667&1 (.78642 (.86(02 (.(26(( (.(6624
""φ) 18.57202 1(.0(572 1(.45157 1(.6&5(5 1(.75505 1(.8&765 1(.8(56& 1(.(&487
""φ) . 8.57202 (.0(572 (.45157 (.6&5(5 (.75505 (.8&765 (.8(56& (.(&487
0.0&7&& 0.12466 0.28286 0.4&246 0.568(2 0.68810 0.786&7 0.86074
0.0&7&1 0.12402 0.27566 0.40817 0.51725 0.6026( 0.666&7 0.71070
2.1&76( 7.10574 15.7(&88 2&.&8652 2(.6&652 &4.5&18& &8.18040 40.72004
2.08& 7.06& 15.476 2&.2&2 2(.&82 &4.&1( &8.108 40.4&2
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245.4 255.4 265.4 275.4 285.4 2(5.4 &05.4 &15.4 &25.4
240 250 260 270 280 2(0 &00 &10 &20
-120 -110 -100 -(0 -80 -70 -60 -50 -40
-120 -110 -100 -(0 -80 -70 -60 -50 -40
ALUE ALUE ALUE ALUE ALUE ALUE ALUE ALUE ALUE
ALUE ALUE ALUE ALUE ALUE ALUE ALUE ALUE ALUE
ALUE ALUE ALUE ALUE ALUE ALUE ALUE ALUE ALUE
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&&5.4 &45.4
&&0 &40
-&0 -20
-&0 -20
ALUE ALUE
ALUE ALUE
ALUE ALUE