godi[na zada^a po predmetot po predmetot fotogrametrija...

Univerzitet “Sveti Kiril i Metodij” Univerzitet “Sveti Kiril i Metodij” Univerzitet “Sveti Kiril i Metodij” Univerzitet “Sveti Kiril i Metodij”

Gradeèn fakultet Gradeèn fakultet Gradeèn fakultet Gradeèn fakultet –––– Skopje Skopje Skopje Skopje

Otsek: GeodezijaOtsek: GeodezijaOtsek: GeodezijaOtsek: Geodezija

GODI[NA ZADAÂGODI[NA ZADAÂGODI[NA ZADAÂGODI[NA ZADAÂ

PO PREDMETOT PO PREDMETOT PO PREDMETOT PO PREDMETOT FOTOGRAMETRIJA 1FOTOGRAMETRIJA 1FOTOGRAMETRIJA 1FOTOGRAMETRIJA 1

U~ebna 2006/2007 U~ebna 2006/2007 U~ebna 2006/2007 U~ebna 2006/2007

Student: Cvetkovski Zoran br.ind. 313Student: Cvetkovski Zoran br.ind. 313Student: Cvetkovski Zoran br.ind. 313Student: Cvetkovski Zoran br.ind. 313

Pregledal:Pregledal:Pregledal:Pregledal:

----------------- Ocenka:Ocenka:Ocenka:Ocenka:

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

ZADA^A BR 1. Numeri~ka relativna orientacija so slikovni

koordinati

1.Dadeni (mereni)golemini:

i1ξ i1η i2ξ i2η s 0ξ

0η

i=1,2,3…n 2.Barani golemini

22121,,,, ωϕϕκκ

3.Matemati~ki model 3.1funkcionalen model

ηη

ωη

ϕηξ

ϕηξ

ξξ

η

η

ii

iiiii

iii

p

dc

cdc

dc

dkdkp

i −−

−−−−−

−−

−=

⋅

++⋅⋅

−⋅⋅

+⋅+⋅−=

21

2

2

2

2

22

1

11

2211

3.2 Stohasti~ki model

R=E

3.3 ravenki na popravki

i

iiiii

iiip pdc

cdc

dc

dd ηωη

ϕηξ

ϕηξ

κξκξν η −

++−++−=

2

2

2

2

22

1

11

2211

V=Ax + f

+

⋅−

⋅−

+

⋅−

⋅−

+

⋅−

⋅−

=

−−−−−−−

−−−−−−−

−−−−−−−

cc

cc

cc

cc

cc

cc

nnnnnnn

A

2

22211

21

2

2222222121

2221

2

1212121111

1211

ηηξηξξξ

ηηξηξξξ

ηηξηξξξ

MMMMM

MMMMM

−

−

−

=

np

p

p

f

η

η

η

M

M

2

1

[ ]22121

ωϕϕ ddddkdkXT=

4.Normalni ravenki

AT Ah+ A

T l=0

Nx + n = 0

nNx 1−−=

5.Ocenka na to~nost

fxAV +⋅=

unVV

T

−⋅

±=σ 0

Qk 1101 σσ ±=

Qk 2202 σσ ±=

Q3301 σσϕ

±=

Q4402 σσ ϕ

±=

Q5502 σσ ω

±=

ξ1ξ1ξ1ξ1 η1η1η1η1 ξ2ξ2ξ2ξ2 η2η2η2η2

1 0,089 -84,333 -103,335 -93,064

2 37,700 -86,292 -65,459 -93,072

3 20,476 -19,479 -84,468 -26,267

4 99,978 -1,129 -6,156 -3,990

5 1,286 4,311 -104,124 -3,437

6 102,394 -83,450 -0,418 -86,700

7 104,372 88,987 -5,205 84,633

8 -2,600 89,421 -109,946 80,201

C= 153,128 mm

ξ0=ξ0=ξ0=ξ0= -0,007 mm

η0=η0=η0=η0= 0,001 mm


1 0,096 -84,334 -103,328 -93,065

2 37,707 -86,293 -65,452 -93,073

3 20,483 -19,480 -84,461 -26,268

4 99,985 -1,130 -6,149 -3,991

5 1,293 4,310 -104,117 -3,438

6 102,401 -83,451 -0,411 -86,701

7 104,379 88,986 -5,198 84,632

8 -2,593 89,420 -109,939 80,200

To~ka S1 S2

To~ka S1 S2

Zada~a 1.)

Da se odredat elementite na relativna orientacija na

A.) Nezavisen stereopar ako se izmereni slikovnite koordinati na osum to~ki i

da se izvr{i ocena na to~nost!

Dadeni elementi:

011

011

ηηη

ξξξ

−=

−=

ii

ii

022

022

ηηη

ξξξ

−=

−=

ii

ii

-0,096 -103,328 -0,05287 -62,79858 209,68914 -8,731 [mm]

-37,707 -65,452 -21,24922 -39,78250 209,69886 -6,780

-20,483 -84,461 -2,60572 -14,48867 157,63409 -6,788

-99,985 -6,149 -0,73783 -0,16026 153,23202 -2,861

-1,293 -104,117 0,03639 -2,33761 153,20519 -7,748

-102,401 -0,411 -55,80603 -0,23271 202,21806 -3,250

-104,379 -5,198 60,65690 2,87287 199,90309 -4,354

2,593 -109,939 -1,51420 57,57998 195,13234 -9,220

-0,0007382 -0,0006931 -0,0000616 -0,0000856 -0,0004035

-0,0006931 -0,0007089 -0,0000573 -0,0000784 -0,0003972

-0,0000616 -0,0000573 -0,0001458 0,0000059 -0,0000352

-0,0000856 -0,0000784 0,0000059 -0,0001287 -0,0000512

-0,0004035 -0,0003972 -0,0000352 -0,0000512 -0,0002302

[rad]

