Rachunek wyrównawczy - skrypt

Tadeusz Gargula

Rachunek wyrównawczy

3U]\NáDG\�RSUDFRZDQLD�üZLF]H�

Kraków 2005

6NU\SW�RSLQLRZDá� SURI��GU�KDE��LQ*��5RPDQ�.DGDM

Autor skryptu jest pracownikiem naukowo-dydaktycznym w Katedrze Geodezji,

Akademia Rolnicza w Krakowie, ul. Balicka 253A, 30-149 Kraków

Copyright © by GEODPIS

Druk, oprawa: P.W. STABIL Kraków Wydawnictwo: GEODPIS Andrzej Jagielski

tel./fax (012) 411-89-43 tel. kom. 505-204-149 e-mail: [email protected]

ISBN 83-922884-1-6

6SLV�WUH�FL 3

6SLV�WUH�FL

:VWS ............................................................................................................................. 5

3UDZR�SU]HQRV]HQLD�VL�EáGyZ��UHGQLFK�REVHUZDFML�QLH]DOH*Q\FK ....................... 6

:\UyZQDQLH�VSRVWU]H*H��EH]SR�UHGQLFK��3XQNW�Z]áRZ\�Z�QLZHODFML ................... 8

8NáDG�UyZQD��OLQLRZ\FK�MHGQR]QDF]QLH�RNUH�ORQ\FK .............................................. 10

8NáDG�UyZQD��OLQLRZ\FK�QDGRNUH�ORQ\FK.................................................................. 12

8NáDG�UyZQD��OLQLRZ\FK�QLHGRRNUH�ORQ\FK............................................................... 15

:\UyZQDQLH�VWDF\MQH�PHWRG��SR�UHGQLF]�F�.......................................................... 17

:\UyZQDQLH�VWDF\MQH�PHWRG��]DZDUXQNRZDQ�....................................................... 21

:\UyZQDQLH�VLHFL�QLZHODF\MQHM�PHWRG��SR�UHGQLF]�F� .......................................... 24

:\UyZQDQLH�VLHFL�QLZHODF\MQHM�PHWRG��]DZDUXQNRZDQ� ....................................... 27

Rozwijanie funkcji nieliniowej w szereg Taylora ...................................................... 30

:\UyZQDQLH�ZFLFLD�ZVWHF]��N�WRZHJR��PHWRG��SR�UHGQLF]�F� .......................... 32

:\UyZQDQLH�ZFLFLD�ZVWHF]��NLHUXQNRZHJR��PHWRG� SR�UHGQLF]�F�................... 37

:\UyZQDQLH�ZFLFLD�ZVWHF]��NLHUXQNRZHJR��PHWRG� SR�UHGQLF]�F��SRSU]H]�HOLPLQDFM�QLHZLDGRPHM�RULHQWDF\MQHM........................................................................ 42

%á�G�SRáR*HQLD�SXQNWX��HOLSVD�EáGX��UHGQLHJR ....................................................... 47

:\UyZQDQLH�ZFLFLD�Z�SU]yG��NLHUXQNRZHJR��PHWRG� SR�UHGQLF]�F� ................. 49

:\UyZQDQLH�F]ZRURERNX�JHRGH]\MQHJR��N�WRZHJR��PHWRG� SR�UHGQLF]�F� ...... 54

:\UyZQDQLH�F]ZRURERNX�JHRGH]\MQHJR��N�WRZHJR��PHWRG� ]DZDUXQNRZDQ�... 61

Wyrównanie czworoboku geodezyjnego (liniowego) PHWRG��SR�UHGQLF]�F� ...... 64

Spis trH�FL 4

Wyrównanie czworoboku geodezyjnego (liniowego) PHWRG��]DZDUXQNRZDQ� ... 69

:\UyZQDQLH�FL�JX�SROLJRQRZHJR�PHWRG��SR�UHGQLF]�F�...................................... 73

:\UyZQDQLH�FL�JX�SROLJRQRZHJR�PHWRG��ZDUXQNRZ� ........................................... 80

3UDZR�SU]HQRV]HQLD�VL�EáGyZ��UHGQLFK�ZLHONR�FL�VNRUHORZDQ\FK.

%á�G��UHGQL�funkcji....................................................................................................... 84

:\UyZQDQLH�WUDQVIRUPDFML�ZVSyáU]GQ\FK dla n>2 punktów dostosowania ......... 87

Dodatek A: Przydatne wzory matematyczne ........................................................... 92

Dodatek B: Podstawowe wzory z rachunku wyrównawczego ............................... 94

'RGDWHN�&��6áRZQLF]HN�ZD*QLHMV]\FK�WHUPinów z rachunku wyrównawczego .... 96

:VWS 5

:VWS

Skrypt przeznaczony jest dla studentów kierunku Geodezja i Kartografia Akademii

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czenia.

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,PL�1D]ZLVNR Nr zestawu .....

û:,&=(1,(�QU�� Temat: 3UDZR�SU]HQRV]HQLD�VL�EáGyZ��UHGQLFK�REVHUZDFML�QLH]DOH*�nych

Zadanie��:� ]DGDQLX� ZFLFLD� Z� SU]yG� GDQH� V�� EH]EáGQH�ZVSyáU]GQH� SXQNWyZ�$� L� %�oraz poPLHU]RQH�ZDUWR�FL�N�WyZ�α i β�ZUD]�]�EáGDPL��UHGQLPL� W\FK�SRPLDUyZ�mα, mβ.

2EOLF]\ü�EáG\��UHGQLH�ZVSyáU]GQ\FK�mx, my�SXQNWX�ZFLQDQHJR�3�RUD]�Eá�G�SRáR*HQLD�µP tego punktu.

Dane:

xA = 1357,20 xB = 1393,22

yA = 1448,22 yB = 2896,53

α = 54°26′55″ mα = ± 10″

β = 63°21′08″ mβ = ± 10″

52=:,�=$1,(

1) :]RU\�QD�REOLF]HQLH�ZVSyáU]GQ\FK�SXQNWX�3

xP = xA + dAP · cosAAP AAP = AAB - α $%$3 GÃ�VLQ�VLQÃ��G •+

= .

yP = yA + dAP · sinAAP

2) Obliczenie azymutu boku AB

40,20836,02

1448,31

û[û\

tgAB

AB ===ABA AAB = 88°34′31″

3) 2EOLF]HQLH�GáXJR�FL�ERNX�AB

��:VSyáU]GQH�SXQNWX�ZFLQDQHJR

��

=+

=+

••

••

��+(.sin)Asin(�sin

dy =y

��+(.sin)Acos(�sin

dx =x

ABABAP

ABABAP

α

α

P76144822 ,yxdABAB

AB =∆+∆=

B A

ββ αα

P

3UDZR�SU]HQRV]HQLD�VL�EáGyZ��UHGQLFK�REVHUZDFML�QLH]DOH*Q\FK 7

��3RFKRGQH�F]�VWNRZH�]�xP i yP�Z]JOGHP�αα i ββ

P1614602

2

,)(sin

)Acos(sind

)(sin

)cos()Acos()sin()Asin(sind

x

ABAB

ABABAB

P

=++

⋅⋅=

=+

+⋅−−+⋅−⋅⋅=

∂∂

βαβ

β

βαβααβαα

βα

P9812462

2

,)(sin

sin)Acos(d

)(sin

)cos(sin)sin(cos)Acos(d

x

ABAB

ABABP

=+

⋅−⋅=

=+

+⋅−+⋅⋅−⋅=

∂∂

βαα

α

βαβαββαβ

αβ

P75778

1

2

2

,)(sin

)Asin(sind

)(sin

)cos()Asin()sin()()Acos(sind

y

ABAB

ABABAB

P

−=++

⋅⋅=

=+

+⋅−−+⋅−⋅−⋅⋅=

∂∂

βαβ

β

βαβααβαα

βα

P128452

2

,)(sin

sin)Asin(d

)(sin

)cos(sin)sin(cos)Asin(d

y

ABAB

ABABP

=+

⋅−⋅=

=+

+⋅−+⋅⋅−⋅=

∂∂

βαα

α

βαβαββαβ

αβ

:]RU\�SRPRFQLF]H�GR�REOLF]HQLD�SRZ\*V]\FK�SRFKRGQ\FK:

αββαβα

αββαβα

sinsincoscos)cos(

cossincossin)sin(

⋅+⋅=+

⋅+⋅=+

6��%áG\��UHGQLH�ZVSyáU]GQ\FK�SXQNWX�3

( ) ( )2

22

2

2222

20626510341638

20626510161460

⋅+

⋅±=

⋅

∂∂+

⋅

∂∂±=

""m,

""m,

"

mx

"

mx PPx ρβρα

βαPP0890,mx ±=

( ) ( )2

22

2

2222

2062651012845

2062651075778

⋅+

⋅−±=

⋅

∂∂+

⋅

∂∂±=

""m,

""m,

"

my

"

my PPy ρβρα

βαPP0550,my ±=

7��%á�G�SRáR*enia punktu P

PPP 105005500890 2222 ,),(),(mmm yxP ±=+±=+±=

:\UyZQDQLH�VSRVWU]H*H��EH]SR�UHGQLFK��3XQNW�Z]áRZ\�Z�QLZHODFML 8


û:,&=(1,(�QU�� Temat: :\UyZQDQLH�VSRVWU]H*H��EH]SR�UHGQLFK�

3XQNW�Z]áRZ\�Z�Qiwelacji

Zadanie��:�Z\QLNX�SRPLDUX�QLZHODF\MQHJR�FL�JyZ�R�GáXJR�FL�d1, d2, d3�X]\VNDQR�Uy*Qi-

ce�Z\VRNR�FL�∆h1, ∆h2, ∆h3. 2EOLF]\ü�Z\VRNR�ü�Z\UyZQDQ��SXQNWX�W PHWRG��REVHUZa-

cji bezpo�UHGQLFK�

Dane:

Nr punktu HR ∆h d

R1 318,097 3.601 1.5

R2 333,078 18.596 2.0

R3 303,900 10.590 2.0

52=:,�=$1,(

1) Zestawienie tabelaryczne

:\VRNR�FL�SXQNWX�:��Li = HRi +(-) ∆h Wyrazy wolne: li = Li – x0 :DUWR�ü�SU]\EOL*RQD�QLHZLDGRPHM��Z\VRNR�FL��x0 = 314,482 m

Poprawki obserwacji: vi = ∆x - li

3U]\URVW�V]XNDQHM�Z\VRNR�FL�� [ ][ ] mm 8,0

1.67

13,38===

p

plû[

Nr L d p l pl pll v pv pvv L + v

1 314.496 1,5 0,67 14 9,38 131,32 -6,0 -4,0 24,12 314,490

2 314.482 2,0 0,50 0 0 0 8,0 4,0 32,00 314,490

3 314.490 2,0 0,50 8 4 32 0,0 0,0 0,00 314,490

suma 1,67 13,38 163,32 2,0 0,0 56,12

R2

R1

R3

W

∆h1

∆h2

∆h3

:\UyZQDQLH�VSRVWU]H*H��EH]SR�UHGQLFK��3XQNW�Z]áRZ\�Z�QLZHODFML 9

��:\UyZQDQD�Z\VRNR�ü�SXQNWX�:

xwyr = x0 + ∆x = 314,482 + 0,008 = 314,490

3) Kontrola ogólna:

[ ] [ ] [ ][ ]p

plpllpvv

2

−=

[ ] 56,12=pvv [ ] [ ][ ] 56,12

1,67

13,38163,32

2

=−=−p

plpll

2

4) 2FHQD�GRNáDGQR�FL

%á�G��UHGQL�REVHUZDFML�� [ ]

kmmm5,32

56,12 ±≈±=−

±=1n

pvvm0

%á�G��UHGQL�QLHZLDGRPHM��Z\UyZQDQHM�Z\VRNR�FL��

[ ] mm4,11,3

5,30 ±≈±=±=p

mmx

8NáDG�UyZQD��OLQLRZ\FK�MHGQR]QDF]QLH�RNUH�ORQ\FK 10


û:,&=(1,(�QU�� Temat: 8NáDG�UyZQD��OLQLRZ\FK�MHGQR]QDF]QLH�RNUH�ORQ\FK

Zadanie�� 'DQH� V�� ZVSyáU]GQH� x, y punktów A,B,C,D.� 2EOLF]\ü� ]D� SRPRF�� UDFKXQNX�macieU]RZHJR�ZVSyáU]GQH�SXQNWX�SU]HFLFLD�VL�SURVW\FK�AC i BD.

Dane:

Nr punktu x y .

A 102,20 102,20

B 703,30 98,90

C 698,90 896,70

D 98,90 903,30

52=:,�=$1,(

1) Równania prostych Wzory: ax + by +c = 0; a = y1 – y2; b = x2 – x1; c = x1(y2 – y1) – y1(x2 – x1) Prosta AC: –794,50.x + 596,70.y + 20215,16 = 0 Prosta BD: –804,40.x – 604,40.y + 625509,68 = 0 ��8NáDG�UyZQD��OLQLRZ\FK�MHGQR]QDF]QLH�RNUH�Oonych

3) =DSLV�XNáDGX�UyZQD��Z�SRVWDFL�PDFLHU]RZHM

A . X = L

=

=

−−

−=

625509,68-

20215.16-

4060440804

7059650794LXA ;

y

x;

,,

,,

P

y

x

D

C B

A

=+⋅−⋅−=+⋅+⋅−

0686255094060440804

016202157059650794

,y,x,

,y,x,

8NáDG�UyZQD��OLQLRZ\FK�MHGQR]QDF]QLH�RNUH�ORQ\FK 11

4) Utworzenie macierzy blokowej B

[ ]

−−

−==

68625509

1620215

4060440804

7059650794

,

,

,,

,,

�� L-AB

5) 5R]NáDG�SURVWRN�WQHM�PDFLHU]\�B na czynniki trapezowe

[ ]L'-GHG'HB TT �⋅=⋅=

−−

⋅

=

−−

−'

'

L

LG

HH

H

,

,

,,

,,

2

112

2221

11

10

10

68625509

1620215

4060440804

7059650794

��

��

−−−⋅

−−−=

641500

44425

10

7510401

535120840804

050794

,

,,

,,

,

��

B

6) 2EOLF]HQLH�ZDUWR�FL�QLHZLDGRP\FK

G . X = L’

=

⋅

−

641500

44425

10

7510401

,

,

y

x,

=

500,641

401,444X

7) :VSyáU]GQH�SXQNWX�SU]HFLFLD�SURVW\FK

xP = 401,444; yP = 500,641

8) Kontrola

≈=+⋅−⋅−≈=+⋅+⋅−

07060686255094060440804

0387016202157059650794

,,,,

,,,,

��

8NáDG�UyZQD��OLQLRZ\FK�nadRNUH�ORQ\FK 12


û:,&=(1,(�QU�� Temat: 8NáDG�UyZQD��OLQLRZ\FK�QDGRNUH�ORQ\FK

Zadanie��'DQH�V��ZVSyáU]GQH�x, y punktów A,B,C,D,E,F.�2EOLF]\ü�]D�SRPRF��UDFKXn-

NX�PDFLHU]RZHJR�ZVSyáU]GQH�SXQNWX�QDMNRU]\VWQLHMV]HJR�GOD�SU]HFLFLD�VL�WU]HFK�SUo-

stych: AB, CD oraz EF.

Dane:

Nr punktu x y .

A 202,64 203,08

B 802,64 1403,08

C 802,86 202,64

D 202,64 1402,42

E 102,86 702,64

F 1302,42 1102,86

R2=:,�=$1,(

1) Równania prostych Wzory: ax + by +c = 0; a = y1 – y2; b = x2 – x1; c = x1(y2 – y1) – y1(x2 – x1)

Prosta AB: –1200,00.x + 600,00.y + 121320,00 = 0

Prosta CD: –1199,78.x – 600,22.y + 1084883,95 = 0

Prosta EF: –400,22.x + 1199,56.y – 801692,21 = 0 ��8NáDG�UyZQD��OLQLRZ\FK�nadRNUH�ORQ\FK

=−⋅+⋅=+⋅⋅=+⋅+⋅

0 801692,21 1199,56 400,22-

0 1084883,95 600,22 - 1199,78-

0 121320,00 600,00 1200,00-

\[\[yx

F

E y

x

D

C B

A

8NáDG�UyZQD��OLQLRZ\FK�QDGRNUH�ORQ\FK 13


A . X = L

−−

=

=

−−−

−=

��

LXA ;y

x;

,,

,,

,,

56119922400

22600781199

00600001200

4) Macierz normalna AT.A

−

−=

=

−−−

−⋅

−

−−−=⋅

2422159208952479955

9524799550973039648

56119922400

22600781199

00600001200

5611992260000600

22400781199001200

,,

,,

,,

,,

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,.,

,,,AAT

��5R]NáDG�PDFLHU]\�Nwadratowej symetrycznej AT.A�QD�F]\QQLNL�WUyMN�WQH

RRAA ⋅=⋅ TT

⋅

=

−

−

22

1211

2212

11

0

0

2422159208952479955

9524799550973039648

R

RR

RR

R

,,

,,

−=

40714430

2902754591743

,

,,R

��2GZURWQR�ü�PDFLHU]\�WUyMN�WQHM��R-1

IRR =⋅− 1

=

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10

01

40714430

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0 22

1211

,

,,

R

RR'

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⋅⋅⋅

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441

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,

,,R

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( )

⋅⋅⋅⋅

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−−

−−

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77

77

44

4

4

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10799794107578800

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,,

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,

,

,,TRRAA

8NáDG�UyZQD��OLQLRZ\FK�nadRNUH�ORQ\FK 14

8) Wektor niewiadomych X

=

=

−−

⋅

−

−−−⋅

⋅⋅⋅⋅

=

=⋅⋅⋅=

−−

−−

558824

750500

801692,21

1084883,95

121320,00

5611992260000600

22400781199001200

108004107570

107570104103

)(

77

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,,

,,

LAAAX

8) �:VSyáU]GQH�Z\UyZQDQH�SXQNWX�SU]HFLFLD�VL�WU]HFK�SURVW\FK

x = 500,750; y = 824,558

8NáDG�UyZQD��OLQLRZ\FK�QLHGRRNUH�ORQ\FK 15

,PL�1Dzwisko Nr zestawu .....

û:,&=(1,(�QU�� Temat: 8NáDG�UyZQD��OLQLRZ\FK�QLHGRRNUH�ORQ\FK

Zadanie��'DQ\�MHVW�QLHGRRNUH�ORQ\�XNáDG�UyZQD��OLQLRZ\FK��6WRVXM�F�UDFKXQHN�PDFLe-U]RZ\�]QDOH(ü�QDMOHSV]H�UR]ZL�]DQLH�ZHGáXJ�PHWRG\�QDMPQLHMV]\FK�NZDGUDWyZ��

Dane:

=−⋅+⋅+=−+⋅+⋅

04233

02223

zyx

zyx

52=:,�=$1,(


A . X = L

=

=

=

4

2

2331

1223LXA ;

z

y

x

;

2) Macierz normalna A .AT

=

⋅

=⋅

53952

52534

231

32

123

2331

1223TAA

��5R]NáDG�PDFLHU]\�NZDGUDWRZHM�V\PHWU\F]QHM�A .AT nD�F]\QQLNL�WUyMN�WQH

RRAA T ⋅=⋅ T

⋅

=

22

1211

2212

11

0

0

53952

52534

R

RR

RR

R

=

107230

250210823

,

,,R

8NáDG�UyZQD��OLQLRZ\FK�niedoRNUH�ORQ\FK 16


IRR =⋅−1

=

⋅

10

01

107230

250210823

0 22

1211

,

,,

R

RR'

''

⋅⋅−⋅

=−

−−−

2

221

1032840

1042140103274

,

,,R

��2GZURWQR�ü�PDFLHU]\�QRUPDOQHM�� ( A .AT)-1

( )

⋅⋅⋅⋅

=

=

⋅⋅⋅

⋅

⋅⋅⋅

=⋅=⋅

−−

−−

−−

−

−

−−−−

33

33

22

2

2

22111-

1087311018230

101820108901

103284104210

0103274

1032840

104210103274 )(

,,

,,

,,

,

,

,,TT RRAA


≈

⋅

⋅⋅⋅⋅

⋅

=

=⋅⋅⋅=

−−

−−

1670

0270

0770

4

2

108731101820

101820108901

231

32

123

33

33

T

,

,

,

,,

,,

T LAAAX ��

��5R]ZL�]DQLH

x =0,077; y = 0,027 z = 0,167

8) Kontrola

≈=−⋅+⋅+≈=−+⋅+⋅

000104167023027030770

000802167002702077023

,,,,

,,,,

WyrówQDQLH�VWDF\MQH�PHWRG��SR�UHGQLF]�F� 17


û:,&=(1,(�QU�� Temat: :\UyZQDQLH�VWDF\MQH�PHWRG��SR�UHGQLF]�F�

Zadanie: Na stanowisku st.I ]RVWDá\�SRPLHU]RQH�]�Uy*Q��GRNáDGQR�FL��N�W\��– 6. Stosu-M�F�P�Q�N��QDOH*\�REOLF]\ü�SRSUDZNL�GR� W\FK�SRPLDUyZ�RUD]�Z\UyZQDQH�ZDUWR�FL�N�WyZ�SU]\MWych jako niewiadome x, y, z, t��D�WDN*H�SU]HSURZDG]Lü�DQDOL]�GRNáDGQR�FL�SR�Zy-równaniu.

Dane:

L1 = 31,3107g p1 = 2 L2 = 07,1832g p2 = 2

L3 = 18,1519g p3 = 2 L4 = 32,1421g p4 = 2 L5 = 88,7871g p5 = 5 L6 = 50,2933g p6 = 7 52=:,�=$1,(

1) Równania obserwacyjne 2)�:DUWR�FL�SU]\EOL*RQH�QLHZLDGRP\FK

L1 + v1 = x x0 = L1 = 31,3107g x = x0 + ∆x

L2 + v2 = y y0 = L2 = 07,1832g y = y0 + ∆y

L3 + v3 = z z0 = L3 = 18,1519g z = z0 + ∆z

L4 + v4 = t t0 = L4 = 32,1421g t = t0 + ∆t L5 + v5 = x + y + z + t L6 + v6 = z + t ��5yZQDQLD�EáGyZ ��1DGRNUH�ORQ\�XNáDG�UyZQD��OLQLRZ\FK

v1 = ∆x

v2 = ∆y

v3 = ∆z

v4 = ∆t

v5 = ∆x + ∆y + ∆z + ∆t + 8cc

v6 = ∆z + ∆t + 7cc

6

5

4

3

2

1

t

z

x y

st.I

=+∆+∆=+∆+∆+∆+∆

=∆=∆=∆=∆

07

08

0

0

0

0

tz

tzyx

t

z

y

x

:\UyZQDQLH�VWDF\MQH�PHWRG��SR�UHGQLF]�F� 18

5) Zapis macierzowy 6) Macierz wagowa P

LXA =⋅

−−

=

∆∆∆∆

⋅

7

8

0

0

0

0

1100

1111

1000

0100

0010

0001

t

z

y

x

=

700000

050000

002000

000200

000020

000002

P

7) Macierz normalna AT.P.A

=⋅⋅

141255

121455

5575

5557

APAT

8��5R]NáDG�PDFLHU]\�NZDGUDWRZHM�V\PHWU\F]QHM�AT.P.A�QD�F]\QQLNL�WUyMN�WQH

RRAPA ⋅=⋅⋅ TT

⋅

=

44

3433

242322

14131211

44342414

332313

2212

11

000

00

0

0

00

000

141255

121455

5575

5557

R

RR

RRR

RRRR

RRRR

RRR

RR

R

=

89561000

498021358300

7715077150851610

88981889818898164582

.

..

...

....

R

9��2GZURWQR�ü�PDFLHU]\�WUyMN�WQHM��R-1

IRR =⋅− 1

=

⋅

1000

0100

0010

0001

89561000

498021358300

7715077150851610

88981889818898164582

000

00

0

44

3433

242322

14131211

.

..

...

....

'R

'R'R

'R'R'R

'R'R'R'R

−−−−−−

=−

52750000

420203189000

0447013290540100

04470132903858037800

1

.

