Uniwersytet Warmińsko-Mazurski Olsztyn, dn. 07.03.2015r.
Wydział Geodezji, Inżynierii Przestrzennej
I Budownictwa
Instytut Geodezji
SPRAWOZANIE 1
Przeliczanie współrzędnych
Małgorzata Faderewska
GIG rok 3, grupa 2
NR 5
Współrzędne prostokątne w układzie globalnym WGS84 przeliczyć na elipsoidalne BLH odniesione do elipsoidy WGS84, a następnie znając parametry transformacji –parametrowej pomiędzy układami przeliczyć XYZ w układzie WGS84 na współrzędne prostokątne odniesione do Krasowskiego42
| Punkt | X | Y | Z |
|---|---|---|---|
| 1 | 3524523,4793 | 1329693,7456 | 5129846,4049 |
| 2 | 3592275,2039 | 1326519,6154 | 5083787,0189 |
| 3 | 3648845,4694 | 1359140,2015 | 5034919,5360 |
Parametry elipsoidy WGS84:
| a | 6378137 |
|---|---|
| b | 6356752,314 |
| e2 | 0,00669438 |
| e'2 | 0,006739497 |
Parametry elipsoidy Krasowskiego:
| a | 6378245 |
|---|---|
| b | 6356863,019 |
| e2 | 0,00669342 |
| e'2 | 0,006738525415 |
Wykorzystane wzory (XYZ->BLH)WGS84
$$L = \text{arctg}\frac{y}{x}$$
$$B = \text{arctg}\frac{z + {e'}^{2}b\sin^{3}\theta}{p - e^{2}a\cos^{3}\theta}$$
p = (x2+y2)1/2
$$\theta = \text{arctg}\left( \frac{z*a}{p*b} \right)$$
$$H = \frac{p}{\text{cosB}} - N$$
$$N = \frac{a}{{(1 - e^{2}\sin^{2}B)}^{2}}$$
$$B = \frac{z + Ne^{2}\text{sinB}}{{(x^{2} + y^{2})}^{1/2}}$$
$$L = \text{arctg}\frac{y}{x}$$
$$H = \frac{z}{\text{sinB}} - N(1 - e^{2})$$
Wykorzystane wzory (XYZ WGS84-> XYZ KRASOWSKI42)
∆x = -23,00
∆y = 124,50
∆z = 82,50
δω = 0’’,5
δψ = 2’’,0
δε = -2’’,5
∆ = 7 * 10-6
| X | Δx | 1+Δ | δω | -δψ | x | |||
|---|---|---|---|---|---|---|---|---|
| Y | = | Δy | + | -δω | 1+Δ | δε | * | y |
| Z | Δz | δψ | -δε | 1+Δ | z |
Ad.1
Metoda nr 1
| Punkt 1 | radiany | stopnie | minuty | sekundy |
|---|---|---|---|---|
| p | 3767008,258 | |||
| ʘ | 53°,80074 | 48’ | 2’’,6660 | |
| B | 53°,89240 | 53’ | 32’’,6270 | |
| L | 20°,66994 | 40’ | 11’’,7860 | |
| H | 187,400 m | |||
| N | 6392117,674 |
| Punkt 2 | radiany | stopnie | minuty | sekundy |
|---|---|---|---|---|
| p | 3829372,720 | |||
| ʘ | 53°,10347 | 6’ | 12’’,4990 | |
| B | 53°,19581 | 11’ | 44’’,9217 | |
| L | 20°,26768 | 16’ | 3’’,6395 | |
| H | 198,362 m | |||
| N | 6391867,965 |
| Punkt 3 | radiany | stopnie | minuty | sekundy |
|---|---|---|---|---|
| p | 3893755,943 | |||
| ʘ | 53°,37650 | 22’ | 35’’,3987 | |
| B | 52°,46950 | 28’ | 10’’,1844 | |
| L | 20°,42961 | 25’ | 46’’,5872 | |
| H | 153,913 m | |||
| N | 6391605,724 |
Metoda nr 2
| Punkt 1 | stopnie | minuty | sekundy | ||
|---|---|---|---|---|---|
| No | 6381486,210 | Bo | 53°,7989 | 47’ | 56’’,10893 |