1454,96745 0,008559 dk1

3781,13882 -0,041314 dk2

n= 95,27828 x= 0,008239 df1

392,29951 0,004414 df2

-9274,71185 0,022618 dw2

[o] ['] ['']

δκ1=δκ1=δκ1=δκ1= 0 29 25,44

δκ2=δκ2=δκ2=δκ2= -2 22 01,62

δφ1=δφ1=δφ1=δφ1= 0 28 19,38

δφ2=δφ2=δφ2=δφ2= 0 15 10,45

δω2=δω2=δω2=δω2= 1 17 45,32

0,002 [mm] 0,006 [mm[

-0,006

0,006 [o] ['] ['']

-0,004 mδκ1=δκ1=δκ1=δκ1= 0 00 34,75

-0,002 mδκ2=δκ2=δκ2=δκ2= 0 00 34,06

0,004 mδφ1=δφ1=δφ1=δφ1= 0 00 15,44

0,001 mδφ2=δφ2=δφ2=δφ2= 0 00 14,51

-0,001 mδω2=δω2=δω2=δω2= 0 00 19,41

V=

A= f=

Vrednosti na nepoznatite

Dobieni popravki Ocena na to~nosta:

Kreirawe na matricite A i f.

Re{enie:

=− −1N

=−

=un

VVm

T

0

ZADA^A BR 2. 1.Dadeni (mereni)golemini:

i1ξ i1η i2ξ i2η s 0ξ 0η i=1,2,3…n

2. Barani golemini

by2 > bz2 > k2 > f2 > w2

3. Matemati~ki model 3.1. Funkciski model

( ) ωϕ dh

yhd

h

ybxdkbxdbz

h

ydbyp

i

i

i

i

ixi

xi

i

i

iy2

2

2222

)(⋅

++⋅

⋅−−⋅−−⋅+=

3.2. Stohasti~ki model

P=E

3.3. Ravenki na popravki

pdh

yhd

h

ybxdkbxdbz

h

ydby

ii yi

i

i

i

ixi

xi

i

i

p−⋅

++⋅

−−⋅−−+= ωϕυ 2

2

2222

))(

(

V=Ax+f

+

⋅−−−

+

⋅−−−

+

⋅−−−

=

n

nn

n

nxnxn

n

n

xx

xx

h

yh

h

ybxbx

h

y

h

yh

h

ybxbx

h

y

h

yh

h

ybxbx

h

y

A

2

2

2

22

2

222

2

2

1

2

11

1

111

1

1

)()(1

)()(1

)()(1

MMMMM

MMMMM

−

−

−

=

ny

y

y

p

p

p

f

M

M

2

1

[ ]22222 ωϕ dddkdbzdbyXT=

3.4. Normalni ravenki

nNx

nxN

fAxAATT

⋅−=

=+⋅

=⋅+⋅⋅

−1^

0

0

3.5. Ocena na to~nosta

fxAV +⋅=^^

unVV

T

−⋅

±=σ 0

Qk 1101 σσ ±= Qk 2202 σσ ±=

Q3301 σσ ϕ

±= Q4402 σσ ϕ

±=

Q5502 σσ ω

±=


1 0,089 -84,333 -103,335 -93,064

2 37,700 -86,292 -65,459 -93,072

3 20,476 -19,479 -84,468 -26,267

4 99,978 -1,129 -6,156 -3,990

5 1,286 4,311 -104,124 -3,437

6 102,394 -83,450 -0,418 -86,700

7 104,372 88,987 -5,205 84,633

8 -2,600 89,421 -109,946 80,201

C= 153,128 mm

ξ0=ξ0=ξ0=ξ0= -0,007 mm

η0=η0=η0=η0= 0,001 mm

h= 279,369 mm


1 0,096 -84,334 -103,328 -93,065

2 37,707 -86,293 -65,452 -93,073

3 20,483 -19,480 -84,461 -26,268

4 99,985 -1,130 -6,149 -3,991

5 1,293 4,310 -104,117 -3,438

6 102,401 -83,451 -0,411 -86,701

7 104,379 88,986 -5,198 84,632

8 -2,593 89,420 -109,939 80,200

To~ka S1 S2

To~ka S1 S2

Zada~a 2.)

Da se odredat elementite na relativna orientacija na

B.) Priklu~en stereopar ako se izmereni slikovnite koordinati na osum to~ki i

da se izvr{i ocena na to~nost!

Dadeni elementi:

011

011

ηηη

ξξξ

−=

−=

ii

ii

022

022

ηηη

ξξξ

−=

−=

ii

ii

0,54812 -0,33313 -103,32800 -62,79858 209,68914 8,731 [mm]

0,54812 -0,33315 -65,45200 -39,78250 209,69886 6,780

0,54812 -0,09403 -84,46100 -14,48867 157,63409 6,788

0,54812 -0,01429 -6,14900 -0,16026 153,23202 2,861

0,54812 -0,01231 -104,11700 -2,33761 153,20519 7,748

0,54812 -0,31035 -0,41100 -0,23271 202,21806 3,250

0,54812 0,30294 -5,19800 2,87287 199,90309 4,354

0,54812 0,28708 -109,93900 57,57998 195,13234 9,220

-27,77359 0,86037 -0,00943 0,00726 0,07855

0,86037 -5,37156 -0,00041 0,03017 -0,00331

-0,00943 -0,00041 -0,00006 0,00001 0,00001

0,00726 0,03017 0,00001 -0,00029 -0,00002

0,07855 -0,00331 0,00001 -0,00002 -0,00023

27,2592 1,696 δβΥ2δβΥ2δβΥ2δβΥ2 [mm]

-2,9846 -1,511 δβΖ2δβΖ2δβΖ2δβΖ2 [mm]

n= -3781,1388 x= 0,049708 δκ2δκ2δκ2δκ2 [rad]