..

...

....

R

WyrówQDQLH�VWDF\MQH�PHWRG��SR�UHGQLF]�F� 19

10) OdwrotQR�ü�PDFLHU]\�QRUPDOQHM�� (AT.P. A)-1

( )

−−−−−−−−−−−−

=

=

−−−−−

−⋅

−−−−−−

=

=⋅=⋅⋅ −−

27830221700236002360

22170278300236002360

02590023603113018870

02360023601887031130

52750420200447004470

0318901329032900

005401038580

00037800

52750000

420203189000

0447013290540100

04470132903858037800

11

....

....

....

....

....

...

..

.

.

..

...

....

TRRAPA �� 7

11) Wektor niewiadomych X 12) Wyrównane niewiadome

−−−−

≅

−−−−

=

=⋅⋅⋅⋅⋅=

13

13

70

70

153

153

710

710

.

.

.

.

.

.

.

.

LPAA)P(AX T-1T

13) Wektor poprawek 14) Obserwacje wyrównane

−−−−

=

−−

−

−−−−

⋅

=

=−⋅=

cc

cc

cc

cc

cc

cc

,

,

,

,

,

,

.

.

.

.

80

40

13

13

70

70

7

8

0

0

0

0

13

13

70

70

1100

1111

1000

0100

0010

0001

LXAV

15) Kontrola ogólna

545545

5617663545

..

..

=−=

⋅⋅⋅−⋅⋅=⋅⋅ XAPLLPLVPV TTT

gccg

gccg

gccg

g

,.ttt

,.zzz

,.yyy

,xxx

32.14179

18.15159

7.18313

31.31063

=−=∆+==−=∆+==−=∆+=

=−=∆+=

13142132

13151918

7018327

7031.3107

0

0

0

ccg0

gccg

gccg

gccg

gccg

gccg

g

,.vL

,.vL

,.vL

,.vL

,.vL

,vL

50.29338

88.78714

32.14179

18.15159

7.18313

31.31063

=+=+

=+=+

=−=+=−=+

=−=+=−=+

80293350

40787188

13142132

13151918

7018327

7031.3107

66

55

44

33

22

ccg11

:\UyZQDQLH�VWDF\MQH�PHWRG��SR�UHGQLF]�F� 20

16) Kontrola generalna /HZD�VWURQD�UyZQD��REVHUZDF\MQ\FK 3UDZD�VWURQD�UyZQD��REVHUZDF\MQ\Fh

17��2FHQD�GRNáDGQR�FL

a) Eá�G��UHGQL�REVHUZDFML

CC0 .

knm 84

2

45.5±≈±=

−⋅

±=VPV T

b) EáG\��UHGQLH�QLHZLDGRP\FK��ZLHONR�FL�Z\UyZQDQ\FK�

Macierz kowariancyjna

2EOLF]HQLH�EáGyZ��UHGQLFK

g

g

g

g

g

g

vL

vL

vL

vL

vL

vL

50.29338

8.78714

32.14179

18.15159

7.18313

31.31063

=+

=+

=+=+=+

=+

66

55

44

33

22

11

8g

g

g

g

g

g

tz

tzyx

t

z

y

x

50.29338

88.78714

32.14179

18.15159

7.18313

31.31063

=+

=+++

====

cc

ttt

cc

zzz

cc

yyy

ccxxx

.Qmm

.Qmm

.Qmm

.Qmm

52

52

72

72

0

0

0

0

±=⋅±=

±=⋅±=

±=⋅±=

±=⋅±=

−−−−−−−−−−−−

=⋅⋅=

27830221700236002360

22170278300236002360

02590023603113018870

02360023601887031130

....

....

....

....

�� 7 APAQ

:\UyZQDQLH�VWDF\MQH�PHWRG��zawarunkowan� 21


û:,&=(1,(�QU�� Temat: :\UyZQDQLH�VWDF\MQH�PHWRG��]DZDUXQNRZDQ�

Zadanie: Na stanowisku st.I ]RVWDá\�SRPLHU]RQH�]�Uy*Q��GRNáDGQR�FL��N�W\��– 6. Prze-prowaG]Lü�Z\UyZQDQLH�ZHGáXJ�PHWRG\�ZDUXQNRZHM�

Dane:

L1 = 31,3107g p1 = 2 L2 = 07,1832g p2 = 2

L3 = 18,1519g p3 = 2 L4 = 32,1421g p4 = 2 L5 = 88,7871g p5 = 5 L6 = 50,2933g p6 = 7

52=:,�=$1,(

1) Liczba warunków w sieci niwelacyjnej

r = n – k = 2

n = 6 – liczba wszystkich obserwacji k = 4 –�OLF]ED�REVHUZDFML�QLH]EGQ\FK 2) Równania warunkowe (L1 + v1) + (L2 + v2) + (L3 + v3) + (L4 + v4) - (L5 + v5) = 0

(L3 + v3) + (L4 + v4) - (L6 + v6) = 0

��5yZQDQLD�EáGyZ��QLHGRRNUH�ORQ\�XNáDG�UyZQD��OLQLRZ\FK�

v1+ v2 + v3 + v4 – v5 + ω1 = 0 ω1 = L1 + L2 + L3 + L4 – L5 = 8cc

v3 + v4 – v6 + ω2 = 0 ω1 = L3 + L4 – L6 = 7cc

6

5

4

3

2

1

st.I


4) Zapis macierzowy ��2GZURWQR�ü�PDFLHU]\�Zagowej WVA =⋅

6) Macierz normalna A.P-1.AT

=⋅⋅ −

1411

1221

.

.TAPA

��5R]NáDG�PDFLHU]\�NZDGUDWRZHM�V\metrycznej A.P-1.AT�QD�F]\QQLNL�WUyMN�WQH

RRAPA ⋅=⋅⋅ − TT1

⋅

=

22

1211

2212

11

0

0

1411

122

R

RR

RR

R

.

.

=

8300

670481

.

..R


IRR =⋅− 1

=

⋅

10

01

8300

670481

0 22

1211

.

..

R

RR'

''

−

=−

211550

06701

..

.R

��2GZURWQR�ü�PDFLHU]\ normalnej: (A.P-1. AT)-1

( )

−

−=

⋅

=

=⋅=⋅⋅ −−−

451660

660750

830670

0481

8300

670481

111

..

..

..

.

.

..

TRRAPA �� 7

=−

14000000

0200000

0050000

0005000

0000500

0000050

1

.

.

.

.

.

.

P

−−

=

⋅

−

−7

8

101100

011111

6

5

4

3

2

1

v

v

v

v

v

v


10) Wektor korelat K

−−

=⋅⋅⋅= −

894

4211

.

.W)AP(AK 1-T

11) Wektor poprawek V

−−−−

=⋅⋅=⋅⋅⋅⋅⋅= −−−

70

30

23

23

70

70

111

.

.

.

.

.

.

KAPW)AP(AAPV T1-TT

12) Obserwacje wyrównane 13) Kontrola ogólna

14) Kontrola generalna 31.3106g + 7.1831g + 18.1516g + 32.1418g – 88.7871g = 0.0000g

18.1516g + 32.1418g – 50.2934g = 0.0000g

1��%á�G��UHGQL�REVHUZDFML

cc

0 .kn

m 842

45.5±≈±=

−⋅

±=VPV T

gccg

gccg

gccg

gccg

gccg

g

.vL

.vL

.vL

.vL

.vL

vL

50.2934

88.7871

32.1418

18.1516

7.1831

31.3106

=+=+

=−=+

=−=+=−=+

=−=+=−=+

1293350

0787188

3142132

3151918

118327

66

55

44

33

22

11FFJ ��

545545 .. =⋅=⋅⋅ KWVPV TT

:\UyZQDQLH�VLHFL�QLZHODF\MQHM�PHWRG��SR�UHGQLF]�F� 24


û:,&ZENIE nr .....

Temat: :\UyZQDQLH�VLHFL�QLZHODF\MQHM�PHWRG��SR�UHGQLF]�F�

Zadanie��:�Z\QLNX�SRPLDUX�QLZHODF\MQHJR�FL�JyZ�R�GáXJR�FL�d1, d2, d3, d4, d5 uzyskano Uy*QLFH�Z\VRNR�FL�∆h1, ∆h2, ∆h3, ∆h4, ∆h5. 6WRVXM�F�PHWRG�SR�UHGQLF]�F��QDOH*\�Zy-UyZQDü�SRPLDU\�RUD]�Z\VRNR�FL�SXQNWyZ�x i y,�D�WDN*H�SU]HSURZDG]Lü�DQDOL]�GRNáDGQo-�FL�SR�Z\UyZQDQLX�

Dane:

52=:,�=$1,(

1) Równania obserwacyjne ��:DUWR�FL�SU]\EOL*RQH�QLHZLDGRP\FK

∆h1 + v1 = R1 – x x0 = R1 – ∆h1 = 104,554 x = x0 + ∆x

∆h2 + v2 = R1 – y y0 = R1 – ∆h2 = 104,002 y = y0 + ∆y

∆h3 + v3 = x – y

∆h4 + v4 = R3 – x

∆h5 + v5 = R2 – y 3) RównanLD�EáGyZ ��1DGRNUH�ORQ\�XNáDG�UyZQD��OLQLRZ\FK

v1 = - ∆x

v2 = - ∆y

v3 = ∆x – ∆y – 6mm

v4 = - ∆x

v5 = - ∆y – 5mm

Nr 5y*QLFH�Z\VRNR�FL�

[m]

'áXJR�FL�FL�JyZ

[km]

:\VRNR�FL�reperów

[m]

1

2

3

4

5

2,722

3,274

0,558

2,277

1,226

1,0

1,0

1,0

0,5

0,5

107,276

105,223

106,831

R1

R2 R3

x

∆h1 ∆h2

∆h3 y

∆h5 ∆h4

=−∆−=∆−

=−∆−∆=∆−=∆−

05

0

06

0

0

y

x

yx

y

x


5) Zapis macierzowy 6) Macierz wagowa P

LXA =⋅ waga 1

i

i

pd

=

− − ∆

⋅ =− ∆ − −

1 0 0

0 1 0

1 1 6

1 0 0

0 1 5

x

y

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 2 0

0 0 0 0 2

=

P

7) 8NáDG�UyZQD��QRUPDOQ\FK: ( ) LPAXAPA TT ⋅⋅=⋅⋅⋅

−

=

⋅

−

−16

6

41

14

y

x

8��5R]NáDG�PDFLHU]\�NZDGUDWRZHM�V\PHWU\F]QHM�AT.P.A�QD�F]\QQLNL�WUyMN�WQH

RRAPA ⋅=⋅⋅ TT

→

9��2GZURWQR�ü�PDFLHU]\�WUyMN�WQHM��R-1

IRR =⋅− 1

→

10��2GZURWQR�ü�PDFLHU]\�QRUPDOQHM�� (AT.P. A)-1

( )− − ⋅ ⋅ = ⋅ = ⋅ =

T -1 1 1 0,5 0,13 0,5 0 0,27 0,067

( ) 0 0,52 0,13 0,52 0,067 0,27

T

A P A R R

11) Wektor niewiadomych X 12) Wyrównane niewiadome

( )

−

=

=⋅⋅=⋅⋅⋅

mm,

mm,

93

50

LPAXAPA TT

− =

1 0,5 0,13

0 0,52R

= + ∆ = + == + ∆ = − =

0

0

104,554 1

104,002 4

x x x m mm

y y y m mm

104,555

103,998

− ⋅ =

11 12

22

' ' 2 0,5 1 0

0 ' 0 1,94 0 1

R R

R

− = ⋅ −

11 11 12

12 22 22

04 1

01 4

R R R

R R R

− =

2 0,5

0 1,94R


13) Wektor poprawek 14) Obserwacje wyrównane

= ⋅ − =

− − − −

= ⋅ − = ≅ − − − − − − − − − −

1 0 0 0,53 1

0 1 0 3,87 40,53

1 1 6 1,60 23,87

1 0 0 0,53 1

0 1 5 1,13 1

mm

mm

mm

mm

mm

V A X L


⋅ ⋅ = ⋅ ⋅ − ⋅ ⋅ ⋅= −

=20,9 86 65,1

20,9 20,9

T T TV P V L P L L P A X

16) Kontrola generalna /HZD�VWURQD�UyZQD��REVHUZDF\MQ\FK 3UDZD�VWURQD�UyZQD��REVHUZDF\MQ\FK



kmmm,

knm0 62

3

20,9±≈±=

−⋅

±=VPV T

b) EáG\��UHGQLH�QLHZLDGRP\FK��ZLHONR�FL�Z\UyZQDQ\FK�



∆ −∆ +∆ −∆ −∆ −

2,722 1

4

2

2,277 1

1,226 1

1 1

2 2

3 3

4 4

5 5

h + v = mm =

h + v = 3,274 mm =

h + v = 0,558 mm =

h + v = mm =

h + v = mm =

2,721

3,278

0,556

2,276

1,225

∆∆∆∆∆

1 1

2 2

3 3

4 4

5 5

h + v =

h + v =

h + v =

h + v =

h + v =

2,721

3,278

0,556

2,276

1,225

− = − =− = − =

− = − =− = − =− = − =

1

1

3

2

107,276 104,555

107,276 103,998

104,555 103,998

106,831 104,555

105,223 103,998

R x

R y

x y

R x

R y

2,721

3,278

0,557

2,276

1,225

= ⋅ ⋅ =

0,27 0,067

0,067 0,27

T -1( ) Q A P A

mm.Qmm

mm,Qmm

yyy

xxx

41

41

0

0

±=⋅±=

±=⋅±=

:\UyZQDQLH�VLHFL�QLZHODF\MQHM�PHWRG��zawarunkowan� 27


û:,&=(1,(�QU�� Temat: :\UyZQDQLH�VLHFL�QLZHODF\MQHM�PHWRG��]DZDUXQNRZDQ� Zadanie��:�Z\QLNX�SRPLDUX�QLZHODF\MQHJR�FL�JyZ�R�GáXJR�FL�d1, d2, d3, d4, d5 uzyskano Uy*QiFH�Z\VRNR�FL�∆h1, ∆h2, ∆h3, ∆h4, ∆h5. 3U]HSURZDG]Lü�Z\UyZQDQLH�VLHFL�QLZHODF\MQHM�ZHGáXJ�PHWRG\�ZDUXQNRZHM�

Dane:

52=:,�=$1,(

1) Liczba warunków w sieci niwelacyjnej

a) ogólna: w = n – p + pr = 5 – 5 + 3 = 3 b) PLG]\�UHSHUDPL��Zr = pr -1 = 3 – 1 = 2 c) dla oczek siatki: wo = 1

2) Równania warunkowe

Dla reperów: R1 – (∆h1 + v1) + ∆h4 + v4 – R3 = 0

R1 – (∆h2 + v2) + ∆h5 + v5 – R2 = 0

Dla oczka: ∆h2 + v2 – (∆h3 + v3) – (∆h1 + v1) = 0 ��5yZQDQLD�RGFK\áHN��QLHGRRNUH�ORQ\�XNáDG�UyZQD��OLQLRZ\FK�

– v1 + v4 + ω1 = 0 gdzie: ω1 = R1 – ∆h1 + ∆h4 – R3 = 0

– v2 + v5 + ω2 = 0 ω2 = R1 – ∆h2 + ∆h5 – R2 = 5 mm

– v1 + v2 – v3 + ω3 = 0 ω3 = – ∆h1 + ∆h2 – ∆h3 = - 6 mm

Nr 5y*QLFH�Z\VRNR�FL�

[m]

'áXJR�FL�FL�JyZ

[km]

:\VRNR�FL�reperów

[m]

1

2

3

4

5

2,722

3,274

0,558

2,277

1,226

1,0

1,0

1,0

0,5

0,5

107,276

105,223

106,831

R1

R2 R3

x

∆h1 ∆h2

∆h3 y

∆h5 ∆h4


4) Zapis macierzowy 5) Macierz wagowa��RGZURWQR�ü�

WVA =⋅ waga 1

i

i

pd

= ; ii dp =−1

−=

⋅

−−−

−

6

5

0

00111

10010

01001

5

4

3

2

1

v

v

v

v

v

=−

500000

050000

00100

00010

00001

1

.

.

P

��8NáDG�UyZQD��QRUPDOQ\FK�NRUHODW� ( ) WKAPA T =⋅⋅⋅ −1

−=

⋅

−−

6

5

0

311

1510

1051

3

2

1

k

k

k

.

.

��5R]NáDG�PDFLHU]\�QRUPDOQHM�A.P-1.AT na czynniki trójk�WQH

RRAPA ⋅=⋅⋅ − TT1

⋅

=

−−

33

2322

131211

332313

2212

11

00

00

00

311

1510

1051

R

RR

RRR

RRR

RR

R

.

.

−=

1.29100

0.8161.2250

0.81601.225

R


IRR =⋅− 1

=

−⋅

100

010

001

1.29100

0.8161.2250

0.81601.225

00

0

33

2322

131211

'R

'R'R

'R'R'R

−=−

775000

516081600

5160081601

.

..

..

R


��2GZURWQR�ü�PDFLHU]\�QRUPDOQHM�� (A.P-1. AT)-1

( )

−−

−−=

−⋅

−=

=⋅=⋅⋅ −−−

604040

4093302670

4026709330

775051605160

081600

008160

775000

516081600

516008160

)( 111-T1

...

...

...

...

.

.

.

..

..

TRRAPA


−−

=⋅⋅⋅= −

601

272

0711

.

.

.

W)AP(AK 1-T


−−−

−

≅

−−−

−

=⋅⋅=⋅⋅⋅⋅⋅= −−−

mm

mm

mm

mm

mm

.

.

.

.

.

1

1

2

4

1

131

530

601

873

530

111 KAPW)AP(AAPV T1-TT

12) Obserwacje wyrównane


920920 .. =

⋅=⋅⋅ KWVPV TT

14) Kontrola generalna 107.276 – 2.721 + 2.276 – 106.831 = 0.000

107.276 – 3.278 + 1.225 – 105.223 = 0.000

3.278 – 0.556 – 2.721 = 0.001


kmmm.

knm0 62

3

20.9±≈±=

−⋅

±=VPV T

∆ −∆ +∆ −∆ −∆ −

2,722 1

4

2

2,277 1

1,226 1

1 1

2 2

3 3

4 4

5 5

h + v = mm =

h + v = 3,274 mm =

h + v = 0,558 mm =

h + v = mm =

h + v = mm =

2,721

3,278

0,556

2,276

1,225

Rozwijanie funkcji nieliniowych w szereg Taylora 30


û:,&=(1,(�QU�� Temat: Rozwijanie funkcji nieliniowej w szereg Taylora Zadanie��5R]ZLQ�ü�Z�V]HUHJ�SRGDQH�IXQNFMH�QLHOLQLRZH�]�SRPLQLFLHP�wyrazów o pot�G]H�Z\*V]HM�QL*��2EOLF]\ü�ZDUWR�ü�IXQNFML�f�RUD]�ZDUWR�FL�MHM�UR]ZLQLFLD�OLQLRZHJR�f’ dla

GDQ\FK�ZDUWR�FL�]PLHQQ\FK��x0, y0) i ich przyrostów (∆x, ∆y��2EOLF]\ü�Uy*QiF�f – f’. Funkcja 1. Dane:

yxf 221 −=

( )��\��[��\��[ �� =∆=∆==

5R]ZLQLFLH�IXQNFML� ( ) y

y

fx

x

fy,xff ' ∆⋅

∂∂

+∆⋅∂∂

+= 110011

yxxyxf ' ∆⋅−∆⋅+−= 222 00

2

01

Funkcja w postaci liniowej f1’ dla ��\��[ �� == :

802201 +∆⋅−∆⋅= yxf '

:DUWR�ü�IXQNFML�f1’ dla ��\��[ =∆=∆ :

60898020250201 ...f ' =+⋅−⋅=

:DUWR�ü�IXQNcji f1 dla 210510 .. =∆+==∆+= \��\\��[��[[ ��

85892102510 21 ...f =⋅−=

5y*QLFD� 25011 .ff ' =−

Funkcja 2. Dane:

ycosxsinf −=2

( )JJJ�

J� ��\��[��\��[ =∆=∆==

5R]ZLQLFLH�IXQNFML� ( ) y:

y

fx:

x

fy,xff gg' ∆⋅

∂∂

+∆⋅

∂∂

+= ρρ 220022

yysin

xxcos

ycosxsinfgg

' ∆⋅+∆⋅+−=ρρ

00002

Rozwijanie funkcji nieliniowej w szereg Taylora 31

Funkcja w postaci liniowej f2’ dla ( )πρ :, gg 200=== J�

J� ��\��[ :

8312540002457001551502 .y.x.f ' −∆⋅+∆⋅=

:DUWR�ü�IXQNFML�f2’ dla JJ ��\��[ =∆=∆ :

829211083125402000245701001551502 ......f ' −=−⋅+⋅=

:DUWR�ü�IXQNFML�f1 dla gg .. 200010100010 =∆+==∆+= \��\\��[��[[ ��

( ) ( ) 82920602101102 ..cos.sinf gg −=−=

5y*QLFD� 622 1050000050 −⋅==− .ff '

Funkcja 3. Dane:

x

yarctgf =3

( )��\��[��\��[ �� =∆=∆==

5R]ZLQLFLH�IXQNFML� ( ) y

y

fx

x

fy,xff gg' ∆⋅

⋅

∂∂

+∆⋅

⋅∂∂

+= ρρ 330033

yyx

xx

yx

y

x

yarctgf gg' ∆⋅

⋅

++∆⋅

⋅

+−= ρρ

20

20

0

20

20

0

0

03

Funkcja w postaci liniowej f3’ dla ( )πρ :, gg 200=== ��\��[ �� :

ggg' .y.x.f 00005018313183133 +∆⋅+∆⋅−=

:DUWR�ü�IXQNFML�f3’ dla ��\��[ =∆=∆ :

gggg' ......f 318350000050201831310183133 =+⋅+⋅−=

:DUWR�ü�IXQNFML�f3 dla 210110 .. =∆+==∆+= \��\\��[��[[ ��

g..

.arctgf 313650

110

2103 ==

5y*QLFD� cc'ff 4733 −=−

:\UyZQDQLH�ZFLFLD�ZVWHF]��N�WRZHJR��PHWRG��SR�UHdQLF]�F� 32


û:,&=(1,(�QU�� Temat: :\UyZQDQLH�ZFLFLD�ZVWHF]��N�WRZHJR��PHWRG� SR�UHdQLF]�F� Zadanie: Na stanowisku P SRPLHU]RQR�]� MHGQDNRZ��GRNáDGQR�FL��N�W\��–��6WRVXM�F�PHWRG� SR�UHGQLF]�F��QDOH*\�REOLF]\ü�Z\UyZQDQH�ZDUWR�FL�SRPLDUyZ�RUD]�ZVSyáU]d-nych x, y punktu P,�D�WDN*H�SU]HSURZaG]Lü�DQDOL]�GRNáDGQR�FL�SR�Z\UyZQaniu.

Dane:

3RPLHU]RQH�N�W\ .�W o / //

1 2 3

62 48 72

39 35 28

137 430 289

:VSyáU]GQH�SXQNWyZ�VWDá\FK Pkt x y 1 2 3 4

7108.75 5953.34 5282.99 5453.62

-3034.50 -2142.62 -3148.85 -4528.10

2EOLF]RQH�ZVSyáU]GQH�SU]\EOL*RQH�punktu P xPo = 6307.48 yPo = -3708.87 52=:,�=$1,(

1) Równania obserwacyjne

P3

P3

P4

P4P34P33

P2

P2

P3

P3P23P22

P1

P1

P2

P2P12P11

xx

yyarctg

xx

yyarctgAAv

xx

yyarctg

xx

yyarctgAAv

xx

yyarctg

xx

yyarctgAAv

−−

−−−

=−=+

−−

−−−

=−=+

−−

−−−

=−=+

β

β

β

��5R]ZLQLFLH�Z�V]HUHJ�7D\ORUD

P

P

P3

P

P4P

P

P3

P

P4P34P33

P

P

P2

P

P3P

P

P2

P

P3P23P22

P

P

P1

P

P2P

P

P1

P

P2P12P11

yy

A

y

Ax

x

A

x

AAAv

yy

A

y

Ax

x

A

x

AAAv

yy

A

y

Ax

x

A

x

AAAv

oo

oo

oo

∆⋅

∂∂

−∂∂

+∆⋅

∂∂

−∂∂

+−=+

∆⋅

∂∂

−∂∂

+∆⋅

∂∂

−∂∂

+−=+

∆⋅

∂∂

−∂∂

+∆⋅

∂∂

−∂∂

+−=+

β

β

β

P (xP=?, yP=?)