| N1 | 6392084,272 | B1 | 53°,8922 | 53’ | 31’’,84037 |
| N2 | 6392117,596 | B2 | 53°,8924 | 53’ | 32’’,62516 |
| N3 | 6392117,674 | B3 | 53°,8924 | 53’ | 32’’,62700 |
| N4 | 6392117,674 | B4 | 53°,8924 | 53’ | 32’’,62700 |
| L | 20°,6699 | 40’ | 11’’,78600 | ||
| H | 187,400 m |
| Punkt 2 | stopnie | minuty | sekundy | ||
|---|---|---|---|---|---|
| No | 6381486,210 | Bo | 53°,1025 | 6’ | 8’’,84199 |
| N1 | 6391834,362 | B1 | 53°,1956 | 11’ | 44’’,10994 |
| N2 | 6391867,884 | B2 | 53°,1958 | 11’ | 44’’,91979 |
| N3 | 6391867,965 | B3 | 53°,1958 | 11’ | 44’’,92174 |
| N4 | 6391867,965 | B4 | 53°,1958 | 11’ | 44’’,92175 |
| L | 20°,2677 | 16’ | 3’’,63952 | ||
| H | 198,362 m |
| Punkt 3 | stopnie | Minuty | sekundy | ||
|---|---|---|---|---|---|
| No | 6381486,210 | Bo | 52°,3764 | 22’ | 34’’,92565 |
| N1 | 6391571,969 | B1 | 52°,4693 | 28’ | 9’’,34715 |
| N2 | 6391605,639 | B2 | 52°,4695 | 28’ | 10’’,18233 |
| N3 | 6391605,723 | B3 | 52°,4695 | 28’ | 10’’,18442 |
| N4 | 6391605,724 | B4 | 52°,4695 | 28’ | 10’’,18442 |
| L | 20°,4296 | 25’ | 46’’,58273 | ||
| H | 153,913 m |
Ad.2
| Δx | -23,00 | |
|---|---|---|
| Δy | 124,50 | |
| Δz | 82,50 | |
| δω | 0’’,5 | 2,42407E-06 |
| δψ | 2’’ | 9,69627E-06 |
| δε | -2’’,5 | -1,21203E-05 |
| Δ | 0,000007 |
| X | -23 | 1,000007 | 2,42407E-06 | -9,69627E-06 | 3524523,4793 | |||
|---|---|---|---|---|---|---|---|---|
| Y | = | 124,5 | + | -2,424E-06 | 1,000007 | -1,21203E-05 | * | 1329693,7456 |
| Z | 82,5 | 9,6963E-06 | 1,21203E-05 | 1,000007 | 5129846,4049 |
| X1 | 3524478,633839 |
|---|---|
| Y1 | 1329756,834277 |
| Z1 | 5130015,104912 |
| X | -23 | 1,000007 | 2,42407E-06 | -9,69627E-06 | 3592275,2039 | |||
|---|---|---|---|---|---|---|---|---|
| Y | = | 124,5 | + | -2,424E-06 | 1,000007 | -1,21203E-05 | * | 1326519,6154 |
| Z | 82,5 | 9,6963E-06 | 1,21203E-05 | 1,000007 | 5083787,0189 |
| X2 | 3592231,271611 |
|---|---|
| Y2 | 1326583,075879 |
| Z2 | 5083956,014964 |
| X | -23 | 1,000007 | 2,42407E-06 | -9,69627E-06 | 3648845,4694 | |||
|---|---|---|---|---|---|---|---|---|
| Y | = | 124,5 | + | -2,424E-06 | 1,000007 | -1,21203E-05 | * | 1359140,2015 |
| Z | 82,5 | 9,6963E-06 | 1,21203E-05 | 1,000007 | 5034919,5360 |
| X3 | 3648802,486010 |
|---|---|
| Y3 | 1359204,345484 |
| Z3 | 5035089,133885 |
Współrzędne BLHkrasowski42
| Punkt 1 | radiany | stopnie | minuty | sekundy |
|---|---|---|---|---|
| p | 3766988,569 | |||
| ʘ | 53°,80177 | 48’ | 6’’,3651 | |
| B | 53°,89341 | 53’ | 36’’,2734 | |
| L | 20°,67108 | 40’ | 15’’,8849 | |
| H | 202,331 m | |||
| N | 6392224,261 |