-392,2995 0,003735 δφ2δφ2δφ2δφ2 [rad]

9274,7119 -0,022896 δω2δω2δω2δω2 [rad]

δβΥ2=δβΥ2=δβΥ2=δβΥ2= 1,696 mm

δβΖ2=δβΖ2=δβΖ2=δβΖ2= -1,511 mm

[o] ['] ['']

δκ2=δκ2=δκ2=δκ2= 2 50 52,94

δφ2=δφ2=δφ2=δφ2= 0 12 50,31

δω2=δω2=δω2=δω2= -1 18 42,72

VtV= 0,000195

-0,008 [mm] m0=0=0=0= 0,008 [mm]

0,010 mβΥ2=βΥ2=βΥ2=βΥ2= 0,042 [mm]

-0,002 mβΖ2=βΖ2=βΖ2=βΖ2= 0,019 [mm]

-0,002 [o] ['] ['']

0,004 mκ2=κ2=κ2=κ2= 0 00 13,03

-0,002 mφ2=φ2=φ2=φ2= 0 00 28,41

0,001 mω2=ω2=ω2=ω2= 0 00 25,31

-0,002

V=

A= f=



=− −1N


Re{enie:

ZADA^A BR 3. Relativna orientacija so modelski koordinati

A.) Na nezavisen stereopar

• Dadeni golemini:

x

y

b

5n n , .... 2, 1,i )P h, y, (x, ≥=i

• Barani golemini 22121 ω ,, ,k ,k ϕϕ

• Matemati~ki model

- Funkcionalen model

22

i

2

ii2

i

xii1

i

ii2xi1yi d)

h

y1(hd

h

)bx(yd

h

yxdk)bx(xdkP ωϕϕ ++

−−

∗+−+−=

- Stohasti~ki model

EP ≡ - Ravenki na popravki

yi

i

ii

i

xii

i

iixiPyi Pd

h

yhd

h

bxyd

h

yxdkbxxdk −++

−−

∗+−+−= 2

2

2121 )1()(

)( ωϕϕν

To~ka Py[mm] x[mm] y[mm] h[mm]

1 -28,25 57,58 -141,57 279,70

2 -24,95 55,17 21,79 279,42

3 -32,42 40,43 174,87 278,57

4 -26,28 179,10 166,52 278,53

5 -25,31 193,67 161,20 278,91

6 -19,15 192,30 -3,77 279,67

7 -18,93 197,76 -15,96 279,96

8 -22,37 206,29 -147,70 279,89

bx= 200,00 mm

Zada~a 3.)

Da se odredat elementite na relativna orientacija na nezavisen stereopar ako se

izmereni modelskite koordinati i y-paralaksi i da se izvr{i ocena na to~nost!

Dadeni elementi:

-57,58 -142,42 -29,144 -72,086 351,356 -28,25 [mm]

-55,17 -144,83 4,302 11,294 281,119 -24,95

-40,43 -159,57 25,380 100,169 388,343 -32,42

-179,10 -20,90 107,075 12,495 378,084 -26,28

-193,67 -6,33 111,934 3,659 372,078 -25,31

-192,30 -7,70 -2,592 -0,104 279,721 -19,15

-197,76 -2,24 -11,274 -0,128 280,870 -18,93

-206,29 6,29 -108,861 3,319 357,832 -22,37

-0,000208 -0,000208 0,000020 0,000025 -0,000123

-0,000208 -0,000235 0,000024 0,000018 -0,000128

0,000020 0,000024 -0,000032 0,000008 0,000014

0,000025 0,000018 0,000008 -0,000075 0,000015

-0,000123 -0,000128 0,000014 0,000015 -0,000075

[rad]

25963,252 -0,013 κ1

13568,748 -0,054 κ2

n= -3055,559 x= 0,007 φ1

-1983,655 0,005 φ2

-67561,387 0,058 ω2

[o] ['] ['']

κ1=κ1=κ1=κ1= -0 44 20,09

κ2=κ2=κ2=κ2= -3 04 56,87

φ1=φ1=φ1=φ1= 0 24 53,76

φ2=φ2=φ2=φ2= 0 18 24,33

ω2=ω2=ω2=ω2= 3 20 00,64

0,000 [mm] VtV= 0,000079

0,001 mo= 0,005 [mm]

0,000 [o] ['] ['']

-0,006 mo= 0 00 18,44

0,006 mκ1=κ1=κ1=κ1= 0 00 15,25

-0,001 mκ2=κ2=κ2=κ2= 0 00 16,19

0,000 mφ1=φ1=φ1=φ1= 0 00 05,95

0,000 mφ2=φ2=φ2=φ2= 0 00 09,17

mω2=ω2=ω2=ω2= 0 00 09,15

V=

A= f=




=− −1N

Re{enie:

ZADA^A BR 4. Relativna orientacija so modelski koordinati

B.) Na priklu~en stereopar

• Dadeni golemini:

x

iy

b

5n n , .... 2, 1,i )P h, y, (x, ≥=

• Barani golemini

222z2y2 ω , ,k ,b ,b ϕ

• Matemati~ki model

- Funkcionalen model

2

2

22

2)22 )()(

()( ωϕ dh

yhd

h

bxydkbxdb

h

ydbP

i

ii

i

xixiz

i

iyyi ++

−−−++=

- Stohasti~ki model

EP ≡ - Ravenki na popravki

yi

i

ii

i

xixiz

i

iyPyi Pd

h

yhd

h

bxydkbxdb

h

ydb −++

−−−++= 2

2

22

2)22 )()(

()( ωϕν

To~ka Py[mm] x[mm] y[mm] h[mm]

1 87,38 467,98 508,44 256,14

2 89,45 369,31 505,50 256,14

3 85,52 363,00 695,85 256,59

4 79,94 572,91 714,77 256,64

5 81,13 583,09 326,24 255,46

6 85,01 387,89 326,96 255,43

bx= 200,00 mm

Zada~a 4.)