3

2

1

1

2

3

4


3) :VSyáF]\QQLNL�NLHUXQNRZH (poFKRGQH�F]�VWNRZH�

),,,i(

;d

xx

y

Ab;

d

yy

x

Aa

iP

PiiP

Pi

iP

PiiP

iP

4321

5206262

P2

P

=

′′=′′′′⋅−

−=′′⋅∂

∂=′′⋅

−=′′⋅

∂

∂= ρρρρρ

Przyrosty :VSyáF]��NLHUXQNRZH Kierunek

P-i 0Pi xx −

0Pi yy −

.ZDGUDW�GáXJR�FL 2

0iPd iPa iPb

P-1 P-2 P-3 P-4

801.27 –354.14

–1024.49 –853.86

674.37 1566.25 560.02

–819.23

1096808.51 2578554.20 1363202.16 1400214.69

126.822 125.288

84.736 –120.680

–150.686 28.329

155.015 125.782

4) :VSyáF]\QQLNL�SU]\�QLHZLDGRP\FK�Z�UyZQDQLDFK�EáGyZ a1 = aP2 – aP1 = -1.534 b1 = bP2 – bP1 = 179.015

a2 = aP3 – aP2 = -40.552 b2 = bP3 – bP2 = 126.686

a3 = aP4 – aP3 = -205.416 b3 = bP4 – bP3 = -29.233

��:\UD]\�ZROQH�UyZQD��EáGyZ $]\PXW\�REOLF]RQH�]H�ZVSyáU]GQ\FK�przybli*RQych

25184223180251844386853

23819

115021511809449328491024

02560

62644102180433517714354

251566

5055040505504027801

37674

P4oP4o

P4

P4

P4o

P3oP3o

P3

P3

P3o

P2oP2o

P2

P2

P2o

P1oP1

P1

P1

P1o

0

0

0

0

0

0

0

0

′′′=+=′′′=−−

=−

−=

′′′=−=′′′=−

=−

−=

′′′=−=′′′=−

=−

−=

′′′==′′′==−

−=

.A;..

.arctg

xx

yyarctg

.A;..

.arctg

xx

yyarctg

.A;..

.arctg

xx

yyarctg

.A;..

.arctg

xx

yyarctg

o

$$$

$$$

$$$

$$

ϕϕ

ϕϕ

ϕϕ

ϕϕ

Wyrazy wolne

//

oo

//

oo

//

oo

.AAl

.AAl

.AAl

27

55

47

3P34P3

2P23P2

1P12P1

=−−=

=−−=

=−−=

β

β

β

6) RyZQDQLD�EáGyZ 7��1DGRNUH�ORQ\�XNáDG�UyZQD��OLQLowych

v1 = -1.534.∆xP + 179.015.∆yP + 7.4

v2 = -40.552.∆xP + 126.686.∆yP + 5.5

v3 = -205.416.∆xP – 29.233.∆yP + 7.2

-1.534.∆xP + 179.015.∆yP + 7.4 = 0

-40.552.∆xP + 126.686.∆yP + 5.5 = 0

-205.416.∆xP – 29.233.∆yP + 7.2 = 0


8) Zapis macierzowy LXA =⋅

−−−

=

∆∆

⋅

−−−

−

27

55

47

23329416205

68612655240

0151795341

.

.

.

y

x

..

..

..

P

P

��8NáDG�UyZQD��QRUPDOQ\FK� AT.A.X = AT.L

−

=

∆∆

⋅

0061811

3831713

28148950946592

94659255143842

.

.

y

x

..

..

P

P

��5R]NáDG�PDFLHU]\�QRUPDOQHM�AT.A�QD�F]\QQLNL�WUyMN�WQH

RRAA ⋅=⋅ TT

⋅

=

22

1211

2212

11

0

0

28148950946592

94659255143842

R

RR

RR

R

..

..;

=

2292210

8322386209

.

..R


IRR =⋅− 1

=

⋅

10

01

2292210

8322386209

0 22

1211

.

..

R

RR'

''

;

⋅−=

−−

00452000

1011360047760 51

.

..R

��2GZURWQR�ü�PDFLHU]\�QRUPDOQHM�� (AT.A)-1

( ) =⋅=⋅ −− T11 RRAA �� 7

⋅⋅−⋅−⋅

=

⋅−

⋅

⋅−=

−−

−−

−

−

57

75

5

5

100432107632

107632102812

0045200101136

00047760

00452000

1011360047760

..

..

..

.

.

..


−

≅

−

⋅

⋅⋅−⋅−⋅

=⋅⋅⋅=−−

−−

0370

0400

0061811

3831713

100432107632

10763210281257

75

.

.

.

.

..

..)( LAA)(AX T1-T

14) Wyrównane niewiadome��ZVSyáU]dne punktu P)

907370803708703708

52063070404806307

0

0

...yyy

...xxx

PPP

PPP

−=−−=∆+==+=∆+= �


15) Wektor poprawek LXAV −⋅=

′′′′−

′′

=

−−−

−

−

⋅

−−−

−=

20

90

60

27

55

47

0370

0400

23329416205

68612655240

051795341

.

.

.

.

.

.

.

.

..

..

...

V



151151

7013585136151

..

...

=−=

⋅⋅−⋅=⋅ XALLLVV TTT

18) Kontrola generalna $]\PXW\�REOLF]RQH�]H�ZVSyáU]GQ\FK�Zyrównanych

44184223180441844390853

19819

312021511807479328531024

06560

53044102180529517718354

291566

6165040616504023801

41674

P4P4

P4

P4P4

P3P3

P3

P3P3

P2P2

P2

P2P2

P1P1

P1

P1P1

′′′=+=′′′=−−

=−−

=

′′′=−=′′′=−

=−−

=

′′′=−=′′′=−

=−−

=

′′′==′′′==−−

=

.A;..

.arctg

xx

yyarctg

.A;..

.arctg

xx

yyarctg

.A;..

.arctg

xx

yyarctg

.A;..

.arctg

xx

yyarctg

$$$

$$$

$$$

$$

ϕϕ

ϕϕ

ϕϕ

ϕϕ

/HZD�VWURQD�UyZQD� 3UDZD�VWURQD�UyZQD��REVHUZDF\MQ\FK obserwacyjnych

129827229288272

142534890435348

3149326713932

33

22

11

′′′=′′+′′′=+

′′′=′′−′′′=+

′′′=′′+′′′=+

...v

...v

...v

$$

$$

$$

��

��

βββ

1298272

1425348

314932

33

22

11

′′′=+

′′′=+

′′′=+

.v

.v

.v

$

$

$

βββ �

12982723120215144184223

84153485304410231202151

9139362616504053044102

P34P

P23P

P12P

′′′=′′′−′′′=−

′′′=′′′−′′′=−

′′′=′′′−′′′=−

...AA

...AA

...AA

$$$

$$$

$$$


19��2FHQD�GRNáDdQR�FL


111

1.15 ′′±≈±=−⋅

±= .kn

mVV T

b) báG\��UHGQLH�QLHZLDGRP\FK��ZVSyáU]GQ\FK�Z\UyZQDQ\FK�


⋅⋅−⋅−⋅=⋅= −−

−−

57

75

100432107632

107632102812

..

..�� 7 AAQ


m.Qmm

m.Qmm

yyy

xxx

0050

0050

±=⋅±=

±=⋅±=

:\UyZQDQLH�ZFLFLD�ZVWHF]��NierunkRZHJR��PHWRG��SR�UHdQLF]�F� 37

�,PL�1D]ZLVNR Nr zestawu .....

û:,&=(1,(�QU�� Temat: :\UyZQDQLH�ZFLFLD�ZVWHF] (kierunkowego��PHWRG� SR�UHGQi-

F]�F� Zadanie: Na stanowisku P SRPLHU]RQR�]�MHGQDNRZ��GRNáDGQR�FL��NLHUXQNL��– 4. Stosu-M�F�PHWRG� SR�UHGQLF]�F�� QDOH*\� REOLF]\ü�Z\UyZQDQH�ZDUWR�FL� SRPLDUyZ� RUD]�ZVSyá�U]GQ\FK�x, y punktu P,�D�WDN*H�SU]HSURZDG]Lü�DQDOL]�GRNáDGQR�FL�SR�Z\Uywnaniu.

Dane:

Pomierzone kierunki Nr o / //

1 2 3 4

94 263 325 26

00 31 18 11

132 082 492

117


3301.17 397.93 858.74

2051.78

-4408.82 -3582.04 -4936.63 -5347.76

2EOLF]RQH�ZVSyáU]GQH�przyblL*RQH�punktu P xPo = 1803.30 yPo = -4126.20 52=:,�=$1,(


zxx

yyarctgzAvK

zxx

yyarctgzAvK

zxx

yyarctgzAvK

zxx

yyarctgzAvK

−−−

=−=+

−−−

=−=+

−−−

=−=+

−−−

=−=+

P4

P44P44

P3

P33P33

P2

P22P22

P1

P11P11

P (xP=?, yP=?)

3

2

1

1

2

3

4

4

:\UyZQDQLH�ZFLFLD�ZVWHF]��NierunkoweJR��PHWRG��SR�UHdQLF]�F� 38


zyy

Ax

x

AzAvK

zyy

Ax

x

AzAvK

zyy

Ax

x

AzAvK

zyy

Ax

x

AzAvK

oo

oo

oo

oo

∆−∆⋅

∂∂

+∆⋅

∂∂

+−=+

∆−∆⋅

∂∂

+∆⋅

∂∂

+−=+

∆−∆⋅

∂∂

+∆⋅

∂∂

+−=+

∆−∆⋅

∂∂

+∆⋅

∂∂

+−=+

P

P

P4P

P

P44P44

P

P

P3P

P

P33P33

P

P

P2P

P

P22P22

P

P

P1P

P

P11P11

3) :VSyáF]\QQLNL�NLHUXQNRZH��SRFKRGQH�F]�VWNRZH�

),,,i(;d

xx

y

Ab;

d

yy

x

Aa

iP

PiiP

i

iP

PiiP

i 43215206262

P2

P

=′′=′′′′⋅−−=′′⋅∂∂

=′′⋅−=′′⋅∂∂

= ρρρρρ


P-i 0Pi xx −

0Pi yy −

'áXJR�ü

0iPd ia ib

P-1 P-2 P-3 P-4

1497.87 -1405.37

-944.56

248.48

-282.62 544.16

-810.43

-1221.56

1524.299 1507.042

1244.584

1246.576

-25.09 49.42

-107.92

-162.14

-132.97 127.63

125.78

-32.98 ��$]\PXW\�REOLF]RQH�]H�ZVSyáU]GQ\FK�SU]\EOL*RQ\FK

05292281360008037848248

561221

24673220180246734056944

43810

900051581801599021371405

16544

953813493601061410871497

62282

P4oP4o

P4

P4

P4o

P3oP3o

P3

P3

P3o

P2oP2o

P2

P2

P2o

P1oP1

P1

P1

P1o

0

0

0

0

0

0

0

0

′′′=−=′′′=−

=−

−=

′′′=+=′′′=−−

=−

−=

′′′=−=′′′=−

=−

−=

′′′=−=′′′=−

=−

−=

.A;..

.arctg

xx

yyarctg

.A;..

.arctg

xx

yyarctg

.A;..

.arctg

xx

yyarctg

.A;..

.arctg

xx

yyarctg

o

$$$

$$$

$$$

$$

ϕϕ

ϕϕ

ϕϕ

ϕϕ

5) Tabela orientacji stanowiska

Kierunek

K

Azymut przybl.

0A

6WDáD�RULHntac.

KAz / −= 00

Kier. przyEOL*� 000 zAK −=

Wyraz wolny

KK −= 0l Stan. Nr

kier. ° ′ ″ ° ′ ″ ° ′ ″ ° ′ ″ ″

P

1 2 3 4

94 00 13.2 263 31 08.2 325 18 49.2 26 11 11.7

349 18 53.9 158 50 00.9 220 37 46.2 281 29 52.0

255 18 40.7 255 18 52.7 255 18 57.0 255 18 40.3

94 00 06.2 263 31 13.2 325 18 58.5 26 11 04.3

-7.0 5.0 9.3 -7.4

�UHGQLD zo = 255 18 47.7


6) RyZQDQLD�EáGyZ 7��1DGRNUH�ORQ\�XNáDG�UyZQD��OLQLowych

v1 = -25.09.∆xP – 132.97.∆yP – ∆z – 7.0

v2 = 49.42.∆xP + 127.63.∆yP – ∆z + 5.0

v3 = -107.92.∆xP + 125.78.∆yP – ∆z + 9.3

v4 = -162.14.∆xP – 32.98.∆yP – ∆z – 7.4

-25.09.∆xP – 132.97.∆yP – ∆z – 7.0 = 0

49.42.∆xP + 127.63.∆yP – ∆z + 5.0 = 0

-107.92.∆xP + 125.78.∆yP – ∆z + 9.3 = 0

-162.14.∆xP – 32.98.∆yP – ∆z – 7.4 = 0


−−

=

∆∆∆

⋅

−−−−

−−−

−−

47

39

5

7

1

1

1

1

983214162

7812592107

631274249

971320925

.

.zyx

..

..

..

..

P

P


−−

−=

∆∆∆

⋅

−−

100

752982

91618

004468773245

46877350878891416

7324589149541007

.

.

.

zyx

...

...

...

P

P

��5R]NáDG�PDFLHU]\�QRUPDOQHM�AT.A�QD�F]\QQLNL�WUyMN�WQH

RRAA ⋅=⋅ TT

⋅

=

−−

33

2322

131211

332313

2212

11

00

00

00

004468773245

46877350878891416

7324589149541007

R

RR

RRR

RRR

RR

R

...

...

...

;

−=

532100

4256042250

213199765202

.

..

...

R


IRR =⋅− 1

=

−⋅

100

010

001

532100

4256042250

213199765202

00

0

33

2322

131211

.

..

...

'R

'R'R

'R'R'R

;

−−=−

6528000

001232000443500

00395400001532000493801

.

..

...

R

:\UyZQDQLH�ZFLFLD�ZVWHF]��NierunkoweJR��PHWRG��SR�UHdQLF]�F� 40


( ) =⋅=⋅ −− T11 RRAA �� 7

−⋅⋅−

−⋅−⋅=

=

−−⋅

−−=

−−

−−

42610000804500025820

00080450101192105535

0025820105535100054

6528000123200039540

0004435000015320

000049380

6528000

001232000443500

0039540000153200049380

56

65

...

...

...

...

..

.

.

..

...


−−−

≅

−−

−⋅

−⋅⋅−

−⋅−⋅=⋅⋅⋅=

−−

−−

80

0600

0080

100

752982

91618

42610000804500025820

00080450101192105535

002582010553510005456

65

.

.

.

.

.

.

...

...

...

)( LAA)(AX T-1T

14) Wyrównane niewiadome


′′−

′′′′−

′′

=

−−

−

−−−

⋅

−−−−

−−−

−−

=

33

53

22

02

47

39

5

7

80

0600

0080

1

1

1

1

983214162

7812592107

631274249

971320925

.

.

.

.

.

..

.

.

..

..

..

..

V


4081126337111126

752813255324981325

006132632220813263

2150094022130094

44

33

22

11

′′′=′′−′′′=+

′′′=′′+′′′=+

′′′=′′−′′′=+

′′′=′′+′′′=+

...vK

...vK

...vK

...vK

$$

$$

$$

$$


71317131

54183252157131

..

...

=−=


946812558074781255

260412606002004126

29218030083001803

0

0

0

′′′=′′−′′′=∆+=

−=−−=∆+==−=∆+=

...zzz

...yyy

...xxx

PPP

PPP

$$

�


18) Kontrola generalna $]\PXW\�REOLF]RQH�]H�ZVSyáU]GQ\FK�Z\Uywnanych

655922813604040378

839732201808397340

752941581803070121

102913493609570410

P4P4

P4

P4P4

P3P3

P3

P3P3

P2P2

P2

P2P2

P1P1

P1

P1P1

′′′=−=′′′=−−

=

′′′=+=′′′=−−

=

′′′=−=′′′=−−

=

′′′=−=′′′=−−

=

.A;.xx

yyarctg

.A;.xx

yyarctg

.A;.xx

yyarctg

.A;.xx

yyarctg

$$$

$$$

$$$

$$

ϕϕ

ϕϕ

ϕϕ

ϕϕ

/HZD�VWURQD�UyZQD� Prawa stroQD�UyZQD��REVHUZDF\MQ\FK obserwacyjnych

19��2FHQD�GRNáDGQo�FL

D��Eá�G��UHGQL�REVHUZDFML

651

31.71 ′′′±≈±=−⋅

±= .kn

mVV T

E��EáG\��UHGQLH�QLHZLDGRP\FK��ZVSyáU]GQ\FK�Z\UyZQDQych)



m.Qmm

m.Qmm

yyy

xxx

0260

0360

±=⋅±=

±=⋅±=

−⋅⋅−

−⋅−⋅=⋅= −−

−−

42610000804500025820

00080450101192105535

002582010553510005456

65

...

...

...

�� 7 AAQ

4081126

75281325

00613263

2150094

44

33

22

11

′′′=+

′′′=+

′′′=+

′′′=+

.vK

.vK

.vK

.vK

$

$

$

$

70811269468125565592281

952813259468125583973220

805132639468125575294158

21500949468125510291349

4P

3P

2P

1P

′′′=′′′−′′′=−

′′′=′′′−′′′=−

′′′=′′′−′′′=−

′′′=′′′−′′′=−

...zA

...zA

...zA

...zA

$$$

$$$

$$$

$$$

Wyrównanie ... SRSU]H]�HOLPLQDFM�QLHZLDGomej orientacyjnej 42


û:,&=(1,(�QU�� Temat: :\UyZQDQLH�ZFLFLD�ZVWHF]��NLHUXQNRZHJR��PHWRG�� SR�UHGQLF]�F��SRSU]H]�HOLPLQDFM�QLHZLDGomej orientacyjnej Zadanie: Na stanowisku P SRPLHU]RQR�]�MHGQDNRZ��GRNáDGQR�FL��NLHUXQNL��– 4. Stosu-M�F�PHWRG� SR�UHGQLF]�F�� QDOH*\� REOLF]\ü�Z\UyZQDQH�ZDUWR�FL� NLHUXQNyZ�RUD]�ZVSyá�U]GQ\FK�x, y punktu P,�D�WDN*H�SU]HSURZDG]Lü�DQDOL]�GRNáDGQR�FL�SR�Z\UyZQDQLX��=a-VWRVRZDü�VSRVyE�HOLPLQDFML�VWDáHM�RULHQWDFML�VWanowiska.

Dane:


1 2 3 4

94 263 325 26

00 31 18 11

132 082 492

117


3301.17 397.93 858.74

2051.78

-4408.82 -3582.04 -4936.63 -5347.76

Obliczone wVSyáU]GQH�SU]\EOL*RQH�punktu P xPo = 1803.30 yPo = -4126.20 52=:,�=$1,(


zxx

yyarctgzAvK

zxx

yyarctgzAvK

zxx

yyarctgzAvK

zxx

yyarctgzAvK

−−−

=−=+

−−−

=−=+

−−−

=−=+

−−−

=−=+

P4

P44P44

P3

P33P33

P2

P22P22

P1

P11P11

P (xP=?, yP=?)

3

2

1

1

2

3

4

4

Wyrównanie ... poprzez elLPLQDFM�QLHZLDGomej orientacyjnej 43


zyy

Ax

x

AzAvK

zyy

Ax

x

AzAvK

zyy

Ax

x

AzAvK

zyy

Ax

x

AzAvK

oo

oo

oo

oo

∆−∆⋅

∂∂

+∆⋅

∂∂

+−=+

∆−∆⋅

∂∂

+∆⋅

∂∂

+−=+

∆−∆⋅

∂∂

+∆⋅

∂∂

+−=+

∆−∆⋅

∂∂

+∆⋅

∂∂

+−=+

P

P

P4P

P

P44P44

P

P

P3P

P

P33P33

P

P

P2P

P

P22P22

P

P

P1P

P

P11P11


),,,i(;d

xx

y

Ab;

d

yy

x

Aa

iP

PiiP

i

iP

PiiP

i 43215206262

P2

P

=′′=′′′′⋅−−=′′⋅∂∂

=′′⋅−=′′⋅∂∂

= ρρρρρ


P-i 0Pi xx −

0Pi yy −

'áXJR�ü

0iPd ia ib

P-1 P-2 P-3 P-4

1497.87 -1405.37

-944.56

248.48

-282.62 544.16

-810.43

-1221.56

1524.299 1507.042

1244.584

1246.576

-25.09 49.42

-107.92

-162.14

-132.97 127.63

125.78

-32.98 ��$]\PXW\�REOLF]RQH�]H�ZVSyáU]GQ\FK�SU]\EOL*RQ\FK

05292281360008037848248

561221

24673220180246734056944

43810

900051581801599021371405

16544

953813493601061410871497

62282

P4oP4o

P4

P4

P4o

P3oP3o

P3

P3

P3o

P2oP2o

P2

P2

P2o

P1oP1

P1

P1

P1o

0

0

0

0

0

0

0

0

′′′=−=′′′=−

=−

−=

′′′=+=′′′=−−

=−

−=

′′′=−=′′′=−

=−

−=

′′′=−=′′′=−

=−

−=

.A;..

.arctg

xx

yyarctg

.A;..

.arctg

xx

yyarctg

.A;..

.arctg

xx

yyarctg

.A;..

.arctg

xx

yyarctg

o

$$$

$$$

$$$

$$

ϕϕ

ϕϕ

ϕϕ

ϕϕ

5) Tabela orientacji stanowiska

Kierunek

K

Azymut przybl.

0A

6WDáD�RULHntac.

KAz i −= 00

Kier. przyEOL*� 000 zAK −=

Wyraz wolny

KK −= 0l Stan. Nr

kier. ° ′ ″ ° ′ ″ ° ′ ″ ° ′ ″ ″

P

1 2 3 4

94 00 13.2 263 31 08.2 325 18 49.2 26 11 11.7

349 18 53.9 158 50 00.9 220 37 46.2 281 29 52.0

255 18 40.7 255 18 52.7 255 18 57.0 255 18 40.3

94 00 06.2 263 31 13.2 325 18 58.5 26 11 04.3

-7.0 5.0 9.3 -7.4

�UHGQLD zo = 255 18 47.7


6) RóZQDQLD�EáGyZ

v1 = -25.09.∆xP – 132.97.∆yP – ∆z – 7.0

v2 = 49.42.∆xP + 127.63.∆yP – ∆z + 5.0

v3 = -107.92.∆xP + 125.78.∆yP – ∆z + 9.3

v4 = -162.14.∆xP – 32.98.∆yP – ∆z – 7.4

��:VSyáF]\QQLNL�]UHGXNRZDQHJR�XNáDGX�UyZQD��EáGyZ

[ ] [ ] [ ]n

llL;

n

bbB;

n

aaA iiiiii −=−=−=

[ ] [ ] [ ] 410468773245 =−==−= n;.l;.b;.a

Kier. Ai Bi Li

1

2

3

4

36.34

110.85

-46.48

-100.71

-154.84

105.77

103.61

-54.85

-7.0

5.0

9.3

-7.4

��=UHGXNRZDQ\�XNáDG�UyZQD��EáGyZ

vi = Ai.∆xP + Bi

.∆yP + Li 9��1DGRNUH�ORQ\�XNáDG�UyZQD��OLQLRZ\FK

v1 = 36.34.∆xP – 154.84.∆yP – 7.0

v2 = 110.85.∆xP + 105.77.∆yP + 5.0

v3 = -46.48.∆xP + 103.61.∆yP + 9.3

v4 = -100.71.∆xP – 54.85.∆yP – 7.4

36.34.∆xP – 154.84.∆yP – 7.0 = 0

110.85.∆xP + 105.77.∆yP + 5.0 = 0

-46.48.∆xP + 103.61.∆yP + 9.3 = 0

-100.71.∆xP – 54.85.∆yP – 7.4 = 0


−−

=

∆∆

⋅

−−−

−

47

39

5

7

855471100

911034846

7710585110

841543436

.