| Punkt 2 | Radiany | stopnie | minuty | Sekundy |
|---|---|---|---|---|
| p | 3829353,492 | |||
| ʘ | 53°,10451 | 6’ | 16’’,2406 | |
| B | 53°,19684 | 11’ | 48’’,6105 | |
| L | 20°,26880 | 16’ | 7’’,6658 | |
| H | 212,426 m | |||
| N | 6391974,591 |
| Punkt 3 | radiany | stopnie | minuty | Sekundy |
|---|---|---|---|---|
| p | 3893738,054 | |||
| ʘ | 53°,37755 | 22’ | 39’’,1681 | |
| B | 52°,47053 | 28’ | 13’’,9010 | |
| L | 20°,43071 | 25’ | 50’’,5662 | |
| H | 167,845 m | |||
| N | 6391712,388 |
Przeliczanie odwrotne:
V + x42 = x84 + x + *x84 + [ob]x84
V = AX + L
V = Δx + δω*Y – δψ* Z + Δ *X
V = Δy - δω*X – δε* Z + Δ *Y
V = Δz + δψ*X – δε* Y + Δ *Z
V = Δx + δω*Y2 – δψ* Z2 + Δ *X2
V = Δy - δω*X2 – δε* Z2 + Δ *Y2
V = Δz + δψ*X2 – δε* Y2 + Δ *Z2
| Δx | 1 | 0 | 0 | Y | -Z | 0 | X | X1WGS84-X1KRAS42 | |||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Δy | 0 | 1 | 0 | -X | 0 | Z | Y | Y1WGS84-Y1KRAS42 | |||
| Δz | A= | 0 | 0 | 1 | 0 | X | -Y | Z | Z1WGS84-Z1KRAS42 | ||
| X= | δω | 1 | 0 | 0 | Y2 | -Z2 | 0 | X2 | L= | X2WGS84-X2KRAS42 | |
| δψ | 0 | 1 | 0 | - X2 | 0 | Z2 | Y2 | Y2WGS84-Y2KRAS42 | |||
| δε | 0 | 0 | 1 | 0 | X2 | -Y2 | Z2 | Z2WGS84-Z2KRAS42 | |||
| Δ | X3WGS84-X3KRAS42 | ||||||||||
| Y3WGS84-Y3KRAS42 | |||||||||||
| Z3WGS84-Z3KRAS42 |
(ATA)X = ATL
X= − (ATA)−1ATL
| 1 | 0 | 0 | 1329693,7456 | -5129846,4049 | 0 | 3524523,4793 | 44,8455 | |||
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 0 | -3524523,4793 | 0 | 5129846,4049 | 1329693,7456 | -63,0887 | |||
| 0 | 0 | 1 | 0 | 3524523,4793 | -1329693,7456 | 5129846,4049 | -168,7000 | |||
| A | 1 | 0 | 0 | 1326519,6154 | -5083787,0189 | 0 | 3592275,2039 | L | 43,9323 | |
| 0 | 1 | 0 | -3592275,2039 | 0 | 5083787,0189 | 1326519,6154 | -63,4605 | |||
| 0 | 0 | 1 | 0 | 3592275,2039 | -1326519,6154 | 5083787,0189 | -168,9961 | |||
| 1 | 0 | 0 | 1359140,2015 | -5034919,5360 | 0 | 3648845,9694 | 42,9834 | |||
| 0 | 1 | 0 | -3648845,4694 | 0 | 5034919,5360 | 1359140,2015 | -64,1440 | |||
| 0 | 0 | 1 | 0 | 3648845,4694 | -1359140,2015 | 5034919,5360 | -169,5979 | |||
| 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | ||
| 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | ||
| 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | ||
| AT | 1329693,746 | -3524523,479 | 0 | 1326519,615 | -3592275,204 | 0 | 1359140,202 | -3648845,469 | 0 | |
| -5129846,405 | 0 | 3524523,479 | -5083787,019 | 0 | 3592275,204 | -5034919,536 | 0 | 3648845,469 | ||
| 0 | 5129846,405 | -1329693,746 | 0 | 5083787,019 | -1326519,615 | 0 | 5034919,536 | -1359140,202 | ||