Da se odredat elementite na relativna orientacija na priklu~en stereopar ako se

izmereni modelskite koordinati i y-paralaksi i da se izvr{i ocena na to~nost!

Dadeni elementi:

1,000 1,985 267,980 -531,942 1265,398 -87,38 [mm]

1,000 1,974 169,310 -334,138 1253,759 -89,45

1,000 2,712 163,000 -442,042 2143,675 -85,52

1,000 2,785 372,910 -1038,594 2247,351 -79,94

1,000 1,277 383,090 -489,232 672,091 -81,13

1,000 1,280 187,890 -240,506 673,951 -85,01

-44,9966 43,9394 0,0670 0,0256 -0,0343

43,9394 -52,0273 -0,0373 -0,0120 0,0463

0,0670 -0,0373 -0,0002 -0,0001 0,0000

0,0256 -0,0120 -0,0001 0,0000 0,0000

-0,0343 0,0463 0,0000 0,0000 0,0000

-508,4 68,599

-1017,0 31,940

n= -129363,7 x= -0,017 κ2

257335,3 0,002 φ2

-697518,9 -0,031 ω2

δβδβδβδβy2222==== 68,599 mm

δβδβδβδβz2222==== 31,940 mm

[o] ['] ['']

κ2=κ2=κ2=κ2= -0 59 50,65

ϕ2=ϕ2=ϕ2=ϕ2= 0 06 09,87

ω2=ω2=ω2=ω2= -1 45 57,04

0,003 [mm] VtV= 0,000018

-0,003 mo= 0,002 [mm]

0,001

-0,001 mββββy2222==== 0,016 [mm]

-0,001 mββββz2222==== 0,017 [mm]

0,001 [o] ['] ['']

0,000 mκ1=κ1=κ1=κ1= 0 00 00,13

0,000 mϕ2=ϕ2=ϕ2=ϕ2= 0 00 00,06

mω2=ω2=ω2=ω2= 0 00 00,06

V=

A= f=




=− −1N

Re{enie:

Zada~a 5.) Da se opredelat parametrite na apsolutnata orientacija kako i nivnata to~nost ako se dadeni modelskite i terenskite koordinati na 3 modelski to~ki. to~ka modelski koord. (mm) terenski koord. (m)

x' y' z' X' Y' Z'

1 420,44 356,24 296,94 791594,03 529461,22 328,95

2 549,15 495,9 304,02 790796,58 529218,07 360,35

3 564,59 638,4 305,9 790421,76 528712,71 368,67

99 515.18 425.85 327.63 1.Pribli`ni vrednosti na nepoznatite:

λo= 4,38963

κo= 149°37'13,7"

ωo= 0°

φo= 0°

2.Ravenki na popravki: 2.1Sveduvawe na te`inite:

x y z

xt 511,393 1 -90,953 -140,607 -5,347

yt 496,847 2 37,757 -0,947 1,733

zt 302,287 3 53,197 141,553 3,613

X Y Z

Xt 790937,457 1 656,573 330,553 -23,707

Yt 529130,667 2 -140,877 87,403 7,693

Zt 352,657 3 -515,697 -417,957 16,013

2.2 Prethodna (nulta) transformacija na modelskiot vo terenskiot koord. sistem:

-3,78689 -2,21992 0

λ°*R°= 2,21992 -3,78689 0

0 0 4,38963

To~ka 1 2 3

X° 656,566 -140,879 -515,688

Y° 330,553 87,402 -417,954

Z° -23,470 7,609 15,861

-0,86269 -0,50572 0

Ro= 0,50572 -0,86269 0

0 0 1

2.3 Linearizirani r-ki na popravki vo 1 – ta iteracija (Vo matri~en oblik) V=A*t – l

1 0 0 656,566 0 -23,470 -330,553

0 1 0 330,553 23,470 0 656,566

0 0 1 -23,470 330,553 -656,566 0

1 0 0 -140,879 0 7,609 -87,402

A= 0 1 0 87,402 -7,609 0 -140,879

0 0 1 7,609 87,402 140,879 0

1 0 0 -515,688 0 15,861 417,954

0 1 0 -417,954 -15,861 0 -515,688

0 0 1 15,861 -417,954 515,688 0

2.4 Re{avawe na normalnite r-ki N = A’*A n = A’*l A’*A*t – A’*l = 0 t = 1/N*n

3 0 0 -1,023E-12 0

-5,0093E-13

-2,84217E-13

0 3 0 2,8422E-13 5,00933E-13 0

-1,02318E-12

0 0 3 -5,009E-13 2,84217E-13 1,0232E-12 0

N= -1,023E-12 2,8422E-13 -5,0093E-13 1009309,98 0 0 0

0 5,0093E-13 2,84217E-13 0 292450,1852 -420250,419 24660,86295

-5,009E-13 0 1,02318E-12 0 -420250,419 717720,105 13722,27279

-2,842E-13

-1,0232E-12

0 0 24660,86295 13722,2728 1008449,679

1,02318E-12

-2,70006E-13

4,43201E-13

n= 19,0010893

-134,4113483

245,5467402

-4,799717808

0,007

0,001

-0,237

0,002

l= 0,001

0,085

-0,009

-0,002

0,152

3,4114E-13 dx°

-9,005E-14 dy°

1,4756E-13 dz°

t= 0,00002 dλ°

0,00021 dω° 0,012229 vo stepeni 0°00’44,0”

0,00047 dφ° 0,026781 0°01’36,4”

-0,00002 dκ° -0,000936 (-)0°00’03,4”