.y

x

..

..

..

..

P

P


−−

=

∆∆

⋅

982984

86612

5348968936791

9367912125911

.

.

y

x

..

..

P

P

Wyrównanie ... poprzez elLPLQDFM�QLHZLDGomej orientacyjnej 45


RRAA ⋅=⋅ TT

⋅

=

22

1211

2212

11

0

0

5348968936791

9367912125911

R

RR

RR

R

..

..;

=

232170

194297160

.

..R


IRR =⋅− 1

=

⋅

10

01

232170

194297160

0 22

1211

.

..

R

RR'

''

;

−=−

00460300

001207000621201

.

..R


( ) =⋅=⋅ −− T11 RRAA �� 7

⋅⋅−⋅−⋅

=

−

⋅

−=

−−

−−

56

65

101192105545

105545100054

00460300012070

00062120

00460300

00120700062120

..

..

..

.

.

..


−−

≅

−

−⋅

⋅⋅−⋅−⋅

=⋅⋅⋅=−−

−−

0600

0080

988429

86612

101192105545

10554510005456

65

.

.

..

.

..

..)( LAA)(AX T1-T

16) Przyrost niewiadomej orientacyjnej

[ ] [ ] [ ] ( ) ( ) 80000600862100804361 ′′−=+−⋅+−⋅−=+∆⋅+∆⋅=∆ ......n

ly

n

bx

n

az PP



′′−

′′′′−

′′

=

−−

−

−−

⋅

−−−

−

=

33

53

22

02

47

39

5

7

0600

0080

855471100

911034846

7710585110

841543436

.

.

.

.

.

..

.

..

..

..

..

V

946812558074781255

26041260600204126

2921803008301803

0

0

0

′′′=′′−′′′=∆+=

−=−−=∆+==−=∆+=

...zzz

...yyy

...xxx

PPP

PPP

$$

�



4081126337111126

752813255324981325

006132632220813263

2150094022130094

44

33

22

11

′′′=′′−′′′=+

′′′=′′+′′′=+

′′′=′′−′′′=+

′′′=′′+′′′=+

...vK

...vK

...vK

...vK

$$

$$

$$

$$

20) Kontrola ogólna 21) Kontrola generalna $]\PXW\�REOLF]RQH�]H�ZVSyáU]GQ\FK�Z\Uywnanych

655922813604040378

839732201808397340

752941581803070121

102913493609570410

P4P4

P4

P4P4

P3P3

P3

P3P3

P2P2

P2

P2P2

P1P1

P1

P1P1

′′′=−=′′′=−−

=

′′′=+=′′′=−−

=

′′′=−=′′′=−−

=

′′′=−=′′′=−−

=

.A;.xx

yyarctg

.A;.xx

yyarctg

.A;.xx

yyarctg

.A;.xx

yyarctg

$$$

$$$

$$$

$$

ϕϕ

ϕϕ

ϕϕ

ϕϕ

/HZD�VWURQD�UyZQD� 3UDZD�VWURQD�UyZQD��REVHUZDF\MQ\FK

obserwacyjnych

70811269468125565592281

952813259468125583973220

805132639468125575294158

21500949468125510291349

4P

3P

2P

1P

′′′=′′′−′′′=−

′′′=′′′−′′′=−

′′′=′′′−′′′=−

′′′=′′′−′′′=−

...zA

...zA

...zA

...zA

$$$

$$$

$$$

$$$



b) EáG\��UHGQLH�QLHZLDGRP\FK��ZVSyáU]GQ\FK�Z\UyZQDQ\FK�



m.Qmm

m.Qmm

yyy

xxx

0260

0360

±=⋅±=

±=⋅±=

⋅⋅−⋅−⋅

=⋅=−−

−−

56

65

10119210555

10555100054

..

..�� 7 AAQ

4081126

75281325

00613263

2150094

44

33

22

11

′′′=+

′′′=+

′′′=+

′′′=+

.vK

.vK

.vK

.vK

$

$

$

$

71317131

54183252157131

..

...

=−=


651

31.71 ′′′±≈±=−⋅

±= .kn

mVV T

%á�G�SRáR*HQLD�SXQNWX��HOLSVD�EáGX��UHGQLHJR 47


û:,&=(1,(�QU�� Temat: %á�G�SRáR*HQLD�SXQNWX��HOLSVD�EáGX��UHGQLHJR Zadanie��1D�SRGVWDZLH�GDQ\FK�]�üZLF]HQLD�Ä:\UyZQDQLH�NLHUXQNRZHJR�ZFLFLD�ZVWHF]” QDOH*\� REOLF]\ü� EáG\� �UHGQLH� ZVSyáU]GQ\FK� SXQNWX� Z\]QDF]DQHJR�� Eá�G� SRáR*HQLD�SXQNWX�RUD]�SDUDPHWU\�HOLSV\�EáGX��UHGQLHJR��:\QLNL� SURV]� SU]HGVWDZLü� UyZQLH*� So-staci rysunku w odpowiedniej skali.

Dane:

%á�G��UHGQL�REVHUZDFML:

Macierz kowariancyjna:

:\UyZQDQH�ZVSyáU]GQH: xp = 1803.292; yp = - 4126.260 Obliczenia: 1) %áG\��UHGQLH�ZVSyáU]GQ\FK

2) %á�G�SRáR*HQLD�SXQNWX 3) .�W�VNUFHQLD�HOLSV\

( ) 58897101086601

101108122

5

5

..

.

QQ

Qtg

yyxx

xy −=⋅⋅−

=−

=−

−

α

ϕ = 30°29′49″

2α = 360° - ϕ = 329°30′11″

α = 164°45′06″

m.Qmm

m.Qmm

yyy

xxx

0260

0360

±=⋅±=

±=⋅±=

⋅⋅−⋅−⋅

=⋅=−−

−−

56

65

10119210555

10555100054

..

..�� 7 AAQ

65 ′′′±=−⋅

±= .kn

mVV T

m.mmm yxP 044022 ±=+±=

%á�G�SRáR*HQLD�SXQNWX��HOLSVD�EáGX��UHGQLHJR 48

��'áX*V]D�SyáR��HOLSV\

m.cossinQsinQcosQma xyyyxx 0360222 ±=⋅⋅⋅+⋅+⋅⋅±= αααα

��.UyWV]D�SyáR��HOLSV\

m.cossinQsinQcosQmb xyxxyy 0250222 ±=⋅⋅⋅−⋅+⋅⋅±= αααα

��6]NLF�HOLSV\�L�RNUJX�EáGX��UHGQLHJR

α

a

b

mP

y’

my

P

x

y

mx

x’

Skala 1:1

WyróZQDQLH�ZFLFLD�Z�SU]yG��NLHUXQNRZHJR��PHWRG��SR�UHGQLF]�F� 49


û:,&=(1,(�QU�� Temat: :\UyZQDQLH�ZFLFLD�Z�SU]yG��NLHUXQNRZHJR��PHWRG�� SR�UHGQLF]�F� Zadanie: Na stanowiskach 1, 2, 3 SRPLHU]RQR�NLHUXQNL�ZFLQDM�FH��–��6WRVXM�F�PHWRG�SR�UHGQLF]�F��QDOH*\�REOLF]\ü�Z\UyZQDQH�ZDUWR�FL�SRPLDUyZ�RUD]�ZVSyáU]GQ\FK�x, y punktu P,�D�WDN*H�SU]HSURZDG]Lü�DQDOL]�GoNáDGQR�FL�SR�Z\UyZQDQLX�

Dane:


1 2 3 4 5 6 7

237 297 91 151 219 0 50

42 11 23 02 11 00 02

068 20 33 10 53 00 37

:VSyáU]GQH�SXQNWyZ�VWDá\FK Pkt x y 1 2 3

2051.78 858.74 397.93

-5347.76 -4936.63 -3582.04

2EOLF]RQH�ZVSyáU]GQH�SU]\EOL*RQH�SXQNWX�3 xPo = 1803.30 yPo = -4126.20 52=:,�=$1,(

1) Równania obserwacyjne �GOD�NLHUXQNyZ�ZFLQDM�F\FK�

3

3

333P77

2

2

222P44

1

1

111P11

zxx

yyarctgzAvK

zxx

yyarctgzAvK

zxx

yyarctgzAvK

P

P

P

P

P

P

−−−

=−=+

−−−

=−=+

−−−

=−=+

7 6

5

1

3

4

2

P (xP=?, yP=?)

1

2

3

:\UyZQDQLH�ZFLFLD�Z�SU]yG��NLHUXQNRZHJR��PHWRG��SR�UHGQLF]�F� 50


( )

( )

( ) P

P

3PP

P

3P33P77

P

P

2PP

P

2P22P44

P

P

1PP

P

1P11P11

yy

Ax

x

AzAvK

yy

Ax

x

AzAvK

yy

Ax

x

AzAvK

o

o

o

∆⋅

∂∂

+∆⋅

∂∂

+−=+

∆⋅

∂∂

+∆⋅

∂∂

+−=+

∆⋅

∂∂

+∆⋅

∂∂

+−=+

��:VSyáF]\QQLNL�NLHUXQNRZH��SRFKRGQH�F]�VWNRZH�

),,i(

;d

xx

y

Ab;

d

yy

x

Aa

Pi

iPPi

i

Pi

iPPi

i

321

5206262

P2

P

=

′′=′′′′⋅−

=′′⋅∂

∂=′′⋅

−−=′′⋅

∂

∂= ρρρρρ


i-P iP xx −

0 iP yy −

0

'áXJR�ü

0Pid ia ib

1-P 2-P 3-P

-248.48 944.56

1405.37

1221.56 810.43

-544.16

1246.58 1244.58

1507.04

-162.14 -107.92

49.24

-32.98 125.77

127.63

4) Azymuty �GOD�NLHUXQNyZ�ZFLQDM�F\FK -�SU]\EOL*RQH)

0.1574288180;5.0895340180

0.1574108180;0.45217181.460

59.1354

5.0895160180;5.51001904.1193

13.411

9.0005338360;1.59902137.1405

16.544

2.467340;2.46734056.944

43.810

0.5292101180;0.08037848.248

56.1221

23322121

2323

23

2323

2121

12

1221

33

3

3

o3

22

2

2

o2

o11

1

1

o1

0

0

0

0

0

0

′′′=+=′′′=+=

′′′=−=′′′=−

=−−

=

′′′=−=′′′=−

=−−

=

′′′=−=′′′=−

=−

−=

′′′==′′′==−

−=

′′′=−=′′′=−

=−

−=

$$$$

$$$

$$$

$$$

$$

$$$

AAAA

Aarctgxx

yyarctg

Aarctgxx

yyarctg

Aarctgxx

yyarctg

Aarctgxx

yyarctg

Aarctgxx

yyarctg

oPoP

P

P

p

oPoP

P

P

p

poP

P

P

p

ϕϕ

ϕϕ

ϕϕ

ϕϕ

ϕϕ

��6WDáH�RULHQWDFML�VWDQRZLVN��

015742880000001574288

7285324902253249351121901574108

5355324933329150895340

54874223021129750895160

6323

2

5232

3212

2211

′′′=′′′−′′′=−=

′′′=⇒

′′′=′′′−′′′=−=

′′′=′′′−′′′=−=

′′′=′′′−′′′=−=

..KAz

.z..KA''z

..KA'z

..KAz

$$$

$$$$

$$$

$$$


6) Wyrazy wolne

0.9

5.7

3.3

7337

4224

1111

′′=−−=

′′=−−=

′′−=−−=

KzAl

KzAl

KzAl

oP

oP

oP

��5yZQDQLD�EáGyZ ��1DGRNUH�ORQ\�XNáDG�UyZQD��OLQLRZ\FK

v1 = -162.14.∆xP – 32.98.∆yP – 3.3

v4 = -107.92.∆xP + 125.78.∆yP + 7.5

v7 = 49.42.∆xP + 127.63.∆yP + 9.0

-162.14.∆xP – 32.98.∆yP – 3.3 = 0

-107.92.∆xP + 125.78.∆yP + 7.5 = 0

49.42.∆xP + 127.63.∆yP + 9.0 =0

9) Zapis macierzowy 10) Macierz wagowa

LXA =⋅

−=

∆∆

⋅

−

−−

0.9

5.7

3.3

63.12742.49

78.12592.107

98.3215.162

P

P

y

x

=

5.000

067.00

005.0

P

��8NáDG�UyZQD��QRUPDOQ\FK� AT.P.A.X = AT. P.L

−

=

∆∆

⋅

−

−80.1260

38.52

36.1928827.3267

27.326716.22169

P

P

y

x

��5R]NáDG�PDFLHU]\�QRUPDOQHM�AT.P.A�QD�F]\QQLNL�WUyMN�WQH

RRAPA ⋅=⋅⋅ TT

⋅

=

−

−

22

1211

2212

11

0

0

36.1928827.3267

27.326716.22169

R

RR

RR

R :��

−=

14.1370

94.2189.148R

13) OdwURWQR�ü�PDFLHU]\�WUyMN�WQHM��R-1

IRR =⋅− 1

=

−⋅

10

01

14.1370

94.2189.148

0 '22

'12

'11

R

RR :��

=−

007292.00

001075.0006716.01R

:\UyZQDQLH�ZFLFLD�Z�SU]yG��NLHUXQNRZHJR��PHWRG��SR�UHGQLF]�F� 52

��2GZURWQR�ü�PDFLHU]\�QRUPDOQHM�� (AT.P.A)-1

( ) =⋅=⋅⋅ −− T-1T )( 11 RRAPA

⋅⋅⋅⋅

=

⋅

=

−−

−−

56

65

103175108357

108357106264

00729200010750

00067160

00729200

00107500067160

..

..

..

.

.

..


−−

≅

−

⋅

⋅⋅⋅⋅

=⋅⋅⋅⋅⋅=−−

−−

m.

m.

.

.

..

..)(

0670

0070

801260

3852

103175108357

10835710626456

65

LPAA)P(AX T1-T



′′′′−

′′

=

−−−

−−=

1.0

1.0

1.0

0.9

5.7

3.3

9.8

6.7

4.3

V


137205010732050

90920151100120151

906242371080624237

77

44

11

′′′=′′+′′′=+

′′′=−′′′=+

′′′=′′+′′′=+

..vK

..vK

...vK

$$

$$

$$


01700170

61583632830170

..

...

=−=


267.4126067.020.4126

293.1803007.030.1803

0

0

−=−−=∆+==−=∆+=

PPP

PPP

yyy

xxx


20) Kontrola generalna Azymuty obliczone ze wspyáU]GQ\FK�Z\UyZQDQ\FK

0529433836000801213631405

227544

63873406387340553944

363810

455921011806040378487248

4931221

33

3

3

3

22

2

2

2

11

1

1

1

′′′=−=′′′=−

=−

−=

′′′==′′′==−

−=

′′′=−=′′′=−

=−

−=

.A;..

.arctg

xx

yyarctg

.A;..

.arctg

xx

yyarctg

.A;..

.arctg

xx

yyarctg

PP

P

P

p

PP

P

P

p

pP

P

P

p

$$$

$$

$$$

ϕϕ

ϕϕ

ϕϕ


��2FHQD�GRNáDGQo�FL�


101

0.017 ′′±≈±=−

⋅⋅±= .

kn

Pm0

VV T

E��EáG\��UHGQLH�ZVSyáU]GQ\Fh



F��Eá�G�SRáR*HQLD�SXQNWX�3

m.mmm yxP 001022 =+=

m.Qmm

m.Qmm

yyy

xxx

0010

0010

0

0

±=⋅±=

±=⋅±=

⋅⋅⋅⋅

=⋅⋅=−−

−−

56

65

103175108357

108357106264

..

..�� 7 APAQ

1372050

90920151

90624237

77

44

11

′′′=+

′′′=+

′′′=+

.vK

.vK

.vK

$

$

$

03720500157428805294338

90920151728532496387340

906242375487422345592101

33P

22P

11P

′′′=′′′−′′′=−

′′′=′′′−′′′=−

′′′=′′′−′′′=−

...zA

...zA

...zA

$$$

$$$

$$$

Wyrównanie F]ZRURERNX�JHRGH]\MQHJR��N�WRZHJR� PHWRG��SR�UHGQLF]�F� 54

D (xD=?, yD=?)

A

7

5

4 3

2

1

C 6

8

B (xB=?, yB=?)


û:,&=(1,(�QU�� Temat: :\UyZQDQLH�F]ZRURERNX�JHRGH]\MQHJR��N�WRZHJR��PHWRG� SR�UHGQiF]�F� Zadanie: Na stanowiskach A, B, C, D SRPLHU]RQR�]�MHGQDNRZ��GRNáDGQR�FL��N�W\��– 8. StosuM�F� PHWRG� SR�UHGQLF]�F�� QDOH*\� REOLF]\ü� Z\UyZQDQH� ZDUWR�FL� SRPLDUyZ� RUD]�ZVSyáU]GQ\FK�x, y punktów B i D. :�UDPDFK�RFHQ\�GRNáDGQR�FL�SURV]�REOLF]\ü�Eá�G��UHGQL� REVHUZDFML�� EáG\� �UHGQLH� ZVSyáU]GQ\FK� Z\UyZQDQ\FK� RUD]� EáG\� SRáR*HQLD�punktów wyznaczanych. Dane: ��3RPLHU]RQH�N�W\

.�W o / //

1 2 3 4 5 6 7 8

81 46 27 54 52 44 29 25

06 18 21 03 16 31 08 13

13 12 45 30 27 21 52 37

��:VSyáU]GQH�SXQNWyZ�VWDá\FK 2EOLF]RQH�ZVSyáU]GQH�SU]\EOL*RQH

xBo = 421.65 yBo = 552.58 xDo = 1852.23 yDo = 133.38

52=:,�=$1,(


Pkt x y A C

1000.00 1000.00

0.00 1000.00

DB

DB

DA

DADBDA

DC

DC

DB

DBDCDB

CA

CA

CD

CDCACD

CB

CB

CA

CACBCA

A

A

D

DBDBC

A

A

D

DBABD

C

C

B

BACAB

AD

AD

AC

ACADAC

xx

yyarctg

xx

yyarctgAAv

xx

yyarctg

xx

yyarctgAAv

xx

yyarctg

xx

yyarctgAAv

xx

yyarctg

xx

yyarctgAAv

xx

yyarctg

xx

yyarctgAAv

xx

yyarctg

xx

yyarctgAAv

xx

yyarctg

xx

yyarctgAAv

xx

yyarctg

xx

yyarctgAAv

−−

−−−

=−=+

−−

−−−

=−=+

−−

−−−

=−=+

−−

−−−

=−=+

−−

−−−

=−=+

−−

−−−

=−=+

−−

−−−

=−=+

−−

−−−

=−=+

88

77

66

55

B

B

B

B44

B

B

B

B33

A

A

A

A22

11

β

β

β

β

β

β

β

β

Wyrównanie czworoboku geodezyjnego (N�WRZHJR��PHWRG��SR�UHGQLF]�F� 55

��5R]ZLQLFLH�Z�V]ereg Taylora

DD

DB

D

DAD

D

DB

D

DAB

B

DBB

B

DBDBDA

DD

DC

D

DBD

D

DC

D

DBB

B

DBB

B

DBDCDB

DD

CDD

D

CDCACD

BB

CBB

B

CBCBCA

DD

BDD

D

BDB

B

BD

B

BCB

B

BD

B

BCBDBC

DD

BDD

D

BDB

B

BA

B

BDB

B

BA

B

BDBABD

BB

ABB

B

ABACAB

DD

ADD

D

ADADAC

yy

A

y

Ax

x

A

x

Ay

y

Ax

x

AAAv

yy

A

y

Ax

x

A

x

Ay

y

Ax

x

AAAv

yy

Ax

x

AAAv

yy

Ax

x

AAAv

yy

Ax

x

Ay

y

A

y

Ax

x

A

x

AAAv

yy

Ax

x

Ay

y

A

y

Ax

x

A

x

AAAv

yy

Ax

x

AAAv

yy

Ax

x

AAAv

0

0

0

0

0

O

∆⋅

∂∂−

∂∂+∆⋅

∂∂−

∂∂+∆⋅

∂∂−+∆⋅

∂∂−+−=+

∆⋅

∂∂−

∂∂+∆⋅

∂∂−

∂∂+∆⋅

∂∂+∆⋅

∂∂+−=+

∆⋅

∂∂

+∆⋅

∂∂

+−=+

∆⋅

∂∂−+∆⋅

∂∂−+−=+

∆⋅

∂∂−+∆⋅

∂∂−+∆⋅

∂∂−

∂∂+∆⋅

∂∂−

∂∂+−=+

∆⋅

∂∂+∆⋅

∂∂+∆⋅

∂∂−

∂∂+∆⋅

∂∂−

∂∂+−=+

∆⋅

∂∂+∆⋅

∂∂+−=+

∆⋅

∂∂−+∆⋅

∂∂−+−=+

0

0

0

0

0

0

88

77

66

55

44

33

22

11

β

β

β

β

β

β

β

β


52062622

′′=′′′′⋅−

−=′′⋅−

=−−

ρρρ ;d

xxb;

d

yya

iSt

Sti

iSt

Sti

Przyrosty :VSyáF]��NLHUXQNRZH Kierunek (St.-Cel)

Sti xx − Sti yy −

'áXJR�ü

iStd − a b

A-D A-B B-A B-D B-C C-B C-D D-C D-B D-A

852,23 -578,35 578,35

1430,58 578,35

-578,35 852,23

-852,23 -1430,58

-852,23

133,38 552,58

-552,58 -419,20 447,42

-447,42 -866,62 866,62 419,20

-133,38

862,60 799,90 799,90

1490,73 731,21 731,21

1215,45 1215,45 1490,73

862,60

36,97 178,14

-178,14 -38,91 172,60

-172,60 -121,00 121,00

38,91 -36,97

-236,24 186,44

-186,44 -132,78 -223,11 223,11

-118,99 118,99 132,78 236,24

4) :VSyáF]\QQLNL�SU]\�QLHZLDGRP\FK�Z�UyZQDQLDFK�EáGyZ

iPPPPLLLLSSSSi lybxaybxaybxav +∆⋅−∆⋅−∆⋅+∆⋅+∆⋅+∆⋅=

LPSLPS bbb,aaa −=−=

aB bB aD bD

1 0 0 L 36,97 L -236,24 2 P -178,14 P -186,44 0 0 3 S 139,23 S 53,66 P 38,91 P 132,78 4 S 211,51 S -90,33 L -38,91 L -132,78 5 L -172,60 L 223,11 0 0 6 0 0 P 121,00 P 118,99 7 P -38,91 P -132,78 S -82,09 S 13,79 8 L 38,91 L 132,78 S -75,88 S 103,46