| 3524523,479 | 1329693,746 | 5129846,405 | 3592275,204 | 1326519,615 | 5083787,019 | 3648845,469 | 1359140,202 | 5034919,536 | ||
| 3 | 0 | 0 | 4015353,563 | -15248552,96 | 0 | 10765644,15 | ||||
| 0 | 3 | 0 | -10765644,15 | 0 | 15248552,96 | 4015353,563 | ||||
| ATA | 0 | 0 | 3 | 0 | 10765644,15 | -4015353,563 | 15248552,96 | |||
| 4015353,563 | -10765644,15 | 0 | 4,40158E+13 | -2,0408E+13 | -5,47143E+13 | 0 | ||||
| -15248552,96 | 0 | 10765644,15 | -2,0408E+13 | 1,16151E+14 | -1,44111E+13 | 0 | ||||
| 0 | 15248552,96 | -4015353,563 | -5,47143E+13 | -1,44111E+13 | 8,28856E+13 | 0 | ||||
| 10765644,15 | 4015353,563 | 15248552,96 | 0 | 0 | 0 | 1,21526E+14 | ||||
| 131,7611411 | ||||||||||
| -190,6931397 | ||||||||||
| -507,2939607 | ||||||||||
| ATL | 860704739,8 | |||||||||
| -2490314640 | ||||||||||
| -290211331,2 | ||||||||||
| -2360992201 | ||||||||||
| 15751,32757 | -44220,90677 | 2740,494592 | -0,004421012 | 0,001738322 | 0,005651975 | -0,000278122 | ||||
| -44220,90677 | 158690,0422 | -9740,183899 | 0,015459226 | -0,004702428 | -0,020278912 | -0,000103734 | ||||
| 2740,494592 | -9740,183899 | 3769,574783 | -0,000949547 | 1,10015E-05 | 0,001349627 | -0,000393935 | ||||
| ATA-1 | -0,004421012 | 0,015459226 | -0,000949547 | 1,55917E-09 | -4,59218E-10 | -1,94065E-09 | 4,67892E-22 | |||
| 0,001738322 | -0,004702428 | 1,10015E-05 | -4,59218E-10 | 2,21073E-10 | 6,00943E-10 | -1,8988E-22 | ||||
| 0,005651975 | -0,020278912 | 0,001349627 | -1,94065E-09 | 6,00943E-10 | 2,61955E-09 | -5,95957E-22 | ||||
| -0,000278122 | -0,000103734 | -0,000393935 | 1,53122E-22 | 5,08953E-23 | -2,24983E-22 | 7,75027E-11 | ||||
| -15751,32757 | 44220,90677 | -2740,494592 | 0,004421012 | -0,001738322 | -0,005651975 | 0,000278122 | ||||
| 44220,90677 | -158690,0422 | 9740,183899 | -0,015459226 | 0,004702428 | 0,020278912 | 0,000103734 | ||||
| -ATA-1 | -2740,494592 | 9740,183899 | -3769,574783 | 0,000949547 | -1,10015E-05 | -0,001349627 | 0,000393935 | |||
| 0,004421012 | -0,015459226 | 0,000949547 | -1,55917E-09 | 4,59218E-10 | 1,94065E-09 | -4,67892E-22 | ||||
| -0,001738322 | 0,004702428 | -1,10015E-05 | 4,59218E-10 | -2,21073E-10 | -6,00943E-10 | 1,8988E-22 | ||||
| -0,005651975 | 0,020278912 | -0,001349627 | 1,94065E-09 | -6,00943E-10 | -2,61955E-09 | 5,95957E-22 | ||||
| 0,000278122 | 0,000103734 | 0,000393935 | -1,53122E-22 | -5,08953E-23 | 2,24983E-22 | -7,75027E-11 | ||||
| -23,0 | ||||||||||
| 124,5 | ||||||||||
| X | 82,5 | |||||||||
| 2,42407E-06 | 0’’,5 | |||||||||
| 9,69627E-06 | 2’’,0 | |||||||||
| -1,21203E-05 | -2’’,5 | |||||||||
| 7E-06 | ||||||||||