ρ°= 57,29577951

2.5 Pribli`ni vrednosti na nepoznatite posle 1 – tata iteracija: R¹=dR¹*R°

1 1,6339E-05 0,00046741

dR¹= -1,634E-05 1 -0,00021344

-0,0004674 0,00021344 1

-0,8626817 -0,5057341 0,00046741

R¹= 0,5057341 -0,86268174 -0,00021344

0,00051117 5,2245E-05 1

λ¹=(1+dλ¹)*λ°= 4,38971264

-3,786924923 -2,220027 0,002051797

λ¹*R¹= 2,220027352 -3,786925 -0,000936946

0,002243897 0,000229 4,389712638

Xo¹= 793976,458

Yo¹= 529877,164

Zo¹= -975,556

2.6 Transformacija posle 1 – tata iteracija: To~ka 1 2 3

X¹ 656,573 -140,876 -515,697

Y¹ 330,553 87,404 -417,957

Z¹ -23,707 7,693 16,013

3. Linearizirani r-ki na popravki vo 2 – tata iteracija (Vo matri~en oblik)

1 0 0 656,573 0 -23,707 -330,553

0 1 0 330,553 23,707 0 656,573

0 0 1 -23,707 330,553 -656,573 0

1 0 0 -140,876 0 7,693 -87,404

A1= 0 1 0 87,404 -7,693 0 -140,876

0 0 1 7,693 87,404 140,876 0

1 0 0 -515,697 0 16,013 417,957

0 1 0 -417,957 -16,013 0 -515,697

0 0 1 16,013 -417,957 515,697 0

0,000212926

0,000321102

3,16511E-07

-0,000186981

l= -0,000830924

-6,88342E-06

-2,59453E-05

0,000509822

6,5669E-06

3.1 Re{avawe na normalnite r-ki N = A’*A n = A’*l A’*A*t – A’*l = 0 t = 1/N*n

3 0 0 0 0 -0,001 0

0 3 0 0 0,001 0 0

0 0 3 -0,001 0 0 0

N= 0 0 -0,001 1009347,97 0 0 0

0 0,001 0 0 292470,4191 -420258,22 24906,99124

-0,001 0 0 0 -420258,22 717755,168 13856,76644

0 0 0 0 24906,99124 13856,7664 1008470,346

0

5,68434E-14

-6,21725E-15

n1= -1,50564E-06

0,002599169

-0,004692729

8,55595E-05

-2,846E-12 dx¹

1,1536E-12 dy¹

-2,57E-15 dz¹

t1= -1,492E-12 dλ¹

-3,404E-09 dω¹ -0,00000020 vo ste. 0°00’00’0”

-8,537E-09 dφ¹ -0,00000049 0°00’00,0”

2,8621E-10 dκ¹ 0,00000002 0°00’00’0”

ρ°= 57,2957795

3.2 Pribli`ni vrednosti na nepoznatite posle 2 – tata iteracija

R²=dR²*R¹

1 -2,8621E-10 -8,5367E-09

dR²= 2,8621E-10 1 3,4041E-09

8,5367E-09 -3,4041E-09 1

-0,8626817 -0,5057341 0,000467402

R²= 0,5057341 -0,86268174 -0,00021344

0,00051116 5,2244E-05 1

-3,78692492 -2,22002735 0,00205176

λ²*R²= 2,22002735 -3,786924924 -0,000936932

0,00224386 0,000229334 4,389712638

λ²= 4,38971264

Xo²= 793976,458

Yo²= 529877,164

Zo²= -975,556

3.3 Transformacija posle 2 – tata iteracija: To~ka 1 2 3

X² 656,573 -140,876 -515,697

Y² 330,553 87,404 -417,957

Z² -23,707 7,693 16,013

4.Definitivni vrednosti na nepoznatite:

To~ka 1 2 3

Xdef 656,573 -140,876 -515,697

Ydef 330,553 87,404 -417,957

Zdef -23,707 7,693 16,013

λdef= 4,38971264

ωdef=arctg(-R23/R33)= 0,012204 0°00'43,9"

φdef=arcsinR13= 0,026757 0°01'36,3"

κdef=arctg(-R12/R11)= -30,380305 -30,22491 149°37'10,9"

4.1 Transformacija na definitivnite vrednosti na nepoznatite vo terenski koordinati

To~ka 1 2 3

Xdef 791594,0298 790796,5802 790421,76

Ydef 529461,2197 529218,0708 528712,709

Zdef 328,9500042 360,3500054 368,66999

5. Kriterium za izlez od ciklusot:

1 2 3

X²-X¹ 0,000 0,000 0,000 m

Y²-Y¹ 0,000 0,000 0,000

Z²-Z¹ 0,000 0,000 0,000

X²-X¹ 0,0 0,0 0,0 cm

Y²-Y¹ 0,0 0,0 0,0 <10

Z²-Z¹ 0,0 0,0 0,0

6.Ocenka na to~nosta To~ka 1 2 3

∆X -0,000213 0,000187 0,000026

∆Y -0,000321 0,000831 -0,000510

∆Z 0,000004 0,000005 -0,000010

Σ∆²= 0,0000011

0,333333

-1,43705E-

12 0 0 4,31116E-

09 2,99152E-

09 -1,47581E-

10

-1,4371E-

12 0,333333 0 0 -7,355E-09 -4,31116E-

09 2,4089E-10

0 0 0,333333 3,3025E-

10 0 0 0

Q= 0 0 3,30246E-

10 0,000001 0 0 0

4,31116E-

09

-7,35504E-

09 0 0 0,000022 1,29335E-

05 -7,2267E-07

2,99152E-

09

-4,31116E-

09 0 0 1,29335E-

05 0,000009 -4,42742E-

07

-1,4758E-

10 2,4089E-

10 0 0 -7,2267E-

07 -4,42742E-

07 0,000001

n-u= 9-7=2

δo= 0,000615

δxo= 0,000355

δyo= 0.000355

δzo= 0,000355

δλo= 0,000001

δωo= 0,000003 0,000172 vo ste. 0°00'00,6"