��:\UD]\�ZROQH�UyZQD��EáGyZ $]\PXW\�REOLF]RQH�]H�ZVSyáU]GQ\FK�SU]\EOL*RQych

1,4235188180

6,0404163180

2,1313341180

2,1313314360;8,46824523,852

62,866

0000270180

8,3334217180

8,333437;8,33343735,578

42,447

6,0404343360;4,55911658,1430

20,419

3,1981316180

3,1981136180;7,40144335,578

58,552

00009000,0

00,1000

1,42358;1,4235823,852

38,133

0

0

0

0

0

0

′′′=+=

′′′=−=

′′′=−=

′′′=−=′′′=−=−

−=

′′′=+=

′′′=+=

′′′==′′′==−

−=

′′′=−=′′′=−

=−

−=

′′′=+=

′′′=−=′′′=−

=−

−=

′′′=→=−−

=

′′′==′′′==−

−=

$$

$$

$$

$$$

$$

$$

$$

$$$

$$

$$$

$

$$

ADoDAo

BDoDBo

CDoDCo

CDoCDo

CD

CD

CDo

ACCA

BCoCBo

BCoBCo

BC

BC

BCo

BDoBDo

BD

BD

BDo

ABoBAo

ABoABo

AB

AB

ABo

AC

AC

ACAC

ADoADo

AD

AD

ADo

AA

AA

AA

Aarctgxx

yyarctg

AA

AA

Aarctgxx

yyarctg

Aarctgxx

yyarctg

AA

Aarctgxx

yyarctg

Aarctgxx

yyarctg

Aarctgxx

yyarctg

O

0

0

O

O

O

ϕϕ

ϕϕ

ϕϕ

ϕϕ

ϕ

ϕϕ

Wyrazy wolne

//

DBDA

//DCDB

//CACD

//

CBCA

//

BDBC

//

BABD

//

ACAB

//

ADAC

,AAl

,AAl

,AAl

,AAl

,AAl

,AAl

,AAl

,AAl

0

0

0

0

0

O

60

60

87

80

80

30

37

94

88

77

66

55

44

33

22

11

0

0

0

0

0

0

=−−=

−=−−=

−=−−=

−=−−=

−=−−=

=−−=

=−−=

=−−=

β

β

β

β

β

β

β

β

6) RyZQDQLD�EáGyZ

v1 = 0 .∆xB +0 .∆yB +36,97 .∆xD -236,24 .∆yD +4,9

v2 = -178,14 .∆xB -186,44 .∆yB +0 .∆xD +0 .∆yD +7,3

v3 = 139,23 .∆xB +53,66 .∆yB +38,91 .∆xD +132,78 .∆yD +0,3

v4 = 211,51 .∆xB -90,33 .∆yB -38,91 .∆xD -132,78 .∆yD -0,8

v5 = -172,60 .∆xB +223,11 .∆yB +0 .∆xD +0 .∆yD -0,8

v6 = 0 .∆xB +0 .∆yB +121,00 .∆xD +118,99 .∆yD -7,8

v7 = -38,91 .∆xB -132,78 .∆yB -82,09 .∆xD +13,79 .∆yD -0,6

v8 = 38,91 .∆xB +132,78 .∆yB -75,88 .∆xD +103,46 .∆yD +0,6


7��1DGRNUH�ORQ\�XNáDG�UyZQD��OLQLRZ\FK

0 .∆xB +0 .∆yB +36,97 .∆xD -236,24 .∆yD +4,9 = 0

-178,14 .∆xB -186,44 .∆yB +0 .∆xD +0 .∆yD +7,3 = 0

139,23 .∆xB +53,66 .∆yB +38,91 .∆xD +132,78 .∆yD +0,3 = 0

211,51 .∆xB -90,33 .∆yB -38,91 .∆xD -132,78 .∆yD -0,8 = 0

-172,60 .∆xB +223,11 .∆yB +0 .∆xD +0 .∆yD -0,8 = 0

0 .∆xB +0 .∆yB +121,00 .∆xD +118,99 .∆yD -7,8 = 0

-38,91 .∆xB -132,78 .∆yB -82,09 .∆xD +13,79 .∆yD -0,6 = 0

38,91 .∆xB +132,78 .∆yB -75,88 .∆xD +103,46 .∆yD +0,6 = 0


−

−−−

=

∆

∆∆∆

⋅

−−−−

−−−−

−−−

6,0

6,0

8,7

8,0

8,0

3,0

3,7

9,4

46,10388,7578,13291,38

79,1309,8278,13291,38

99,11800,12100

0011,22360,172

78,13291,3833,9051,211

78,13291,3866,5323,139

0044,18614,178

24,23697,3600

D

D

B

B

y

x

yx


=

∆

∆∆∆

⋅

−−−

−−−

84,1885

09,716

83,1291

07,1243

50,11612516,701237,3102622,09,61

16,701210,3153285,642602,2571

37,3102685,642618,13084349,6600

22,610902,257149,660022,128675

D

D

B

B

y

x

yx


⋅⋅−⋅−⋅⋅−⋅⋅−⋅⋅−⋅−⋅⋅⋅⋅⋅⋅

=⋅

−−−−

−−−−

−−−−

−−−−

6667

6567

6667

7776

10288,910611,110108,210006,3

10611,110235,310181,110094,5

10108,210181,110216,810977,2

10006,310094,510977,210811,7

)( 1-T AA


≅

⋅

⋅⋅−⋅−⋅⋅−⋅⋅−⋅⋅−⋅−⋅⋅⋅⋅⋅⋅

=

=⋅⋅⋅=

−−−−

−−−−

−−−−

−−−−

m

m

m

m

014,0

019,0

006,0

011,0

84,1885

09,716

83,1291

07,1243

10288,910611,110108,210006,3

10611,110235,310181,110094,5

10108,210181,110216,810977,2

10006,310094,510977,210811,7

)(

6667

6567

6667

7776

LAA)(AX T-1T


12) Wyrównane niewiadome��ZVSyáU]GQH�SXQNWyZ�B i D)


−−−−

=

−

−−−

−

−

−−

−−

=

8,1

2,3

8,3

4,1

6,1

8,4

2,4

3,2

6,0

6,0

8,7

8,0

8,0

3,0

3,7

9,4

2,1

6,2

0,4

6,0

8,0

5,4

1,3

6,2

V


////////

88

////////77

////////66

////////55

////////

44

////////

33

////////

22

////////11

8,3813258,1371325

8,4808292,3520829

2,1731448,3213144

6,2516524,1271652

4,2803546,1300354

8,4921278,4452127

2,1618462,4121846

3,1506813,2130681

°=+°=+

°=−°=+

°=−°=+

°=−°=+

°=−°=+

°=+°=+

°=+°=+

°=+°=+

v

v

v

v

v

v

v

v

β

β

β

β

β

β

β

β


80,7841,78

43,6123,14041,78

≅−=


///371325°

394,133014,038,133

249,1852019,023,1852

586,552006,058,552

661,421011,065,421

0

0

0

0

=+=∆+==+=∆+=

=+=∆+==+=∆+=

DDD

DDD

BBB

BBB

yyy

xxx

yyy

xxx


16) Kontrola generalna $]\PXW\�REOLF]RQH�]H�ZVSyáU]GQ\FK�Z\UyZQDQ\FK

7,4435188180

0,0604163180

2,1713341180

2,1713314360;8,428245

4,3434217180

4,343437;4,343437

0,0604343360;0,549116

2,1681316180

2,1681136180;8,431443

7,44358;7,44358

′′′=+=

′′′=−=

′′′=−=

′′′=−=′′′=−

−=

′′′=+=

′′′==′′′=−−

=

′′′=−=′′′=−

−=

′′′=+=

′′′=−=′′′=−

−=

′′′==′′′=−

−=

$$

$$

$$

$$$

$$

$$

$$$

$$

$$$

$$

ADDA

BDDB

CDDC

CDoCDo

CD

CD

CD

BCCB

BCoBCo

BC

BCBC

BDoBDo

BD

BD

BD

ABBA

ABoABo

AB

AB

AB

ADoADo

AD

AD

AD

AA

AA

AA

Axx

yyarctg

AA

Axx

yyarctg

Axx

yyarctg

AA

Axx

yyarctg

Axx

yyarctg

ϕϕ

ϕϕ

ϕϕ

ϕϕ

ϕϕ


///

///

///

6

///

///

///

///

///

8,381325

8,480829

2,173144

6,251652

4,280354

8,492127

2,161846

3,150681

°=−

°=−

°=−

°=−

°=−

°=−

°=−

°=−

DBDA

DCDB

CACD

CBCA

BDBC

BABD

ACAB

ADAC

AA

AA

AA

AA

AA

AA

AA

AA

///88

///77

///66

///

55

///44

///33

///22

///11

8,381325

8,480829

2,173144

6,251652

4,280354

8,492127

2,161846

3,150681

°=+

°=+

°=+

°=+

°=+

°=+

°=+

°=+

v

v

v

v

v

v

v

v

β

β

β

β

β

β

β

β




4,44

78,41 ′′±≈±=−⋅

±=kn

mVV T

E��EáG\��UHGQLH�QLHZLDGRP\FK��ZVSyáU]GQ\FK�Z\UyZQDQ\FK�



F��Eá�G�SRáR*HQLD�SXQNWyZ�B i D

m.mmm

m.mmm

DDD

BBB

yxP

yxP

0280

0180

22

22

=+=

=+=

mQmm

mQmm

mQmm

mQmm

DD

DD

BB

BB

yyy

xxx

yyy

xxx

013.0

025.0

013.0

012.0

±=⋅±=

±=⋅±=

±=⋅±=

±=⋅±=

⋅⋅−⋅−⋅⋅−⋅⋅−⋅⋅−⋅−⋅⋅⋅⋅⋅⋅

=⋅=

−−−−

−−−−

−−−−

−−−−

6667

6567

6667

7776

10288,910611,110108,210006,3

10611,110235,310181,110094,5

10108,210181,110216,810977,2

10006,310094,510977,210811,7

)( 1-T AAQ

Wyrównanie czworoboku geodezyjnego (N�WRZHJR��PHWRG��zawarunkowan� 61

D

A

7

5

4 3

2

1

C 6

8

B


û:,&=(1,(�QU�� Temat: :\UyZQDQLH�F]ZRURERNX�JHRGH]\MQHJR��N�WRZHJR��PHWRG�� zawarunkowaQ� Zadanie: Na stanowiskach A, B, C, D SRPLHU]RQR�]�MHGQDNRZ��GRNáDGQR�FL��N�W\��÷8. 6WRVXM�F�PHWRG� ]DZDUXQNRZDQ�� QDOH*\� REOLF]\ü�Z\UyZQDQH�ZDUWR�FL� SRPLarów oraz ZDUWR�ü�EáGX��UHGQLHJR�REVHUZDFML� Dane: ��3RPLHU]RQH�N�W\

.�W o / //

1 2 3 4 5 6 7 8

81 46 27 54 52 44 29 25

06 18 21 03 16 31 08 13

13 12 45 30 27 21 52 37

52=:,�=$1,(

1) Liczba warunków

a) liczba ogólna warunków: w = r = 4, (r – liczba obserwacji nadwymiarowych) b) OLF]ED�ZDUXQNyZ�WUyMN�WyZ��Ztr = 3 c) liczba warunków sinusowych: ws = 1

2) Równania warunkowe

:DUXQNL�WUyMN�WyZ� (β1 + v1) + (β2 + v2) + (β3 + v3)+ (β8 + v8) = 180° (β4 + v4) + (β5 + v5) + (β6 + v6)+ (β7 + v7) = 180° (β2 + v2) + (β3 + v3) + (β4 + v4)+ (β5 + v5) = 180°

Warunek sinusowy: ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 1

88664422

77553311 =+⋅+⋅+⋅++⋅+⋅+⋅+

vsinvsinvsinvsinvsinvsinvsinvsin

ββββββββ

��5yZQDQLD�RGFK\áHN��QLHGRRNUH�ORQ\�XNáDG�UyZQD��OLQLRZ\FK�

v1 + v2 + v3 + v8 = ω1

v4 + v5 + v6 + v7 = ω2

v2 + v3 + v4 + v5 = ω3

48877665544332211 ωββββββββ =⋅−⋅+⋅−⋅+⋅−⋅+⋅−⋅ vctgvctgvctgvctgvctgvctgvctgvctg

WyrówQDQLH�F]ZRURERNX�JHRGH]\MQHJR��N�WRZHJR��PHWRG��]DZDUXQNRZDQ� 62

��2GFK\áNL

ω1 = 180° – (β1 + β2 + β3 + β8) = 13″ ω2 = 180° – (β4 + β5 + β6 + β7) = -10″

ω3 = 180° – (β2 + β3 + β4 + β5) = 6″

ω4 = (1 – F0).ρ″ = -0.015″ ≅ 0″,

gdzie: 0000000718642

75310 ,

sinsinsinsinsinsinsinsin

F =⋅⋅⋅⋅⋅⋅= ββββ

ββββ

5) Zapis macierzowy WVA =⋅

−=

⋅

−−−− 0

6

10

13

12327931017177407250932195601570

00011110

01111000

10000111

8

7

6

5

4

3

2

1

v

v

v

vv

v

v

v

........

��8NáDG�UyZQD��QRUPDOQ\FK�NRUHODW� ( ) WKAA T =⋅⋅

−

=

⋅

−

−

0

6

10

13

5521402518250990

0251422

8250240

990204

4

3

2

1

k

k

k

k

....

.

.

.

��2GZURWQR�ü�PDFLHU]\�QRUPDOQHM�� (A. AT)-1

−−−−

−−

=⋅

0739.00409.00052.0039.0

0409.0523.0253.0271.0

0052.0253.0375.0128.0

039.0271.0128.0395.0

)( 1-TAA

8) Wektor korelat K

−

=⋅⋅=

2060

142

613

232

.

.

.

.

W)A(AK 1-T

Wyrównanie czworoboku geodezyjnego (N�WRZHJR��PHWRG��zawarunkowan� 63


−−−−

≅

−

⋅

−

−

−

−

=⋅=⋅⋅⋅=

//

//

//

//

//

//

//

//

8.1

2.3

8.3

3.1

6.1

8.4

2.4

3.2

206.0

14.2

61.3

23.2

123.2001

793.1010

017.1010

774.0110

725.0110

932.1101

956.0101

157.0001

KAW)A(AAV T1-TT


////////

88

////////77

////////66

////////

55

////////44

////////33

////////

22

////////11

8,3813258,1371325

8,4808292,3520829

2,1731448,3213144

7,2516523,1271652

4,2803546,1300354

8,4921278,4452127

2,1618462,4121846

3,1506813,2130681

°=+°=+

°=−°=+

°=−°=+

°=−°=+

°=−°=+

°=+°=+

°=+°=+

°=+°=+

v

v

v

v

v

v

v

v

β

β

β

β

β

β

β

β


9.771.78 ≅⋅=⋅ KWVV TT

12) Kontrola generalna

(81°06′15,3″) + (46°18′16,2″) +(27°21′49,8″) + (25°13′38,8″) = 180°00′00,1″ (54°03′28,4″) + (52°16′25,7″) +(44°31′17,2″) + (29°08′48,8″) = 180°00′00,1″ (46°18′16,2″) +(27°21′49,8″) + (54°03′28,4″) + (52°16′25,7″) = 180°00′00,1″

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 00000051

17493662501749367060

8383125217134442830542168146

8488029725615284912273156081,

,,

,sin,sin,sin,sin

,sin,sin,sin,sin ==′′′°⋅′′′°⋅′′′°⋅′′′°′′′°⋅′′′°⋅′′′°⋅′′′°


444

78,1 ′′±≈±=⋅±= ,r

m VV T

Wyrównanie czworoboku geodezyjnego (liniRZHJR��PHWRG��SR�UHGQLF]�F� 64

4

2

D (xD=?, yD=?)

A

5

3

1

C

B (xB=?, yB=?)


û:,&=(1,(�QU�� Temat: Wyrównanie czworoboku geodezyjnego (liniowego) PHWRG��SR�UHGQiF]�F� Zadanie: W czworoboku geodezyjnym SRPLHU]RQR�]�MHGQDNRZ��GRNáDGQR�FL��GáXJR�FL��–� �� 6WRVXM�F� PHWRG� SR�UHGQLF]�F�� QDOH*\� REOLF]\ü� Z\UyZQDQH� ZDUWR�FL� SRPLDUyZ�oraz wspóáU]GQ\FK�x, y punktów B i D. :�UDPDFK�RFHQ\�GRNáDGQR�FL�SURV]�REOLF]\ü�Eá�G��UHGQL�REVHrZDFML��EáG\��UHGQLH�ZVSyáU]GQ\FK�Z\UyZQDQ\FK�RUD]�EáG\�SRáR*HQLD punktów wyznaczanych. Dane: ��3RPLHU]RQH�GáXJR�FL

�:VSyáU]GQH�SXQNWyZ�VWDá\FK 2EOLF]RQH�ZVSyáU]GQH�SU]\EOL*RQH

xBo = 421.65 yBo = 552.58 xDo = 1852.23 yDo = 133.38

52=:,�=$1,(


( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )22

55

22

44

22

33

22

22

22

11

DADADA

CDCDCD

BCBCBC

BDBDBD

ABABAB

yyxxdvd

yyxxdvd

yyxxdvd

yyxxdvd

yyxxdvd

−+−==+

−+−==+

−+−==+

−+−==+

−+−==+

Nr 'áXJR�ü [m] 1 2 3 4 5

799.82 1490.64 731.26 1215.40 862.64

Pkt x y A C

1000.00 1000.00

0.00 1000.00

Wyrównanie czworoboku geodezyjnego (linoZHJR��PHWRG��SR�UHGQLF]�F� 65


DD

DAD

D

DADA

DD

CDD

D

CDCD

BB

BCB

B

BCBC

DD

BDD

D

BDB

B

BDB

B

BDBD

BB

ABB

B

ABAB

yy

dx

x

ddvd

yy

dx

x

ddvd

yy

dx

x

ddvd

yy

dx

x

dy

y

dx

x

ddvd

yy

dx

x

ddvd

∆⋅

∂∂+∆⋅

∂∂+=+

∆⋅

∂∂+∆⋅

∂∂+=+

∆⋅

∂∂+∆⋅

∂∂+=+

∆⋅

∂∂+∆⋅

∂∂+∆⋅

∂∂+∆⋅

∂∂+=+

∆⋅

∂∂+∆⋅

∂∂+=+

0

0

0

0

0

55

44

33

22

11

3��:\UD]\�ZROQH�UyZQD��EáGyZ 'áXJR�FL�SU]\EOL*RQH

( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) m,yyxxd

m,yyxxd

m,yyxxd

m,yyxxd

m,yyxxd

00

00

00

DADADA

CDCDCD

BCBCBC

BDBDBD

ABABAB

604862

4531215

214731

7341490

896799

22

22

22

22

22

000

0

000

000

=−+−=

=−+−=

=−+−=

=−+−=

=−+−=

Wyrazy wolne

mddl

mddl

mddl

mddl

mddl

DA

CD

BC

BD

AB

036,0640,862604,862

053,0400,1215453,1215

046,0260,731214,731

094,0640,1490734,1490

076,0820,799896,799

55

44

33

22

11

0

0

0

0

0

−=−=−=

=−=−=

−=−=−=

=−=−=

=−=−=

4) $]\PXW\�SU]\EOL*RQH

1423518818014235823852

38133

21313314360846824523852

62866

8333437833343735578

42447

604043433604559116581430

20419

31981136180740144335578

58552

0

0

0

0

0

0

′′′=+=′′′=−−

=−

−=

′′′=−=′′′=−

=−

−=

′′′==′′′==−

−=

′′′=−=′′′=−

=−

−=

′′′=−=′′′=−

=−

−=

,A;,,

,arctg

xx

yyarctg

,A;,,

,arctg

xx

yyarctg

,A;,,

,arctg

xx

yyarctg

,A;,,

,arctg

xx

yyarctg

,A;,,

,arctg

xx

yyarctg

DAoDAo

DA

DA

DAo

CDoCDo

CD

CD

CDo

BCoBCo

BC

BC

BCo

BDoBDo

BD

BD

BDo

ABoABo

AB

AB

ABo

0

O

0

0

O

O

$$$

$$$

$$

$$$

$$$

ϕϕ

ϕϕ

ϕϕ

ϕϕ

ϕϕ



ikjkkjkjjkjjki lyAsinxAcosyAsinxAcosv +∆⋅+∆⋅+∆⋅−∆⋅−=

55

44

33

22

11

sincos

sincos

sincos

sincossincos

sincos

lyAxAv

lyAxAv

lyAxAv

lyAxAyAxAv

lyAxAv

DDADDA

DCDDCD

BBCBBC

DBDDBDBBDBBD

BABBAB

+∆⋅−∆⋅−=

+∆⋅+∆⋅=+∆⋅−∆⋅−=

+∆⋅+∆⋅+∆⋅−∆⋅−=+∆⋅+∆⋅=

v1 = -0,723032 .∆xB +0,690815 .∆yB +0 .∆xD +0 .∆yD +0,076

v2 = -0,959648 .∆xB +0,281204 .∆yB +0,959648 .∆xD -0,281204 .∆yD +0,094

v3 = -0,790945 .∆xB -0,611887 .∆yB +0 .∆xD +0 .∆yD -0,046

v4 = 0 .∆xB +0 .∆yB +0,701162 .∆xD -0,713002 .∆yD +0,053

v5 = 0 .∆xB +0 .∆yB +0,987973 .∆xD 0,154625 .∆yD -0,036


-0,723032 .∆xB +0,690815 .∆yB +0 .∆xD +0 .∆yD +0,076 = 0

-0,959648 .∆xB +0,281204 .∆yB +0,959648 .∆xD -0,281204 .∆yD +0,094 = 0

-0,790945 .∆xB -0,611887 .∆yB +0 .∆xD +0 .∆yD -0,046 = 0

0 .∆xB +0 .∆yB +0,701162 .∆xD -0,713002 .∆yD +0,053 = 0

0 .∆xB +0 .∆yB +0,987973 .∆xD 0,154625 .∆yD -0,036 = 0


−

−−

=

∆∆

∆∆

⋅

−−−

−−−

036,0

053,0

046,0

094,0

076,0

154625,0987973,000

713002,0701162,000

00611887,0790945,0

281204,0959648,0281204,0959648,0

00690815,0723032,0

D

D

B

B

y

x

yx


−−

=

∆

∆∆∆

⋅

−−−−−−

−−

0699,0

0918,0

1071,0

1088,0

6114,06170,007918,02699,0

6170,03886,22699,09209,0

07908,02699,09307,02853,0

2699,09209,02853,00693,2

D

D

B

B

y

x

yx

:\UyZQDQLH�F]ZRURERNX�JHRGH]\MQHJR��OLQLRZHJR��PHWRG��SR�UHGQiF]�F� 67

��2GZURWQR�ü�PDFierzy normalnej: (AT.A)-1

−−

−−

=⋅

2157,25553,001505,003973,0

5553,06505,007817,02063,0

01505,007817,01351,11198,0

03973,02063,01198,05968,0

)( 1-T AA


−

≅

−−

⋅

−−

−−

=

=⋅⋅⋅=

m

m

m

m

098,0

010,0

100,0

030,0

0699,0

0918,0

1071,0

1088,0

2157,25553,001505,003973,0

5553,06505,007817,02063,0

01505,007817,01351,11198,0

03973,02063,01198,05968,0

)( LAA)(AX T-1T

11) Wyrównane niewiadome��ZVSyáU]GQH�SXQNWyZ�B i D)


−−−

−

=

−

−−

−

−

−−

=

m

m

m

m

m

011,0

010,0

009,0

019,0

015,0

036,0

053,0

046,0

094,0

076,0

025,0

063,0

037,0

075,0

091,0

V


629,862011,064,862

390,1215010,0400,1215

251,731009,0260,731

659,1490019,0640,1490

805,799015,0820,799

55

44

33

22

11

=−=+=−=+

=−=+

=+=+=−=+

vd

vd

vd

vd

vd


000920000890

019910020830000890

,,

,,,

≅−=


478,133098,038,133

240,1852010,023,1852

480,552100,058,552

680,421030,065,421

0

0

0

0

=+=∆+==+=∆+=

=−=∆+==+=∆+=

DDD

DDD

BBB

BBB

yyy

xxx

yyy

xxx


15) Kontrola generalna 'áXJR�FL�REOLF]RQH�]H�ZVSyáU]GQ\FK�wyrównanych

( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) myyxxd

myyxxd

myyxxd

myyxxd

myyxxd

DADADA

CDCDCD

BCBCBC

BDBDBD

ABABAB

629,862478,133240,852

390,1215522,866240,852

251,731520,44732,578

659,1490002,419560,1430

805,799480,55232,578

2222

2222

2222

2222

2222

=−+−=−+−=

=−+=−+−=

=−+−=−+−=

=−+=−+−=

=+−=−+−=

/HZD�VWURQD�UyZQD� Prawa stronD�UyZQD��REVHUZDF\MQ\FK obserwacyjnych


a) Eá�G��UHGQL�REVHUZDFML b) EáG\��UHGQLH�QLHZLDGRP\FK��ZVSyáU]GQ\FK�Z\UyZQDQ\FK�



F��Eá�G�SRáR*HQLD�SXQNtów B i D

mmmmmmmmDDDBBB yxPyxP 051.0;039.0 2222 =+==+=

mQmm

mQmm

mQmm

mQmm

DD

DD

BB

BB

yyy

xxx

yyy

xxx

045.02157,2030,0

024.06505,0030,0

032.01351,1030,0

023.05968,0030,0

±=⋅=⋅±=

±=⋅=⋅±=

±=⋅=⋅±=

±=⋅=⋅±=

−−

−−

=⋅=

2157,25553,001505,003973,0

5553,06505,007817,02063,0

01505,007817,01351,11198,0

03973,02063,01198,05968,0

)( 1-T AAQ

md

md

md

md

md

DA

CD

BC

BD

AB

629,862

390,1215

251,731

659,1490

805,799

==

===

mvd

mvd

mvd

mvd

mvd

629,862

390,1215

251,731

659,1490

805,799

55

44

33

22

11

=+=+

=+=+=+

m,kn

m 03001

0.00089 ±≈±=−⋅±= VV T

Wyrównanie czworoboNX�JHRGH]\MQHJR��OLQLRZHJR��PHWRG��]DZDUXQNRZDQ� 69

γ1 γ2

γ3

α1

α2

α3

β1

β2

β3

r1

(1)

r2 (2)

r3 (3)

(5) s1

(4) s2

s3

1

2

3

5

4

6


û:,&=(1,(�QU�� Temat: Wyrównanie czworoboku geodezyjnego (liniowego) PHWRG��]DZDUXQNRZaQ� Zadanie: W czworoboku geodezyjnym SRPLHU]RQR�]�MHGQDNRZ��GRNáDGQR�FL��GáXJo�FL��– 5.�3RPLDU\�QDOH*\�Z\UyZQDü�PHWRG��]DZDUXQNRZDQ�. :�UDPDFK�RFHQ\�GRNáDGQR�FL�SURV]�REOiF]\ü�Eá�G��UHGQL�REVHUZDFML� Dane: ��3RPLHU]RQH�GáXJR�FL

'áXJR�ü�ED]\��]H�ZVSyáU]GQ\FK��G6 = 1000.00m 52=:,�=$1,(

1) Liczba warunków

w = r = 1, (r – liczba obserwacji nadwymiarowych)

2) Równanie warunkowe

0321 =−+www

γγγ

gdzie:

( ) ( ) ( )( ) ( )

21

121

21

2

1

2

2

2

1

21

21

22

21

1

2

2

rr

srr

vrvr

vsvrvrarccos

rr

srrarccos

ww

www

w

+⋅+⋅+−+++

=

=⋅⋅−+

=γ

( ) ( ) ( )( ) ( )

32

232

32

2

2

2

3

2

2

32

22

23

22

2

2

2

rr

srr

vrvr

vsvrvrarccos

rr

srrarccos

ww

www

w

+⋅+⋅+−+++

=

=⋅⋅−+

=γ

( ) ( )( ) ( )

31

31

31

23

2

3

2

1

31

23

23

21

3 22 rr

rr

vrvr

svrvrarccos

rr

srrarccos

ww

ww

w +⋅+⋅−+++

=⋅⋅−+

=γ

Nr 'áXJR�ü [m] 1 2 3 4 5

799.82 1490.64 731.26 1215.40 862.64

Wyrównanie czworoboku geodezyjnego (liniRZHJR��PHWRG��]DZDUXQNRZDQ� 70

3) Doprowadzenie do postaci liniowej

( ) ( ) ( ) 0332211 =+−+++ γγγγγγ ddd

gdzie: 121

1

1

2

1

1

11 srr v

sv

rv

rd ⋅

∂∂+⋅

∂∂+⋅

∂∂= γγγγ

2322

2

3

2

2

22 srr v

sv

rv

rd ⋅

∂∂+⋅

∂∂+⋅

∂∂= γγγγ

313

3

1

33 rr v

rv

rd ⋅

∂∂+⋅

∂∂= γγγ

5yZQDQLH�RGFK\áNL�Z�SRVWDFL�OLQLRZHM:

021321 21321 =+⋅+⋅+⋅+⋅+⋅ ωssrrr vBvBvAvAvA

��:DUWR�FL�SU]\EOL*RQH�N�WyZ

g...

...arccosrs

rrsarccos 55682141

827996486226414908279964862

2

222

11

22

21

21

1 =⋅⋅

−+=⋅⋅−+=α

g...

...arccosrs

rrsarccos 3912532

641490401215226731641490401215

2

222

22

23

22

22

2 =⋅⋅

−+=⋅⋅−+=α

g...

...arccosrs

rrsarccos 0748758

2673100100028279926731001000

2

222

33

21

23

23

3 =⋅⋅

−+=⋅⋅−+=α

g...

...arccosrs

rrsarccos 0328328

6414906486228279964149064862

2

222

21

21

22

21

1 =⋅⋅

−+=⋅⋅−+=β

g...

...arccosrs

rrsarccos 54115107

26731401215264149026731401215

2

222

32

22

23

22

2 =⋅⋅

−+=⋅⋅−+=β

g...

...arccosrs

rrsarccos 4550351

8279900100022673182799001000

2

222

13

23

21

23

3 =⋅⋅

−+=⋅⋅−+=β

g...

...arccosrr

srrarccos 4103530

64149082799240121564149082799

2

222

21

21

22

21

1 =⋅⋅

−+=⋅⋅−+=γ

g...

...arccosrr

srrarccos 0676060

26731641490240121526731641490

2

222

32

22

23

22

2 =⋅⋅

−+=⋅⋅−+=γ

g...

...arccosrr

srrarccos 4701190

267318279920010002673182799

2

222

31

23

23

21

3 =⋅⋅

−+=⋅⋅−+=γ

��3U]\EOL*RQH�ZDUWR�FL�SyO�WUyMN�WyZ

D��WUyMN�W� 121 s,r,r :

21211 6967274051

2

410353064149082799

2m,

.sin..sinrrF

g

=⋅⋅

=⋅⋅

=γ

E��WUyMN�W� 232 s,r,r :

22322 1259882545

2067606026731641490

2m..sin..sinrr

Fg

=⋅⋅=⋅⋅= γ

Wyrównanie czworoboNX�JHRGH]\MQHJR��OLQLRZHJR��PHWRG��]DZDUXQNRZDQ� 71

��:VSyáF]\QQLNL�SU]\�QLHZLDGRP\FK��SRSUDZNDFK�

( ) ( )m

..ctg.ctgm.

ctgctgr

Acc

ccggcc 95413684550351556821418279911

311

1 =⋅−⋅−=⋅−⋅−= ρρβα

( ) ( )m

..ctg.ctgm.

ctgctgr

Acc

ccggcc 042167203283283912532641490

1112

22 −=⋅+⋅−=⋅+⋅−= ρρβα

( ) ( )m

..ctg.ctgm.

ctgctgr

Acc

ccggcc 2637775411510707487582673111

233

3 =⋅+−⋅−=⋅+−⋅−= ρρβα

m.

m.m.

F

sB

cccccc 976500

3934548103264862

2 21

11 =⋅

⋅=⋅

⋅= ρρ

m.

m.m.

F

sB

cccccc 362438

12598825452401215

2 22

22 =⋅

⋅=⋅

⋅= ρρ

7) OGFK\áND�ωω

ccggg .... 478470119006760604103530321 =−+=−+= γγγω

��2VWDWHF]QD�OLQLRZD�SRVWDü�UyZQDQLD�RGFK\áNL

04783624389765002637770421672954136821321

=+⋅+⋅+⋅+⋅−⋅ ccssrrr .v.v.v.v.v.

9) Zapis macierzowy: WVA =⋅

[ ] [ ]47836243897650026377704216729541368

5

4

3

2

1

.

v

v

v

v

v

..... −=

⋅−

��8NáDG�UyZQD��QRUPDOQ\FK�NRUHODW� ( ) WKAA T =⋅⋅

[ ] [ ] [ ]4780455717036 1 .k. −=⋅

��2GZURWQR�ü�PDFLHU]\�QRUPDOQHM�� (A. AT)-1

[ ]7107491 −⋅=⋅ .�� 7AA


[ ]5103711 −⋅−=⋅⋅= .W)A(AK -1T

Wyrównanie czworoboku geodezyjnego (liniRZHJR��PHWRG��]DZDUXQNRZDQ� 72


[ ]

−−−

−

≅⋅−⋅

−

=⋅=⋅⋅⋅= −

m.

m.

m.

m.

m.

.

.

.

.

.

.

0060

0070

0110

0230

0190

103711

362438

976500

263777

0421672

9541368

5KAW)A(AAV T1-TT

14) Obserwacje wyrównane ( )( )( )( )( ) 634862006064862

39312150070401215

249731011026731

66314900230641490

801799019082799

551

442

333

222

111

...vd:s

...vd:s

...vd:r

...vd:r

...vd:r

w

w

w

w

w

=−=+

=−=+

=−=+

=+=+

=−=+


00107500010750 .. ≅

⋅=⋅ KWVV TT

16) Kontrola generalna

.�W\�Z\UyZQDQH� g.

.....arccos

rr

srrarccos

ww

www

w4065330

6631490801799239312156631490801799

2

222

21

21

22

21

1 =⋅⋅

−+=⋅⋅−+

=γ

g...

...arccosrr

srrarccos

ww

www

w0651360

2497316631490239312152497316631490

2

222

32

22

23

22

2 =⋅⋅

−+=⋅⋅−+

=γ

g...

...arccosrr

srrarccos

ww

ww

w4722590

2497318017992001000249731801799

2

222

31

23

23

21

3 =⋅⋅

−+=⋅⋅−+

=γ

Równanie warunkowe: ccggg ....

www95472259006513604065330321 −=−+=−+ γγγ


m.r

m 03301

0.001075 ±≈±=⋅±= VV T

:\UyZQDQLH�FL�JX�SROLJRQRZHJR�PHWRG��SR�UHGQLF]�F� 73

d3 d2 d1

β4

β3 β2

β1

A C

B D 1

2

�,PL�1D]ZLVNR Nr zestawu .....

û:,&=(1,(�QU�� Temat: :\UyZQDQLH�FL�JX�SROLJRQRZHJR�PHWRG��SR�UHGQLF]�F� Zadanie��:� FL�JX� SROLJRQRZ\P� QDZL�]DQ\P� GZXVWURQQLH� N�WRZR� L� OLQLRZR� QDOH*\� Z\-UyZQDü� �PHWRG�� SR�UHGQLF]�F�� SRPLHU]RQH�ZLHONR�FL� RUD]�ZVSyáU]GQH� SXQNWyZ�Z\-znaczanych. :� UDPDFK� RFHQ\� GRNáDGQR�FL� SURV]� REOLF]\ü� Eá�G� �UHGQL� MHGQRVWNRZ\��EáG\��UHGQLH�ZVSyáU]dQ\FK�Z\UyZQDQ\FK�RUD]�EáG\�SRáR*HQLD�SXQNWyZ� Dane: :VSyáU]GQH�SXQNWyZ�VWDá\FK 3RPLHU]RQH�GáXJR�FL ��3RPLHU]RQH�N�W\

.�W o / //

1 2 3 4

195 142 179 218

13 10 50 05

18.8 39.4 10.0 40.0

%á�G��UHGQL�SRPLDUX�GáXJR�FL�md = ± 1cm

%á�G��UHGQL�SRPLDUX�NDWD�mβ = ± 10″

52=:,�=$1,(

1)�:VSyáU]GQH�SU]\EOL*RQH�SXQNWyZ�1 i 2

Przyrosty WVSyáU]GQH Nr pkt

.�W�β

° ′ ″ Azymut A

° ′ ″ 'áXJR�ü�d

[m] ∆x ∆y X Y Nr pkt

A 1027.932 1027.521 A 56-23-18.2 -

B 195-13-18.8 1130.285 181.507 B 71-36-37.0 162.400 51.234 154.107

1 142-10-39.4 1181.519 1335.614 1 33-47-16.4 185.264 153.973 103.029

2

1335.492 1438.643 2

Pkt x y A B C D

1027.932 1130.285 1490.281 1541.380

1027.521 1181.507 1541.544 1696.033

Nr 'áXJR�ü [m] 1 2 3

162.400 185.264 185.836

Wyrównanie FL�JX�SROLJRQRZHJR�PHWRG��SR�UHGQLF]�F� 74


( ) ( )( ) ( )( ) ( )22

273

2

2

2

21262

2

1

2

1151

C2

C2

C

C44

21

21

2

2233

1

1

12

121222

1

1111

2C2CC

11

BBB

D

DC2CD

C

C21C

B

B1B

BA

BA

B

BBAB

yyxxdvd

yyxxdvd

yyxxdvd

xx

yyarctg

xx

yyarctgAAv

xx

yyarctg

xx

yyarctgAAv

xx

yyarctg

xx

yyarctgAAv

xxyy

arctgxxyy

arctgAAv

−+−==+

−+−==+

−+−==+

−−−

−−=−=+

−−−

−−=−=+

−−−

−−=−=+

−−−

−−=−=+

β

β

β

β


22C

22C

C

212

212

112

112

1B1

1B1

B

22

C22

2

C2CCD

22

21

2

2C2

2

21

2

2C1

1

211

1

212C

22

122

2

121

1

1B

1

121

1

1B

1

12B

11

B11

1

B1BAB

yy

dx

x

ddvd

yy

dx

x

dy

y

dx

x

ddvd

yy

dx

x

ddvd

yy

Ax

x

AAAv

yy

A

y

Ax

x

A

x

Ay

y

Ax

x

AAAv

yy

Ax

x

Ay

y

A

y

Ax

x

A

x

AAAv

yy

Ax

x

AAAv

0

0

∆⋅

∂∂+∆⋅

∂∂+=+

∆⋅

∂∂+∆⋅

∂∂+∆⋅

∂∂+∆⋅

∂∂+=+

∆⋅

∂∂+∆⋅

∂∂+=+

∆⋅

∂∂−+∆⋅

∂∂−+−=+

∆⋅

∂∂−

∂∂+∆⋅

∂∂−

∂∂+∆⋅

∂∂−+∆⋅

∂∂−+−=+

∆⋅

∂∂+∆⋅

∂∂+∆⋅

∂∂−

∂∂+∆⋅

∂∂−

∂∂+−=+

∆⋅

∂∂+∆⋅

∂∂+−=+

22273

22111262

11151

244

1233

11222

111

0

0

0

0

00

0

β

β

β

β

��3RFKRGQH�F]�VWNRZH

iSt

Sti

iSt

Sti

iSt

Sti

iSt

Sti

d

xxAcos;

d

yyAsin;

d

xxb;

d

yya

−−−−

−=

−=′′⋅

−−=′′⋅

−= ρρ

22

Przyrosty :VSyáF]��NLHU�

Kier. Sti xx −

Sti yy − 'áXJR�ü

iStd − a b cosA sinA Kier.

B-A

B-1 1-B

1-2

2-1 2-C

C-2

C-D

-102.353

51.234 -51.234 153.973

-153.973 154.789

-154.789 51.099

-153.986

154.107 -154.107 103.029

-103.029 102.901

-102.901 154.489

184.899

162.400

162.400

185.264

185.264

185.872

185.872

162.720

-929.04

1205.24

-1205.24

619.16

-619.16

614.36

-614.36

1203.48

617.53

-400.69

400.69

-925.31

925.31

-924.14

924.14

-398.06

0.315479

0.831102

0.832773

0.948933

0.556120

0.553614

B-A

B-1 1-B

1-2

2-1 2-C

C-2

C-D


5) :VSyáF]\QQLNL�SU]\�QLHZLDGRP\FK�Z�UyZQDQLDFK�EáGyZ

D��N�W: iPPPPLLLLSSSSi lybxaybxaybxav +∆⋅−∆⋅−∆⋅+∆⋅+∆⋅+∆⋅= ,

LPSLPS bbb,aaa −=−=

E��GáXJR�ü� ikjkkjkjjkjjki lyAsinxAcosyAsinxAcosv +∆⋅+∆⋅+∆⋅−∆⋅−=

6) Wyra]\�ZROQH�UyZQD��EáGyZ $]\PXW\�REOLF]RQH�]H�ZVSyáU]GQ\FK�SU]\EOL*RQych

0521471

2556333

4167433

0376371

2183256

0

0

0

2

12

1

′′′°=′′′°=′′′°=′′′°=′′′°=

.A

.A

.A

.A

.A

CD

C

B

AB

'áXJR�FL�SU]\EOL*RQH

( ) ( )( ) ( )( ) ( ) m.yyxxd

m.yyxxd

m.yyxxd

00

00

2C2CC

11

BBB

872185

264185

400162

22

2

2

2

2

212

2

1

2

11

0

000

000

=−+−=

=−+−=

=−+−=

Wyrazy wolne

m...ddl

m...ddl

m,..ddl

....AAl

....AAl

....AAl

....AAl

C

B

///////////CCD

///////////2C

///////////

1B

///////////BAB

0

0

0

0360836185872185

0000264185264185

0000400162400162

24304005218255362130524171

23101050179416472132553633

0043910142037362514164733

0081813195218232360373671

327

2126

115

424

3123

2122

111

0

0

0

0

0

0

=−=−=

=−=−=

=−=−=

−=°−°−°=−−=

−=°−°−°=−−=

=°−°−°=−−=

=°−°−°=−−=

β

β

β

β

Nr x1 y1 x2 y2

1 P -1205.24 P 400.69 0 0

2 S 1824.40 S -1326.00 P -619.16 P 925.31

3 L -619.16 L 925.31 S 1233.52 S -1849.46

4 0 0 L -614.36 L 924.14

5 k 0.315479 k 0.948933 0 0

6 p -0.831102 p -0.556120 k 0.831102 k 0.556120

7 0 0 p -0.832773 p -0.553614



v1 = -1205.24 .∆x1 +400.69

.∆y1 +0 .∆x2 +0

.∆y2 +0

v2 = 1824.40 .∆x1 -1326.00

.∆y1 -619.16 .∆x2 +925.31

.∆y2 +0

v3 = -619.16 .∆x1 +925.31

.∆y1 +1233.52 .∆x2 -1849.46

.∆y2 -31.2

v4 = 0 .∆x1 +0

.∆y1 -614.36 .∆x2 +924.14

.∆y2 -43.2

v5 = 0.315479 .∆x1 +0.948933

.∆y1 +0 .∆x2 +0

.∆y2 +0

v6 = -0.831102 .∆x1 -0.556120

.∆y1 +0.831102 .∆x2 +0.556120

.∆y2 +0

v7 = 0 .∆x1 +0

.∆y1 -0.832773 .∆x2 -0.553614

.∆y2 +0.036


-1205.24 .∆x1 +400.69

.∆y1 +0 .∆x2 +0

.∆y2 +0 = 0

1824.40 .∆x1 -1326.00

.∆y1 -619.16 .∆x2 +925.31

.∆y2 +0 = 0

-619.16 .∆x1 +925.31

.∆y1 +1233.52 .∆x2 -1849.46

.∆y2 -31.2 = 0

0 .∆x1 +0

.∆y1 -614.36 .∆x2 +924.14

.∆y2 -43.2 = 0

0.315479 .∆x1 +0.948933

.∆y1 +0 .∆x2 +0

.∆y2 +0 = 0

-0.831102 .∆x1 -0.556120

.∆y1 +0.831102 .∆x2 +0.556120

.∆y2 +0 = 0

0 .∆x1 +0

.∆y1 -0.832773 .∆x2 -0.553614

.∆y2 +0.036 = 0


−

=

∆∆∆∆

⋅

−−−−

−−−

−−−

0360

0

0

243

231

0

0

5536140832773000

5561200831102055612008311020

0094893303154790

149243661400

4618495212333192516619

3192516619001326401824

0069400241205

2

2

1

1

.

.

.

y

x

yx

..

....

..

..

....

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10) Macierz wagowa

=

10000000000

01000000000

00100000000

000010000

000001000

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000000010

.

.

.

.

P


��8NáDG�UyZQD��QRUPDOQ\FK� AT.P.A.X = AT.P.L

−

=

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⋅

−−−−−−

−−

17320

26417

70288

18193

8857464732498765324756223710

7324987943666505150027325840

6532475051500288398475527134

6223710732584055271348059546

2

2

1

1

.

.

.

.

y

x

yx

....

....

....

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⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

=⋅⋅

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−−−−

−−−−

−−−−

5556

5565

5655

6555

109933109361109212108125

109361109274104799107991

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108125107991109051100973

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≅

−

⋅

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=

=⋅⋅⋅⋅⋅=

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−−−−

−−−−

−−−−

m.

m.

m.

m.

.

.

.

.

....

....

....

....

)(

0160

0200

0180

0070

17320

26417

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14) Wyrównane niewiadome��ZVSyáU]GQH�SXQNWyZ�1 i 2)


′′−

′′−

′′−

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=

−

−

−

−−

=

m.

m.

m.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0100

0100

0190

740

823

78

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0

0

243

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0190

52

47

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21

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659143801606431438

512133502004921335

632133501806141335

526118100705191181

222

222

111

111

0

0

0

0

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...xxx

...yyy

...xxx

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m...vd

m...vd

m...vd

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...v

...v

...v

////////

////////

////////

////////

8461850100836185

2741850100264185

4191620190400162

3590421874004005218

2464917982301050179

730101427843910142

617131952181813195

73

62

51

44

33

22

11

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°=−°=+

°=−°=+

°=−°=+

β

β

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67286128

511218416128

..

...

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18) Kontrola generalna $]\PXW\�REOLF]RQH�]H�ZVSyáU]GQ\FK�Z\UyZQDQ\FK

0521471

7526333

5067433

8356371

2183256

2

12

1

′′′°=

′′′°=

′′′°=

′′′°=

′′′°=

.A

.A

.A

.A

.A

CD

C

B

AB


m.vd

m.vd

m.vd

.v

.v

.v

.v

///

///

///

///

846185

274185

419162

35904218

24649179

7301042

61713195

73

62

51

44

33

22

11

=+

=+

=+

°=+

°=+

°=+

°=+

β

β

β

β

( ) ( )( ) ( )( ) ( ) m.yyxxd

m.yyxxd

m.yyxxd

...AA

...AA

...AA

...AA

2C2CC

11

BBB

/////////C2CD

/////////21C

/////////

1B

/////////BAB

846185

274185

419162

35904218752362130524171

24649179506472137523633

7301042835362515064733

61713195218232368353671

22

2

2

2

2

212

2

1

2

11

2

12

1

=−+−=

=−+−=

=−+−=

°=°−°=−

°=°−°=−

°=°−°=−

°=°−°=−


19��2FHQD�GRNáDdQR�FL

a) Eá�G��UHGQL�REVHUZDFML��MHGQRVWNRZ\�

08833

28.61 .kn

m0 ±≈±=−

⋅⋅±= VPV T

b) EáG\��UHGQLH�QLHZLDGRP\FK��ZVSyáU]GQych wyrównanych)



F��Eá�G�SRáR*HQLD�SXQNWyZ�1 i 2

m.mmm

m.mmm

yxP

yxP

0290

0290

22

22

222

111

=+=

=+=

m.Qmm

m.Qmm

m.Qmm

m.Qmm

yyy

xxx

yyy

xxx

0200

0220

0240

0170

22

22

11

1 1

±=⋅±=

±=⋅±=

±=⋅±=

±=⋅±=

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

=⋅⋅=

−−−−

−−−−

−−−−

−−−−

5556

5565

5655

6555

109933109361109212108125

109361109274104799107991

109212104799108315109051

108125107991109051100973

....

....

....

....