δφo= 0,000002 0,000115 0°00'00,4"

δκo= 0,000001 0,000057 0°00'00,2"

ρ°= 57,2957795

Definitivni koordinati za to~ka 99

=

+

=

897.463

907.529407

786.791080

**

99Z

Y

X

R

Zo

Yo

Xo

Z

Y

Xdefdef

def

λ

Zada~a6.) Da se opredelat parametrite na apsolutnata orientacija kako i nivnata to~nost ako se dadeni modelskite i terenskite koordinati na 3 modelski to~ki. (Model M43)

to~ka modelski koord. (mm) terenski koord. (m) x' y' z' X' Y' Z'

1 420,44 356,24 296,94 791594,03 529461,22 328,95

2 549,15 495,9 304,02 790796,58 529218,07 360,35

3 564,59 638,4 305,9 790421,76 528712,71 368,67

99 515.18 425.85 327.63

xt 511,39 Xt 790937,46

yt 496,85 Yt 529130,67

zt 302,29 Zt 352,66

1 Iteracija a)Polo`beno izramnuvawe Barani golemini Xo,Yo,λ,κ 1.1 R-ki na popravki vo matri~en oblik

1 0 420,44 -356,24 791594,03

0 1 356,24 420,44 529461,22

A= 1 0 549,15 -495,9 l= 790796,58

0 1 495,9 549,15 529218,07

1 0 564,59 -638,4 790421,76

0 1 638,4 564,59 528712,71

1.2 Formirawe i re{avwe na normalnite r-ki

3 0 1534,18 -1490,54 2372812,37

N= 0 3 1490,54 1534,18 n= 1587392

1534,18 1490,54 1577475,692 0 2001932657

-1490,54 1534,18 0 1577475,692 -367026045,4

λ°= 4,389646

κ°= 149° 37' 13,2"

1.3 Transformacija posle polo`benoto izramnuvawe

793977,04 Xo

529876,91 Yo

0 Zo

793977,0435

t= 529876,9093

-3,786918343

2,219964572

-0,862693 -0,505728 0

Rκ°= 0,505728 -0,862693 0

0 0 1

To~ka 1 2 3

X° 791594,03 790796,58 790421,76

Y° 529461,22 529218,07 528712,71

Z° 1303,46 1334,54 1342,79

b) Visinsko izramnuvawe Barani golemini Zo,ω,φ 1.4 R-ki na popravki vo matri~en oblik

1 -415,69 2383,01 974,51

A= 1 -658,84 3180,47 l= 974,19

1 -1164,20 3555,28 974,12


3 -2238,72227 9118,761585 2922,824372

N= -2238,722275 1962217,156 -7225048,33 n= -2180996,269

9118,761585 -7225048,33 28434150,01 8883933,78

-975,5380594 Zo

t= 0,000213649 dω

0,000468058 dφ

1 0 0

dRω= 0 1 -0,00021365

0 0,000213649 1

1 0 0,000468058

dRφ= 0 1 0

-0,000468058 0 1

-0,862693 -0,505728 0,000468058

R°ˇ= 0,505727914 -0,86269305 -0,00021365

0,000511839 0,0000524 1

Xo° 793976,4228

Yo° 529877,1912

Zo° -975,538059

To~ka 1 2 3

X°ˇ 791594,0205 790796,5804 790421,769

Y°ˇ 529461,2241 529218,0685 528712,7074

Z°ˇ 328,95 360,35 368,67

2 Iteracija a)Polo`beno izramnuvawe 2.1 R-ki na popravka

1 0 791594,02 -529461,22 791594,03

0 1 529461,22 791594,02 529461,22

A= 1 0 790796,58 -529218,07 l= 790796,58

0 1 529218,07 790796,58 529218,07

1 0 790421,77 -528712,71 790421,76

0 1 528712,71 790421,77 528712,71


3 0 2372812,37 -1587392 2372812,37

N= 0 3 1587392 2372812,37 n= 1587392

2372812,37 1587392 2,71668E+12 0 2,71668E+12

-1587392 2372812,37 0 2,71668E+12 -11,13305664

-12,57803448

t= 4,224659646

1,000008518

-0,00001104

2.3Transformacija posle polo`benoto izramnuvawe

1 0,000011 0

dκ¹= -0,000011 1 0 1+dλ¹= 1,000008518

0 0 1

λ¹=(1+dλ¹)λ°= 4,38968339

-0,862687437 -0,50573749 0,000468056 κ¹= (-)0° 00' 02,3"

R¹= 0,505737403 -0,86268749 -0,00021365 269° 59' 57,7"

0,000511839 0,0000524 1 0° 00' 02,3"