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Wyrównanie FL�JX�SROLJRQRZHJR�PHWRG��ZDUXQNRZ� 80

d3 d2 d1

β4

β3 β2

β1

A C

B D 1

2


û:,&=(1,(�QU�� Temat: :\UyZQDQLH�FL�JX�SROLJRQRZHJR�PHWRG��ZarunkoZ� Zadanie��:� FL�JX� SROLJRQRZ\P� QDZL�]aQ\P�GZXVWURQQLH� N�WRZR� L� OLQLRZR� QDOH*\� Z\-UyZQDü��PHWRG��ZDUXQNRZ��SRPLHU]RQH�N�W\�L�GáXJR�FL� :�UDPDFK�RFHQ\�GRNáDGQR�FL�SURV]�REOLF]\ü�Eá�G��UHGQL�MHGQRVtkowy. Dane: :VSyáU]GQH�SXQNWyZ�VWDá\FK PomieU]RQH�GáXJR�FL ��3RPLHU]RQH�N�W\

.�W o / //

1 2 3 4

195 142 179 218

13 10 50 05

18.8 39.4 10.0 40.0



52=:,�=$1,(

1) Liczba warunków

w = r = 3, (r – liczba obserwacji nadwymiarowych) 2) Równania warunkowe :DUXQHN�VXP\�N�WyZ� [�]pr = [�]teor

� ( ) ( ) ( ) ( ) $180444332211 ⋅+−=+++++++ ABCD AAvvvv ββββ

Warunki sumy przyrostów: [∆x]pr = [∆x]teor; [∆y]pr = [∆y]teor

� ( ) ( ) ( ) BCCB xxAcosvdAcosvdAcosvd −=⋅++⋅++⋅+ 2731262151

� ( ) ( ) ( ) BCCB yyAsinvdAsinvdAsinvd −=⋅++⋅++⋅+ 2731262151

gdzie:

Pkt x y A B C D

1027.932 1130.285 1490.281 1541.380

1027.521 1181.507 1541.544 1696.033

Nr 'áXJR�ü [m] 1 2 3

162.400 185.264 185.836

( )( ) ( )( ) ( ) ( ) $

$

$

1803

1802

180

3322112

221112

111

⋅−++++++=

⋅−++++=

−++=

vvvAA

vvAA

vAA

ABC

AB

ABB

βββ

ββ

β

:\UyZQDQLH�FL�JX�SROLJRQRZHJR�PHWRG��warunkow� 81

��5yZQDQLD�RGFK\áHN��SR�GRSURZDG]eniu do postaci liniowej)

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) 27026012501302

201

10

27026012501302

201

10

14321

ωρρρ

ωρρρ

ω

=⋅+⋅+⋅+⋅′′∆

+⋅′′∆

+⋅′′∆

=⋅+⋅+⋅+⋅′′∆

−⋅′′∆

−⋅′′∆

−

=+++

vAsinvAsinvAsinvx

vx

vx

vAcosvAcosvAcosvy

vy

vy

vvvv

CBCCBC

CBCCBC

4) Obliczenie azymutów A]\PXW\�ERNyZ�QDZL�]DQLD�

0521471052147109951489154

21832562183256353102986153

′′′==′′′==−−=

′′′==′′′==−−=

.A;...arctg

xx

yyarctg

.A;...arctg

xx

yyarctg

CDCDCD

CDCD

ABABAB

ABAB

$$

$$

ϕϕ

ϕϕ

$]\PXW\�SU]\EOL*RQH�SR]RVWDá\FK�ERNyZ�FL�JX� ( )( )( )

4267333

5400100517943901142818311952183256540

416743336043901142818311952183256360

0376371180818311952183256180

32102

21012

101

′′′=

=−′′′+′′′+′′′+′′′=−+++=

′′′=−′′′+′′′+′′′=−++=

′′′=−′′′+′′′=−+=

.

....AA

....AA

...AA

ABC

AB

ABB

$

$$$$$$

$$$$$$

$$$$$

βββ

ββ

β

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( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) 041360

934205

905102

951359

717308

744154

02301220110

023012201

02302

02301220110

023012201

02302

.AsindAsindAsindy

.AsindAsindy

.Asindy

.AcosdAcosdAcosdx

.AcosdAcosdx

.Acosdx

CBBC

CC

CC

CBBC

CC

CC

=⋅+⋅+⋅=∆

=⋅+⋅=∆

=⋅=∆

=⋅+⋅+⋅=∆

=⋅+⋅=∆

=⋅=∆

��2GFK\áNL�&i

( )( ) m.yyy

m.xxx

.AA

BCBC

BCBC

ABCD

0040

0450

4741804

03

02

43211

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ω

ωββββω $

��2VWDWHF]QD�SRVWDü�XNáDGX�UyZQD��RdFK\áHN

0.0040.5537400.5561200.9489330.0007500.0014970.001745

0.0450.8326890.8311020.3154790.0004990.0009980.001746

74.4

−=⋅+⋅+⋅+⋅+⋅+⋅

=⋅+⋅+⋅+⋅−⋅−⋅−

−=+++

765321

765321

4321

vvvvvv

vvvvvv

vvvv

Wyrównanie FL�JX�SROLJRQRZHJR�PHWRG��ZDUXQNRZ� 82

8) Zapis macierzowy: WVA =⋅

−

−=

⋅

−−−

0.004

0.045

74.4

0.5537400.5561200.94893300.0007500.0014970.001745

0.8326890.8311020.31547900.0004990.0009980.001746

0001111

7

6

5

4

3

2

1

v

v

v

v

v

v

v

��0DFLHU]�ZDJRZD��RGZURWQR�ü�

waga: 2

1

i

im

p = ; 21

ii mp =−

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−

−=

⋅

−−−

−

0040

0450

474

000736000036903992010

000369000057803242820

39920103242820000000400

3

2

1

.

.

.

k

k

k

...

...

...

��2GZURWQR�ü�PDFLHU]\�QRUPDOQHM��

−

−=⋅⋅ −

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82156943666125232401912702

6567612191270200692801

...

...

...

�� 7APA


−

−=

−

−⋅

−

−=

=⋅⋅⋅= −

696055206

81585017

4067270

0040

0450

474

91974930168215694366567612

82156943666125232401912702

656761219127020069280

1

.

.

.

.

.

.

...

...

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W)AP(AK -1T

=−

00010000000

00001000000

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1

.

.

.

P

:\UyZQDQLH�FL�JX�SROLJRQRZHJR�PHWRG��warunkow� 83


′′−′′−

′′−′′−

=⋅⋅=⋅⋅⋅⋅⋅= −−−

m.

m.

m.

.

.

.

.

0100

0100

0190

740

324

08

51

111 KAPW)AP(AAPV T1-TT


m...vd

m...vd

m...vd

...v

...v

...v

...v

////////

////////

////////

////////

8461850100836185

2741850100264185

4191620190400162

3590421874004005218

7454917932401050179

431101420843910142

317131955181813195

73

62

51

44

33

22

11

=+=+

=+=+

=+=+°=−°=+

°=−°=+

°=−°=+

°=−°=+

β

β

β

β

15) Kontrola ogólna 16) Kontrola generalna Azymuty wyrównane: ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) 6526333540

9067433360

5356371180

3322112

221112

111

′′′°=°−++++++=

′′′°=°−++++=

′′′°=°−++=

.vvvAA

.vvAA

.vAA

ABC

AB

ABB

βββββ

β

Równania warunkowe:

/////////

ABCD

///////////////

...AAP

.....L

83318735720218235605241711804

7331873535904218745491794311014231713195

°=°+°−°=⋅+−=

°=°+°+°+°=$

( ) ( ) ( )

m...xxP

m..cos..cos..cos.L

BC 99635928511302811490

996359652633384618590674332741855356371419162

=−=−==′′′°⋅+′′′°⋅+′′′°⋅=

( ) ( ) ( )

m...yyP

m..sin..sin..sin.L

BC 03736050711855441541

037360652633384618590674332741855356371419162

=−=−==′′′°⋅+′′′°⋅+′′′°⋅=

1��%á�G��UHGQL�REVHrwacji

09433

28.72 .kn

m0 ±≈±=−

⋅±= VPV T

72287228 .. =⋅=⋅⋅ KWVPV TT

3UDZR�SU]HQRV]HQLD�VL�EáGyZ��UHGQLFK�ZLHONR�FL�VNRUHORZDQ\FK 84

d3 d2 d1

β4 β3

β2

β1

A C

B D 1

2


û:,&=(1,(�QU�� Temat: 3UDZR�SU]HQRV]HQLD�VL�EáGyZ��UHGQLFK�ZLHONR�FL�VNRUHORZa-

nych. %á�G��UHGQL�IXQNFML� Zadanie��:\NRU]\VWXM�F�Z\QLNL�]�]DGDQLD�ÄWyrównaQLH�FL�JX�SROLJRQRZHJR�PHWRG��So-

�UHGQLF]�F�´�QDOH*\�REOLF]\ü�EáG\��UHGQLH�N�WyZ�β1 i β3�RUD]�GáXJR�FL�d1 i d3 po wyrów-naniu. Dane:



Równania obserwacyjne dla wybranych N�WyZ�ββ1 i ββ3�RUD]�GáXJR�FL�d1 i d3

( ) ( )( ) ( )22

273

2

1

2

1151

21

21

2

2233

1

1111

2C2CC

BBB

C

C21C

BA

BA

B

BBAB

yyxxdvd

yyxxdvd

xx

yyarctg

xx

yyarctgAAv

xx

yyarctg

xx

yyarctgAAv

−+−==+

−+−==+

−−−

−−=−=+

−−−

−−=−=+

β

β

RyZQDQLD�EáGyZ N�WyZ�ββ1 i ββ3�RUD]�GáXJR�FL�d1 i d3

v1 = -1205.24 .∆x1 +400.69 .∆y1 +0 .∆x2 +0 .∆y2 +0

v3 = -619.16 .∆x1 +925.31 .∆y1 +1233.52 .∆x2 -1849.46 .∆y2 -31.2

v5 = 0.315479 .∆x1 +0.948933 .∆y1 +0 .∆x2 +0 .∆y2 +0

v7 = 0 .∆x1 +0 .∆y1 -0.832773 .∆x2 -0.553614 .∆y2 +0.036

%á�G��UHGQL�REVHUZDFML��MHGQRVWNRZ\�

08833

28.61 .kn

m0 ±≈±=−

⋅⋅±= VPV T


Macierz kowariancyjna (teoretyczna) niewiadomych x1, y1, x2, y2

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅

=⋅⋅=

−−−−

−−−−

−−−−

−−−−

5556

5565

5655

6555

109933109361109212108125

109361109274104799107991

109212104799108315109051

108125107991109051100973

....

....

....

....

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52=:,�=$1,(

Macierz kowariancyjna empiryczna Qx

⋅⋅

⋅⋅

=⋅=

−

−

−

−

000380800001846000027860105425

000184600004698010038900017160

000469801038090005560000018170

105425000171600001817000029530

5

5

5

5

20

....

....

.....

....

m QQx

%áG\��UHGQLH�IXQNFML: fQf ⋅⋅= x

T

Fm2

Funkcja 1

[ ]00694002412052

1

2

1

1

1

1

11 ..

y

F

x

F

y

F

x

FT −=

∆∂∂

∆∂∂

∆∂∂

∆∂∂=f

[ ] 76342

112

1

.

m x

T

F

=

⋅

⋅=

=⋅⋅=

−

⋅⋅

⋅⋅

−−

−

−

−

0

0

400.69

1205.24

0.00038080.00018460.0002786105.542

0.00018460.0004698109.0380.0001716

0.0004698109.0.380.00055600.0001817

105.5420.00017160.00018170.0002953

00400.691205.24

5

5

5

5

fQf

5181

′′= .mF

F1 = v1 = -1205.24 .∆x1 +400.69

.∆y1 +0 .∆x2 +0

.∆y2


Funkcja 3

[ ]46184952123331925166192

3

2

3

1

3

1

33 ....

y

F

x

F

y

F

x

FT −−=

∆∂∂

∆∂∂

∆∂∂

∆∂∂=f

[ ]

92673

332

3

.

m x

T

F

=

=

⋅

⋅=

=⋅⋅=

−

−

⋅⋅

⋅⋅

−−−

−

−

−

1849.46

1233.52

925.31

1619.16

0.00038080.00018460.0002786105.542

0.00018460.0004698109.0380.0001716

0.0004698109.0.380.00055600.0001817

105.5420.00017160.00018170.0002953

1849.461233.52925.31619.16

5

5

5

5

fQf

0263

′′= .mF

Funkcja 5

[ ]00948933031547902

5

2

5

1

5

1

55 ..

y

F

x

F

y

F

x

FT =

∆∂∂

∆∂∂

∆∂∂

∆∂∂=f

[ ] 0006390

552

5

.

m x

T

F

=

⋅

⋅=

=⋅⋅=

−

−

−

−

⋅⋅

⋅⋅

0

0

0.948933

0.315479

0.00038080.00018460.0002786105.542

0.00018460.0004698109.0380.0001716

0.0004698109.0.380.00055600.0001817

105.5420.00017160.00018170.0002953

000.9489330.315479

5

5

5

5

fQf

m.mF 02505

=

Funkcja 7

[ ]55361408327730002

7

2

7

1

7

1

77 ..

y

F

x

F

y

F

x

FT −−=

∆∂∂

∆∂∂

∆∂∂

∆∂∂=f

[ ] 0006130

772

7

.

m x

T

F

=

=

=⋅⋅=

⋅

⋅⋅

⋅⋅

⋅−−−

−

−

−

0.553614-

0.832773-

0

0

0.00038080.00018460.0002786105.542

0.00018460.0004698109.0380.0001716

0.0004698109.0.380.00055600.0001817

105.5420.00017160.00018170.0002953

0.5536140.83277300

5

5

5

5

fQf

m.mF 02507

=

F3 = v3 = -619.16 .∆x1 +925.31

.∆y1 +1233.52 .∆x2 -1849.46

.∆y2 -31.2

F5 = v5 = 0.315479 .∆x1 +0.948933

.∆y1 +0 .∆x2 +0

.∆y2

F7 = v7 = 0 .∆x1 +0

.∆y1 -0.832773 .∆x2 -0.553614

.∆y2 +0.036

:\UyZQDQLH�WUDQVIRUPDFML�ZVSyáU]GQ\FK 87


û:,&=(1,(�QU�� Temat: :\UyZQDQLH�WUDQVIRUPDFML�ZVSyáU]GQ\FK dla n>2 punktów dostosowania Zadanie��'DQH�V��ZVSyáU]GQH�SXQNWyZ�GR�WUDQVIRUPDFML1÷5�Z�XNáDG]LH�SLHUZRWQ\P�xy RUD]�ZVSyáU]GQH�WU]HFK�SXQNWyZ�GRVWRVRZDQLD�10, 20, 30 Z�XNáDG]LH�SLHUZRWQ\P�xy i wtórnym XY��1DOH*\�Z\UyZQDü��ZHGáXJ�PHWRG\�QDMPQLHMV]\FK�NZDGUDWyZ��ZVSyáU]GQH�punktów dostosoZDQLD� L� REOLF]\ü� ZVSyáU]GQH� SXQNWyZ� WUDQVIRUPRZDQ\FK� Z� XNáDG]LH�wtórnym. W ramach oceQ\� GRNáDGQR�FL� SURV]� REOLF]\ü� EáG\� �UHGQLH� ZVSyáU]GQ\FK�SXQNWyZ�GRVWRVRZDQLD�RUD]�Eá�G��UHGQL�WUDQVIRUPDFML� Dane:

52=:,�=$1,( ��:VSyáU]GQH�ELHJXQD�WUDQVIRUPDFML�x0, y0�Z�XNáDG]LH�SLHUZRtnym Liczba punktów dostosowania: s = 3

siys

yxs

xs

i

s

i ,...,1;667.10661

;667.7661

1

0

1

0 ===== ∑∑

2) Wzory transformacyjne

( ) ( )( ) ( ) SxxCyyYY

SyyCxxXX

⋅−−⋅−=−⋅−+⋅−=−

000

000

αα sin;cos ⋅=⋅= kSkC

8NáDG�SLHUZRWQ\ 8NáDG�ZWyUQ\ Nr pkt x y X Y

10 400.000 600.000 1800.000 2700.000

20 1500.000 600.000 1993.444 1609.818

30 400.000 2000.000 3185.625 2945.539

1 400.000 200.000

2 700.000 500.000

3 600.000 900.000

4 200.000 1000.000

5 100.000 400.000

Y0

[y]

xP

[x]

YP

XP

yP

α

[Y]

[X]

P

X0

Wyrównanie transforPDFML�ZVSyáU]GQ\FK 88

3) Równania „obserwacyjne” dla punktów dostosowania

X10 + 10XV = (x10 – x0)

.C + (y10 – y0).S + X0

Y10 + 10YV = (y10 – y0)

.C – (x10 – x0).S + Y0

X20 + 20XV = (x20 – x0)

.C + (y20 – y0).S + X0

Y20 + 20YV = (y20 – y0)

.C – (x20 – x0).S + Y0

X30 + 30XV = (x30 – x0)

.C + (y30 – y0).S + X0

Y30 + 30YV = (y30 – y0)

.C – (x30 – x0).S + Y0

4) 8NáDG�UyZQD��EáGyZ z nieznanymi parametrami C, S, X0, Y0

10XV = -366.667.C – 466.667.S + X0 – 1800.000

10YV = -466.667.C + 366.667.S + Y0 – 2700.000

20XV = 733.333.C – 466.667.S + X0 – 1993.444

20YV = -466.667.C – 733.333.S + Y0 – 1609.818

30XV = -366.667.C + 933.333.S + X0 – 3185.625

30YV = 933.333.C + 366.667.S + Y0 – 2945.539

��5HGXNFMD�XNáDGX�R�SDUDPHWU\�X0, Y0 – ZVSyáU]GQH�ELHJXQD�Z�XNáDG]Le wtórnym

Równanie sumowe „X-ów”: ∑ =−⋅+⋅+⋅=s

X XSCVi

1

0 0069.69793000.0000.0 :��

35623263

06969791

1

0 ..Xs

Xs

i === ∑

Równanie sumowe „Y-ów”: ∑ =−⋅+⋅+⋅=s

Y YSCVi

1

0 0357.72553000.0000.0 :

45224183

35772551

1

0 ..Ys

Ys

i === ∑

8NáDG�]UHGXNRZDQ\

10XV = -366.667.C – 466.667.S + 526.356

10YV = -466.667.C + 366.667.S – 281.548

20XV = 733.333.C – 466.667.S + 332.912

20YV = -466.667.C – 733.333.S + 808.634

30XV = -366.667.C + 933.333.S – 859.268

30YV = 933.333.C + 366.667.S – 527.086


6) Zapis macierzowy LXAV −⋅=

−−

−

−

⋅

−−−−

−−−

=

086527

268859

634808

912332

548281

356526

667366333933

333933667366

333733667466

667466333733

667366667466

667466667366

30

30

20

20

10

10

.

.

.

.

.

.

S

C

..

..

..

..

..

..

V

V

V

V

V

V

Y

X

Y

X

Y

X


=

⋅

3682092473

638371717

33321133330

03332113333

.

.

S

C

.

.


⋅

⋅=⋅ −

−

7

7

10731940

01073194

.

.�� 7 AA


=

≅

⋅

⋅

⋅=

=⋅⋅⋅=

−

−

S

C

.

.

.

.

.

.

)(

9901290

1758920

3682092473

638371717

10731940

010731947

7

LAA)(AX T-1T

10) Parametry transformacji k, αα

005631199012901758920 2222 ...SCk =+=+=

984580005631199012901749070

00563111758920 .

.

.kSsin;.

.

.kCcos ====== αα :�

α = 88.80752g 11) Wektor poprawek LXAV −⋅=

−−−

=

−−

−

−

−−

−

=

1270

3590

4570

1610

5830

1980

086527

268859

634808

912332

548281

356526

213527

627859

178808

073333

965280

554526

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

V

Wyrównanie transforPDFML�ZVSyáU]GQ\FK 90

��:\UyZQDQH�ZVSyáU]GQH�SXQNWyZ�GRVWRVRZDQLD�Z�XNáDG]LH�ZWyUQ\P

666294512705392945

983318535906253185

275161045708181609

283199316104441993

417269958300002700

802179919800001800

30

30

20

20

10

10

3030

3030

2020

2020

1010

1010

...VYY

...VXX

...VYY

...VXX

...VYY

...VXX

Y

X

Y

X

Y

X

=+=+=

=+=+=

=+=+=

=−=+=

=−=+=

=−=+=

13) Kontrola ogólna 14) Kontrola generalna /HZD�VWURQD�UyZQD� 3UDZD�VWURQD�UyZQD��REVHUZDF\MQ\FK obserwacyjnych

15) Ocena dokáDGQR�FL transformacji

D��EáG\��UHGQLH�ZVSyáU]GQ\FK�SXQNWyZ�GRVWRVRZDQLD

43403

0.565195

25403

0.193313

.s

M

.s

M

Y

X

±≈±=±=

±≈±=±=

∑

∑

�

�

L<

L;

9

9

E��UHGQL�Eá�G�WUDQVIRUPDFML

503022 .MMM YXt ±=+±=

75907590

324213720208221372027590

..

...

=−=


( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) 665.2945

982.3185

275.1610

283.1993

417.2699

802.1799

003003030

003003030

002002020

002002020

001001010

001001010

=+⋅−−⋅−=

=+⋅−+⋅−=

=+⋅−−⋅−=

=+⋅−+⋅−=

=+⋅−−⋅−=

=+⋅−+⋅−=

YSxxCyyY

XSyyCxxX

YSxxCyyY

XSyyCxxX

YSxxCyyY

XSyyCxxX

6662945

9833185

2751610

2831993

4172699

8021799

30

30

20

20

10

10

30

30

20

20

10

10

.VY

.VX

.VY

.VX

.VY

.VX

Y

X

Y

X

Y

X

=+

=+

=+

=+

=+

=+


��:VSyáU]GQH�SXQNWyZ�GR�WUDQVIRUPDFML�Z�XNáDG]LH�Ztórnym

( ) ( )( ) ( ) 5,4,3,2,1;000

000

=+⋅−−⋅−=

+⋅−+⋅−=

jYSxxCyyY

XSyyCxxX

jjj

jjj

277.2961667.666667.666

008.1549667.666667.666

799.2967667.566667.66

676.2160667.66667.566

158.2554667.166667.166

019.2132667.166667.166

789.2384667.66667.566

556.1753667.566667.66

060.2629667.366667.866

750.1403667.866667.366

05

05

04

04

03

03

02

02

01

01

=+⋅+⋅−==+⋅−⋅−=

=+⋅+⋅−=

=+⋅−⋅−=

=+⋅+⋅−=

=+⋅−⋅−=

=+⋅+⋅−=

=+⋅−⋅−=

=+⋅+⋅−==+⋅−⋅−=

YSCY

XSCX

YSCY

XSCX

YSCY

XSCX

YSCY

XSCX

YSCY

XSCX

��=HVWDZLHQLH�Z\QLNyZ�REOLF]H�

Nr 5y*QLFH�ZVSyárz.

WspóáU]GQH wyrównane

�XNáDG�ZWyUQ\� 3RSUDZNL�ZVSyáU]� pkt x - xo y - yo X Y Vx Vy

10 -366.667 -466.667 1799.802 2699.417 -0.198 -0.583

20 733.333 -466.667 1993.283 1610.275 -0.161 0.457

30 -366.667 933.333 3185.982 2945.665 0.359 0.127

1 -366.667 -866.667 1403.750 2629.060 k = 1.005631

2 -66.667 -566.667 1753.556 2384.789 α = 88.80752g

3 -166.667 -166.667 2132.019 2554.158 Mx = 0.254

4 -566.667 -66.667 2160.676 2967.799 My = 0.434

5 -666.667 -666.667 1549.008 2961.277 Mt = 0.503

Przydatne wzory matematyczne 92

Dodatek A: Przydatne wzory matematyczne Pochodne funkcji

-�)XQNFMD�VWDáD�� ( ) ( )Rx;c:cxf / ∈=•= 0

-�)XQNFMD�SRWJRZD�� ( ) ( )+− ∈⇒∈∈⇒∈⋅=• RxWn,RxNn;xnx n/n 1

-�)XQNFMD�Z\NáDGQLF]D�� ( ) ;ee x/n =• e – podstawa logarytmów naturalnych

- Funkcja logarytmiczna: ( ) ( ) ( )}{Ra;alnx

xlog;x

xln/

a

/111 −∈

⋅=•=• +

- Funkcje trygonometryczne: ( ) ( ) ;xsinxcos;xcosxsin// −=•=•

( ) ( ) ( ) ( )011011 2

2

2

2 ≠−−=−=•≠+==• xsin;xctgxsin

xctg;xcos;xtgxcos

xtg//

- Funkcje cyklometryczne: ( ) ( ) ( )111

1

1

122

<<−−−=•

−=• x;

xxarccos;

xxarcsin

//

( ) ( ) ( )Rx;x

xarcctg;x

xarctg// ∈

+−=•

+=•

22 11

11

-�)XQNFMH�]áR*RQH�� ( ) ( )( ) ( );xfxfgfg /// ⋅=• $ g –�I��]HZQWU]QD��I�–�I��ZHZQWU]QD

-�-H*HOL�IXQNFMH� ( )xf i ( )xg �V��Uy*QLF]NRZDOQH�Z�GDQ\P�SXQNFLH��SU]HG]LDOH��WR� ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( );Ra,xfaxfa;xgxfxgxf ///// ∈⋅=⋅•±=±•

( ) ( )[ ] ( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )[ ]2xg

xgxfxgxfxgxf

;xgxfxgxfxgxf///

/// ⋅−⋅=

•⋅+⋅=⋅•

Trygonometria

-�=ZL�]NL�PLG]\�IXQNFMDPL�� ;ctgtg;cossin 1122 =⋅•=+• αααα

;tg

tgtg;sinsincoscos;cossinsin

αααααααααα 2

222

1

221222

−⋅=•−=−=•⋅⋅=•

ααα

ααααααα

sincos

ctg;sincos

tg;cos

cos;cos

sin+=•−=•+±=•−±=• 1

21

221

221

2

-�6XP\�L�Uy*QLFH�

( ) ( ) ;sincoscossinsin;sincoscossinsin βαβαβαβαβαβα ⋅−⋅=−•⋅+⋅=+•

( ) ( ) βαβαβαβαβαβα sinsincoscoscos;sinsincoscoscos ⋅+⋅=−•⋅−⋅=+•

- Wzory redukcyjne:

ϕ α α−°180 α+°180 α−

ϕsin αsin αsin αsin− αsin− ϕcos αcos αcos− αcos− αcos

ϕtg αtg αtg− αtg αtg−

ϕctg αctg αctg− αctg αctg−

ϕ α−°90 α+°90 α−°270 α+°270

ϕsin αcos αcos αcos− αcos− ϕcos αsin αsin− αsin− αsin

ϕtg αctg αctg− αctg αctg−

ϕctg αtg αtg− αtg αtg−

Przydatne wzory matematyczne 93

3ROH�WUyMN�WD

( ) ( ) ( ) ( )

++⋅=−⋅−⋅−⋅=

=⋅⋅⋅=

=⋅⋅=∆

cbap;cpbpapp

sinba

haP

21

21

21

γ

Wzór sinusów i cosinusów

αγβα coscbcba;sin

csin

bsin

a ⋅⋅⋅−+=•==• 2222

Szereg Taylora (dla funkcji jednej zmiennej)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ...x!

xfx

!

xfx

!

xfxfxxf

//////

+∆⋅+∆⋅+∆⋅+=∆+ 3020000 321

:VSyáF]\QQLN�NLHUXQNRZ\�SURVWHM �Z�XNáDG]LH�JHRGH]\MQ\P�

( )1212

12 xx;xx

yytgm ≠

−−== α

2EUyW�XNáDGX�R�N�W�ϕϕ

ϕϕϕϕ

cosysinx'y

sinycosx'x

⋅+⋅=⋅−⋅=

)XQNFMD�SRWJRZD

( ) ( )xxxyxyxyx

y

xyxyxn mn

m

n

n baba;aa;aaa;aaa;aa;

aa ⋅=⋅•=•=•=⋅•=•=• ⋅−+− 1

6WDáH�Patematyczne

π = 3,141 592 654...

ρ° = 57°,295 779 513...

ρ′ = 3 437′,746 771...

ρ″ = 206 264″,806...

ρg = 63g,661 977 237...

ρc = 6 366c,197 724 ...

ρcc = 636 619cc, 772 ..

α

�

β c

a

b

h

α

(x1,y1)

(x2,y2) x

Podstawowe wzory z rachunku wyrównawczego 94

Dodatek B: Podstawowe wzory z rachunku wyrównawczego 3UDZR�SU]HQRV]HQLD�VL�EáGyZ��UHGQLFK�REVHUZDFML�QLH]DOH*Q\FK��SU]\NáDG��

( ) 2222

dx mdxmx ⋅

∂∂+

⋅

∂∂±=→=

ραα αPG�I�[

%á�G��UHGQL�REVHUZDFML��SRPLDU�MHGQHM�ZLHONR�FL��

[ ] [ ]1n

pvvm;

1nvv

m 0 −±=•

−±=•

�UHGQLD�DU\WPHW\F]QD�]Z\NáD�L�ZD*RQD�

[ ] [ ][ ]ppL

x;nL

x =•=•

%á�G��UHGQL��UHGQLHM�DU\WPHW\F]QHM��]Z\NáHM�L�ZD*RQHM��

[ ]p

mm;

n

mm xx

0±=•±=•

Kontrola ogólna�Z\UyZQDQLD�REVHUZDFML�EH]SR�UHGQLFK��

[ ] [ ] [];

nl

llvv2

−=• [ ] [ ] [ ][ ]ppl

pllpvv2

−=•

5R]ZL�]DQLH�QDGRNUH�ORQHJR�L�QLHGRRNUH�ORQHJR�XNáDGX�UyZQD��OLQLRZ\FK�

LAAAXLAAAX ⋅⋅⋅=•⋅⋅⋅=• ��7 �� T; TT

5R]NáDG�PDFLHU]\�QRUPDOQHM�QD�F]\QQLNL�WUyMN�WQH�L�MHM�RGZURWQR�ü�

;N T RRR →⋅=• ;11 −− →=⋅• RIRR ( )TN 11 −− ⋅=• RR�

2EOLF]HQLH�QLHZLDGRP\FK�L�SRSUDZHN�Z�PHWRG]LH�SR�UHGQLF]�FHM� ;lub LPAA)P(AXLAA)(AX T-1TT-1T ⋅⋅⋅⋅⋅=•⋅⋅⋅=• LXAV −⋅=•

Kontrola ogólna w metodzie�SR�UHGQLF]�FHM� ;XALLLVV TTT ⋅⋅−⋅=⋅• XAPLLPLVPV TTT ⋅⋅⋅−⋅⋅=⋅⋅•

Bá�G��UHGQL�REVHUZDFML (pomiar k�ZLHONR�FL�� ;

knm

−⋅±=• VV T

knm0 −

⋅⋅±=• VPV T

Macierz wariancyjno-kowariancyjna niewiadomych:

;��7 �� AAQ ⋅=• ��7 �� APAQ ⋅⋅=•

%á�G��UHGQL�QLHZLDGRPHM�

;Qmm iiix ⋅±= 0 (Qii –�HOHPHQW�SU]HN�WQLRZ\�PDFLHU]\�NRZDULDQF\MQHM�

Obliczenie wektora korelat w metodzie warunkowej:

;W)A(AK -1T ⋅⋅=• W)AP(AK -1T ⋅⋅⋅=• −1

Podstawowe wzory z rachunku wyrównawczego 95

Obliczenie wektora poprawek w metodzie warunkowej:

;KAW)A(AAV T-1TT ⋅=⋅⋅⋅=• KAPW)AP(AAPV T-1TT ⋅⋅=⋅⋅⋅⋅⋅=• −−− 111

Kontrola ogólna w metodzie zawarunkowanej:

;KWVV TT ⋅=⋅• KWVPV TT ⋅=⋅⋅•

5yZQDQLH�REVHUZDF\MQH�GOD�N�WD�

SL

SL

SP

SPSLSP xx

yyarctg

xx

yyarctgAAv

−−−

−−=−=+ 11β

Równanie obserwacyjne dla kierunku:

zxx

yyarctgzAvK −

−−=−=+

12

121211

RównaQLH�REVHUZDF\MQH�GOD�GáXJR�FL�

( ) ( )22

jkjkjkii yyxxdvd −+−==+

5yZQDQLH�EáGX�GOD�N�WD��SR�UR]ZLQLFLX�Z�V]HUHJ�7D\ORUD��

;lybxaybxaybxav iPPPPLLLLSSSSi +∆⋅−∆⋅−∆⋅+∆⋅+∆⋅+∆⋅=•

;d

xxb;

d

yya

)P(LS

S)P(L)P(L

)P(LS

S)P(L)P(L ρρ ⋅

−−=•⋅

−=•

−−22 LPSLPS bbb;aaa −=•−=•

5yZQDQLH�EáGX�GOD�GáXJR�FL��SR�UR]ZLQLFLX�Z�V]HUHJ�7D\ORUD��

ikjkkjkjjkjjki lyAsinxAcosyAsinxAcosv +∆⋅+∆⋅+∆⋅−∆⋅−=

%á�G�SRáR*HQLD�SXQNWX:

22

yxP mmm +±=

3DUDPHWU\�HOLSV\�EáGX��UHGQLHJR� ( ) ;

QQ

Qtg

yyxx

xy

−=•

22α ;cossinQsinQcosQma xyyyxx αααα ⋅⋅⋅+⋅+⋅⋅±=• 222

αααα cossinQsinQcosQmb xyxxyy ⋅⋅⋅−⋅+⋅⋅±=• 222

%á�G��UHGQL funkcji��XRJyOQLRQH�SUDZR�SU]HQRV]HQLD�VL�EáGyZ��UHGQLFK�:

;m x

T

F fQf ⋅⋅=• 2 ( ) ;APAm T 12

0

−⋅⋅⋅=• xQ

∂∂

∂∂

∂∂=• �

321 xF

xF

xFT

f

5yZQDQLH�SRSUDZNL��ÄREVHUZDF\MQH´��GOD�ZVSyáU]GQ\FK�SXQNWyZ�GRVWRVRZDQLD�SU]\�wyrównaniu transformacji:

( ) ( )( ) ( ) 000

000

YsinkxxcoskyyVY

XsinkyycoskxxVX

iiYii

iiXii

−⋅⋅−−⋅⋅−=+−⋅⋅−+⋅⋅−=+

αααα

2FHQD�GRNáDGQR�FL�Z\UyZQDQLD�WUDQVIRUPDFML�

;s

M;s

M YX

∑∑ ±=•±=•��L<L;

9922YXt MMM +±=•

6áRZQLF]HN�ZD*QLHMV]\FK�WHUPLQyZ�]�UDFKXQNX�Z\UyZQDZF]HJR 96

Dodatek C: 6áRZQLF]HN� ZD*QLHMV]\FK� WHUPLQyZ� ]� UDFKXQNX� Z\Uyw-nawczego

%á�G�SR]RUQ\ –�SRSUDZND�GR�REVHUZDFML��ZLHONR�ü�Z\UyZQDQD�PLQXV�Z\QLN�SRPLDUX�� %á�G�SU]\SDGNRZ\ –�Eá�G�SRPLDURZ\��NWyUHJR�ZDUWR�FL�DQL�]QDNX�QLH�PR*QD�XVWDOLü� %á�G� �UHGQL –� SHZQD�PLDUD� GRNáDGQR�FL� GDQHM� ZLHONR�FL��PR*H� GRW\F]\ü� SRMHG\QF]HM�REVHUZDFML��]ELRUX�REVHUZDFML��ZLHONR�FL�QLHZLDGRP\FK��QS��ZVSyáU]GQ\FK��IXQNFML�QLe-wiadomych itp.

(OLSVD� EáGX� �UHGQLHJR� –� SRND]XMH� NLHUXQNL� Z\VW�SLHQLD� PDNV\PDOQHM� �Pinimalnej) warto�FL�EáGyZ��UHGQLFK�ZVSyáU]GQ\FK� Kontrola generalna wyrównania –� SROHJD� QD� VSUDZG]HQLX� ]JRGQR�FL� Z\UyZQDQHJR�XNáDGX� REVHUZDFML� SRSU]H]� �]DOH*QLH�RG�PHWRG\�Z\UyZQDQLD�� D�� SRGVWDZLHQLH�Z\Uyw-nanych niewiadomych i obserwacji do UyZQD��REVHUZacyjnych, b) podstawienie wy-równanych obserwacji do UyZQD��ZDUXQNRZ\FK.

/LF]ED�REVHUZDFML�QLH]EGQ\FK – liczba obserwacji koniecznych do jednoznacznego okre�OHQLD��UR]ZL�]DQLD��]DGDQLD��UyZQD�MHVW�OLF]ELH�QLHZLDGRP\FK� Macierz wariancyjno-kowariancyjna niewiadomych – macierz kwadratowa syme-WU\F]QD� �RGZURWQR�ü� PDFLHU]\� QRUPDOQHM�� R� Z\PLDU]H� UyZQ\P� OLF]ELH� QLHZLDGRP\FK��Elementy przeN�WQLRZH��ZDULDQFMH��VáX*��GR�REOLF]HQLD�EáGyZ��UHGQLFK�QLHZLDGRP\FK��QDWRPLDVW�HOHPHQW\�SR]DSU]HN�WQLRZH��NRZDULDQFMLH��VWDQRZL��SRGVWDZ�REOLF]HQLD�Eá�GyZ� �UHGQLFK� ZLHONR�FL� ]aOH*Q\FK�� QS�� EáGyZ� �UHGQLFK� IXQNFML� OXE� SDUDPHWUyZ� HOLSV\�EáGX��UHGQLHJR� 0HWRGD� SR�UHGQLF]�FD – wyrównanie zbioru obserwacji (na podstawie met. najmn. NZ��Z�RSDUFLX�R�Z\UyZQDQH�ZF]H�QLHM�SDUDPHWU\�SR�UHGQLF]�FH��QLHZLDGRPH�. 0HWRGD�Z\UyZQDQLD�REVHUZDFML�EH]SR�UHGQLFK – dotyczy wyrównania wielokrotnego poPLDUX�SRMHG\QF]HM�ZLHONR�FL��:LHONR�ü�Z\UyZQDQD��QD�SRGVWDZLH�PHW��QDMPQ��NZ��WR��UHGQLD�DU\WPHW\F]QD�]Z\NáD lub ZD*RQD.

Metoda zawarunkowana (warunkowa) – wyrównanie zbioru obserwacji (na podstawie PHW��QDMPQ��NZ��EH]SR�UHGQLR�EH]�XG]LDáX�QLHZLDGRP\FK��2EOLF]HQLH�Z\UyZQDQ\FK�QLe-ZLDGRP\FK�PR*OLZH� MHVW�SR� ]DNR�F]HQLX�SURFHVX�Z\UyZQDQLD�REVHUZDFML��Z�RSDUFLX�R�wyrównane obserwacje.

Obserwacja – wynik pomLDUX��VSRVWU]H*HQLH� Obserwacje jednorodne –�Z\QLNL�SRPLDUX� MHGQHJR� URG]DMX�ZLHONR�FL��QS��VDP\FK�N��WyZ�OXE�VDP\FK�GáXJR�ci; - niejednorodne –� ]ELyU�REVHUZDFML� Uy*QHJR�URG]DMX�ZLHONo-�FL��QS��Z�FL�JX�SROLJRQRZ\P�Z\VWSXM��]DUyZQR�REVHUZDFMH�N�WRZH�MDN�L�OLQiowe.

2GFK\áND�UyZQDQLD�RGFK\áHN –�Uy*QLFD�SRPLG]\�ZDUWR�FL��WHRUHW\F]Q��UyZQDQLD�Za-UXQNRZHJR� L� MHM�ZDUWR�FL��SU]\EOL*RQ�� REOLF]RQ��SR�SRGVWDZLHQLX�QLHZ\UyZQDQ\FK�Rb-serwacji).

3DUDPHWU\�HOLSV\�EáGX��UHGQLHJR –�ZDUWR�FL�ZLNV]HM��a) i mniejszej (b��SyáRVi elipsy (obliF]RQH�]�SUDZD�SU]HQ��VL�Eá��U��ZLHONR�FL�VNRUHORZDQ\FK�GOD�ZVSyáU]GQ\FK�SXQNWX��RUD]�N�W�VNUFHQLD�α��REOLF]RQ\�MDNR�ZDUXQHN�Z\VW�SLHQLD�HNVWUHPDOQ\FK�ZDUWR�FL�a i b).

3UDZR�SU]HQRV]HQLD�VL�EáGyZ��UHGQLFK��SR]ZDOD�RNUH�OLü�ZDUWR�ü�EáGX��UHGQLHJR�ZLHONR�FL�REOLF]RQHM�QD�SRGVWDZLH�SRPLDUyZ��NWyU\FK�Z\QLNL�REDUF]RQH�V��]QDQ\PL��Zy-OLF]RQ\PL�OXE�]DáR*RQ\PL��EáGDPL��UHGQLPL�� Redundancja – liczba obserwacji QDGZ\PLDURZ\FK� �QDGOLF]ERZ\FK�� QDGRNUH�ODM��cych) równa liczbie wszystkich obserwacji PLQXV�OLF]ED�REVHUZDFML�QLH]EGQ\FK��5yZQD�jest liczELH�QLH]DOH*Q\FK�warunków�]DFKRG]�F\FK�SRPLG]\�REVHUZDFMDPL�

6áRZQLF]HN�ZD*QLHMV]\FK�WHUPLQyZ�]�UDFKXQNX�Z\UyZQDZF]HJR 97

5yZQDQLD� EáGyZ –� SU]HNV]WDáFRQH� UyZQDQLD� REVHUZDF\MQH� GR� SRVWDFL� OLQLRZHM�� 3o-SUDZNL�Z\UD*RQH�V�� MDNR�IXQNFMH�OLQLRZH�SU]\URVWyZ�QLHZLDGRP\FK��:VSyáF]\QQLNL�SU]\�QLHZLDGRP\FK�RWU]\PXMHP\�Z�Z\QLNX�]Uy*QLF]NRZDQLD��Z]JOGHP�SRV]F]HJyOQ\FK�QLe-wiaGRP\FK��IXQNFML�]�UyZQD��REVHUZDF\MQ\FK� Równania obserwacyjne (poprawek) –� REVHUZDFMH�Z\UD*RQH� MDNR� IXQNFMH� QLHZLDGo-P\FK� �SDUDPHWUyZ�SR�UHGQLF]�F\FK��:\UyZQDQH�ZDUWR�FL� REVHUZDFML� L� QLHZLDGRP\FK�SRZLQQ\�VSHáQLDü� WH�]ZL�]NL�Z�VSRVyE��FLVá\� �kontrola generalna wyrównania). Rów-QD��REVHUZDF\jnych jest tyle, ile wszystkich obserwacji.

Równania RGFK\áHN –� SU]HNV]WDáFRQH� UyZQDQLD� ZDUXQNRZH� GR� SRVWDFL� OLQLRZHM�� 5RO�nieZLDGRP\FK� Z� W\FK� UyZQDQLDFK� SHáQL�� SRSUDZNL�� :VSyáF]\QQLNL� SU]\� SRSUDZNDFK�otrzymuMHP\�Z�Z\QLNX�]Uy*QLF]NRZDQLD��Z]JOGHP�SRV]F]HJyOQ\FK�REVHUZDFML��UyZQD��warunkowych.

Równania warunkowe (warunki) –��FLVáH�]ZL�]NL�PDWHPDW\F]QH��MDNLH�SRZLQQ\�]DFKo-dzLü� SRPLG]\� REVHUZDFMDPL� :\UyZQDQH� ZDUWR�FL� REVHUZDFML� SRZLQQ\� VSHáQLDü� WH�]ZL�]NL� Z� VSRVyE� �FLVá\� �kontrola generalna wyrównania). Liczba warunków równa jest liczbie obserwacji nadwymiarowych.

6WDáD�RULHQWDFML�VWDQRZLVND – D]\PXW�Ä]HUD´�OLPEXVD��Uy*QLFD�SRPLG]\�D]\PXWHP�Ga-QHJR� NLHUXQNX� D�ZDUWR�FL�� WHJR� NLHUXQNX�� F]\OL� RGF]\WHP�QD� OLPEXVLH�� 3U]\� SRPLDU]H�kierunków wyVWSXMH�MHGQD�VWDáD��QLHZLDGRPD��RULHQWDF\MQD�QD�ND*G\P�VWDQRZLVNX� Transformacja Helmerta (wyrównanie) – liczba punktów dostosowania s > 2. Wyrów-nanie polega na obliczeniu poprawek Vx, Vy�GR�ZVSyáU]GQ\FK�SXQNWyZ�GRVWRVRZDQLD��5RO�QLHZLDGRP\FK�Z�SURFHVLH�Z\UyZQDQLD�PHWRG�� SR�UHGQLF]�F��SHáQL��SDUDPHWU\�transformacji: wekWRU� SU]HVXQLFLD�X0, Y0�� N�W� VNUFHQLD�α� XNáDGX�ZWyUQHJR�Z]JOGHP�pierZRWQHJR�RUD]�ZVSyáF]\QQLN�]PLDQ\�VNDOL�k��/LF]ED�UyZQD��SRSUDZHN�UyZQD�MHVW�So-dwójnej liczbie punktów dostosowania.

8NáDG�QDGRNUH�ORQ\�UyZQD�� OLQLRZ\FK�–�XNáDG��Z�NWyU\P�OLF]ED�UyZQD�� MHVW�ZLNV]D�RG�OLF]E\�QLHZLDGRP\FK��3U]\NáDGHP�XNáDGX�QDGRNUH�ORQHJR�MHst XNáDG�UyZQD��EáGyZ (przyrówQDQ\�GR�]HUD��Z�PHWRG]LH�SR�UHGQLF]�FHM� 8NáDG�QiedoRNUH�ORQ\�UyZQD�� OLQLRZ\FK�–�XNáDG��Z�NWyU\P�OLF]ED�UyZQD�� MHVW�PQLHj-V]D�RG� OLF]E\�QLHZLDGRP\FK��3U]\NáDGHP�XNáDGX�QLHGRRNUH�ORQHJR� MHVW�XNáDG� UyZQD��odchyáHN w metodzie zawarunkowanej.

Waga obserwacji –�UDQJD�SRPLDUX��OLF]RQD�QDMF]�FLHM�MDNR�RGZURWQR�ü�NZDGUDWX�EáGX��UHGQLHJR�REVHUZDFML�� ,P�ZLNV]D�ZDJD� �PQLHMV]\�Eá�G�� W\P�ZLNV]\�ZSá\Z�GDQHM�Rb-serwacji na wyniki wyrównania.

:\UD]�ZROQ\�UyZQDQLD�EáGyZ – obliczamy jako ró*QLF�SRPLG]\�ZDUWR�FL��REOLF]RQ��obVHUZDFML��QD�SRGVWDZLH�QLHZLDGRP\FK�SU]\EOL*RQ\FK��D�Z\QLNLHP�SRPLDUX� :\UyZQDQLH��FLVáH –� ��ZV]\VWNLH� REVHUZDFMH�Z\UyZQ\ZDQH�V�� MHGQRF]H�QLH��So-SUDZNL� GR� ZV]\VWNLFK� REVHUZDFML� SRZLQQ\� E\ü� GREUDQH� ZHGáXJ� ]DVDG\� QDMPniejszych kwadratów. W wyrównaniu SU]\EOL*RQ\P�*DGHQ�]�W\FK�ZDUXQNyZ�QLH�PXVL�E\ü�VSHáQLo-ny.

Zasada najmniejszych kwadratów –�VXPD�NZDGUDWyZ�SRSUDZHN�SRZLQQD�VL� UyZQDü�minimum [ ]( ).minvvvvv =+++= �2

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UWAGA! 3RZ\*V]H�GHILQLFMH�QLH�V��FLVáH��PDM��RQH�FKDUDNWHU�RJyOQ\��6áRZQLF]HN�PD�QD� FHOX� SU]\SRPQLHQLH� SRMü� L� ]DJDGQLH�� MX*� ZF]H�QLHM� RSDQRZDQ\FK� SU]H]� VWXGHQWD��%DUG]LHM�V]F]HJyáRZH� LQIRUPDFMH�V��SRGDZDQH�Z� UDPDFK�Z\NáDGyZ�RUD]�Z�SRGUF]Qi-kach z rachunku wyrównawczego.

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Rachunek wyrównawczy - skrypt