Xo¹ 793976,46

Yo¹ 529877,16

Zo¹ -975,55

To~ka 1 2 3

X¹ 791594,03 790796,58 790421,76

Y¹ 529461,22 529218,07 528712,71

Z¹ 328,95 360,35 368,67

b) Visinsko izramnuvawe 2.4 R-ki na popravki vo matri~en oblik

1 -415,94 2382,43 -0,0002

A= 1 -659,09 3179,88 l= 0,0001

1 -1164,45 3554,70 0,0001


3 -2239,49247 9117,000635 0

N= -2239,49247 1963368,527 -7226076,26 n= -0,118030255

9117,000635 -7226076,26 28423444,3 0,212154683

0,001044528 Z1

t= -1,4024E-07 dω¹

-3,78156E-07 dφ¹

1 0 0

dRω¹= 0 1 0,0000001

0 -0,0000001 1

1 0 -0,0000004

dRφ¹= 0 1 0

0,0000004 0 1

-0,86268744 -0,50573749 0,000467677

R¹ˇ= 0,505737403 -0,862687488 -0,000213514

0,000511442 5,23259E-05 1

Xo¹ˇ 793976,43

Yo¹ˇ 529877,16

Zo¹ˇ -975,54

To~ka 1 2 3

X¹ˇ 791594,00 790796,55 790421,73

Y¹ˇ 529461,21 529218,06 528712,70

Z¹ˇ 328,96 360,36 368,68

3. Kriterium za izlez od ciklusot

-0,017 -0,026 -0,035

-0,011 -0,004 -0,004

0,011 0,011 0,011

-2 -3 -3

-1 0 0 < 10 cm

1 1 1

4. Definitivni vrednosti na nepoznatite

To~ka 1 2 3

Xdef 791594,00 790796,55 790421,73

Ydef 529461,21 529218,06 528712,70

Zdef 328,96 360,36 368,68

λdef= 4,38968339

ωdef= 0°00”44”

φdef= 0°01’36,4”

κdef= 149°37’10,8”

5. Ocena na to~nost

To~ka 1 2 3

∆X -0,02610877 -0,02568862 -0,025859447 m

∆Y -0,00665156 -0,005538389 -0,006885605

∆Z 0,011313389 0,011313389 0,011313389

δop = ±0,0346410

δzp = ±0,0122474

897970,866 -0,00034183 -0,784307491 0,524695275

6,81466E-05 897970,8661 -0,524695274 -0,784307491

Qop= -0,78430749 -0,52469527 9,91618E-07 -2,56334E-16

0,524695274 -0,78430749 1,96394E-16 9,91618E-07

36,90695248 -0,02288798 -0,017656938

Qoz= -0,02288798 2,21125E-05 1,29631E-05

-0,01765694 1,29631E-05 8,99436E-06

δXo= 32,82627

δYo= 32,82627

δλ= 0,000034

δκ= 0,000034 0°00'07,0"

δZo= 0,074404

δω= 0,000057 0°00'11,7"

δφ= 0,000038 0°00'07,8"

ρ°= 57,29577951

=

+

=

868.463

917.529407

776.791080

**

99Z

Y

X

R

Zo

Yo

Xo

Z

Y

Xdefdef

def

λ

Σ∆X²= 0,0027

Σ∆Y²= 0,0003

Σ∆Z²= 0,0003

n - u = 9 - 7 = 2

Snimka 1

To~ka ξ[mm] η[mm] To~ka X[m] Y[m] Z[m]

1 -101,420 -99,095 1 600020,04 453223,33 454,80

2 -104,496 96,088 2 598760,98 453309,76 465,48

3 -10,062 -94,888 3 600040,50 453817,85 456,11

4 -15,401 98,977 4 598784,77 453881,35 466,94

c= 152,465

ξξξξo= 0,012

ηηηηo= 0,007

1.) Sveduvawe na te`i{teξο[mm] ηο[mm] Xt Yt Zt

-57,845 0,270 599401,57 453558,07 460,83 [m]

1 -43,575 -99,366 1 618,47 -334,74 -6,03

2 -46,651 96,088 2 -640,59 -248,31 4,65

3 47,783 4,477 3 638,93 259,78 -4,72

4 42,444 2,889 4 -616,80 323,28 6,11

2.)Helmertova transformcija

1 0 -101,420 99,095 -618,47

0 1 -99,095 -101,420 334,74

1 0 -104,496 -96,088 640,59

0 1 96,088 -104,496 248,31

1 0 -10,062 94,888 f= -638,93

0 1 -94,888 -10,062 -259,78

1 0 -15,401 -98,977 616,80

0 1 98,977 -15,401 -323,28

4,000 0,000 -231,379 -1,082

0,000 4,000 1,082 -231,379

-231,379 1,082 59396,767 0,000

-1,082 -231,379 0,000 59396,767

0,3227214 0,0000000 0,0012572 0,0000059 0,000

0,0000000 0,3227214 -0,0000059 0,0012572 0,000

0,0012572 -0,0000059 0,0000217 0,0000000 -23943,593

0,0000059 0,0012572 0,0000000 0,0000217 -296820,672

31,8458 Xo λ = 6,47183681

373,0089 Yo dec,zapis

0,5204 a k = 85,38811519

6,4509 b

Xo Yo Zo

599433,418 453931,081 1447,5611

t=

N-1=

Modelski koordinati Terenski koordinati

n=

N=

A=

3.) Ravenka na rotacija Rk

0,08041 -0,99676 0

Rk= 0,99676 0,08041 0

0 0 1

Xd Yd Zd

1 586,6217 -707,751 -992,7611

2 -672,4383 -621,321 -982,0811

3 607,0817 -113,231 -991,4511

4 -648,6483 -49,731 -980,6211

To~ka Zx Zy N

1 -658,292 -641,630 -992,761

2 -673,378 620,303 -982,081

3 -64,052 -614,221 -991,451

4 -101,725 642,549 -980,621

4.) Koeficienti vo matrica A

a1 a2 a3 a4 a5 a6

1 -0,01235 -0,15308 0,10184 -224,04559 47,47988 -98,53937

2 -0,01248 -0,15474 0,08691 -210,35347 -210,19819 96,30014

3 -0,01236 -0,15328 0,00993 -218,10926 -83,83820 -94,45462

4 -0,01250 -0,15497 -0,11054 -152,15174 -217,90888 99,90229

b1 b2 b3 b4 b5 b6

1 0,15308 -0,01235 0,09926 -153,09629 -6,22780 -101,09835

2 0,15474 -0,01248 0,08290 48,66599 -209,80746 -104,53975

3 0,15328 -0,01236 0,09527 -152,77345 -22,72085 -9,84989

4 0,15497 -0,01250 -0,11438 -7,19259 -218,05340 -15,81606

1 2 3 4 5 6-0,0123 -0,1531 0,1018 -224,0456 47,4799 -98,5394 0,3336

0,1531 -0,0123 0,0869 -153,0963 -210,1982 -101,0984 0,5626

-0,0125 -0,1547 0,0869 -218,1093 -83,8382 96,3001 -0,0318

0,1547 -0,0125 -0,1105 48,6660 -217,9089 -104,5398 0,2191

A= -0,0124 -0,1533 0,0099 -153,0963 -6,2278 -94,4546 f= 0,2241

0,1533 -0,0124 0,0829 -152,7735 -209,8075 -9,8499 0,4404

-0,0125 -0,1550 -0,1105 -152,7735 -22,7209 99,9023 -0,4031

0,1550 -0,0125 -0,1144 -7,1926 -218,0534 -15,8161 0,9323

0,096 0,000 -0,010 -31,145 -131,028 -35,680

0,000 0,096 -0,013 118,453 20,817 2,051

-0,010 -0,013 0,070 -56,932 13,366 -9,885

N= -31,145 118,453 -56,932 193744,590 67270,014 12280,408

-131,028 20,817 13,366 67270,014 193072,220 35112,270

-35,680 2,051 -9,885 12280,408 35112,270 59382,473

1404,19148 -821,30931 426,45206 0,57076 0,74297 0,38573

-821,30931 658,20492 -353,27054 -0,48749 -0,38971 -0,24379

426,45206 -353,27054 212,92061 0,27130 0,19404 0,13304

N-1= 0,57076 -0,48749 0,27130 0,00037 0,00026 0,00017

0,74297 -0,38971 0,19404 0,00026 0,00041 0,00019

0,38573 -0,24379 0,13304 0,00017 0,00019 0,00013

0,3306 -7,211 dXo

-0,0448 2,338 dYo

0,0325 -1,586 dZo

n= -190,0146 x= -0,00126 dw

-435,4326 -0,00240 df

-196,2414 0,00031 dk

0,0419 Vt*V 0,436

-0,0419

0,0640

V -0,3216 Xo Yo Zo

0,1183 599426,207 453933,420 1445,975

-0,1328

-0,2223

0,4938

ω = -0,00126

φ = -0,00240

k = 85,38842

σo = 0,47

σx = 17,50

σy = 11,98

σz = 6,81

σω = 0,01

σφ = 0,01

σk = 0,01

Kreirawe na potrebnite matriciSnimka 1

Snimka 2

To~ka ξ[mm] η[mm] To~ka X[m] Y[m] Z[m]

1 -102,180 -100,631 1 600040,50 453817,85 456,11

2 -107,245 93,622 2 598784,77 453881,35 466,94

3 7,276 -106,996 3 600132,22 454524,22 465,77

4 12,775 91,049 4 598857,83 454651,25 467,15

c= 152,465

ξξξξo= 0,012

ηηηηo= 0,007

1.) Sveduvawe na te`i{teξο[mm] ηο[mm] Xt Yt Zt

-47,344 -5,739 599453,83 454218,67 463,99 [m]

1 -54,837 -94,892 1 586,67 -400,82 -7,88

2 -59,902 93,622 2 -669,06 -337,32 2,95

3 54,620 -12,104 3 678,39 305,55 1,78

4 60,119 -2,573 4 -596,00 432,58 3,16

2.)Helmertova transformcija

1 0 -102,180 100,631 -586,67

0 1 -100,631 -102,180 400,82

1 0 -107,245 -93,622 669,06

0 1 93,622 -107,245 337,32

1 0 7,276 106,996 f= -678,39

0 1 -106,996 7,276 -305,55

1 0 12,775 -91,049 596,00

0 1 91,049 12,775 -432,58

4,000 0,000 -189,374 22,956

0,000 4,000 -22,956 -189,374

-189,374 -22,956 60788,125 0,000

22,956 -189,374 0,000 60788,125

0,2939990 0,0000000 0,0009159 -0,0001110 0,000

0,0000000 0,2939990 0,0001110 0,0009159 0,000

0,0009159 0,0001110 0,0000193 0,0000000 -24577,100

-0,0001110 0,0009159 0,0000000 0,0000193 -333406,733

-14,5066 Xo λ = 6,46752742

308,0955 Yo dec,zapis

0,4755 a k = 85,78406297

6,4500 b

Xo Yo Zo

599439,323 454526,763 1450,0641

t=

N-1=

Modelski koordinati Terenski koordinati

n=

N=

A=

3.) Ravenka na rotacija Rk

0,07352 -0,99729 0

Rk= 0,99729 0,07352 0

0 0 1

Xd Yd Zd

1 601,1766 -708,913 -993,9541

2 -654,5534 -645,413 -983,1241

3 692,8966 -2,543 -984,2941

4 -581,4934 124,487 -982,9141

To~ka Zx Zy N

1 -662,799 -651,666 -993,954

2 -691,786 605,334 -983,124

3 48,403 -691,209 -984,294

4 81,401 589,072 -982,914

4.) Koeficienti vo matrica A

a1 a2 a3 a4 a5 a6

1 -0,01128 -0,15298 0,10229 -224,04559 47,47988 -99,96061

2 -0,01140 -0,15466 0,08691 -215,12502 -210,19819 93,87655

3 -0,01139 -0,15448 -0,00762 -218,10926 -83,83820 -107,06670

4 -0,01140 -0,15470 -0,11054 -155,07118 -217,90888 91,37403

b1 b2 b3 b4 b5 b6

1 0,15298 -0,01128 0,10057 -153,09629 -6,22780 -101,66831

2 0,15466 -0,01140 0,08290 50,42064 -209,80746 -107,28374

3 0,15448 -0,01139 0,10878 -152,77345 -22,72085 7,49745

4 0,15470 -0,01140 -0,11438 -22,78116 -218,05340 12,62658

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