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УНИВЕРЗИТЕТ „ГОЦЕ ДЕЛЧЕВ“ Факултет за природни и технички науки ШТИП РИСТО РИБАРОСКИ Г Е О Д Е З И Ј А ШТИП, 2014 г.

kniga Geodezija

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  • , 2014 .

  • 2

    .

    " "-

    -

    . -

    CIP - ". ", 528(075.8) , [ ] / . - , . , . - : " ", , 2014 (URL): http://e-lib.ugd.edu.mk/naslovna.php. - . - 02.10.2014. - : . 260-261 ISBN 978-608-244-101-6 ) - COBISS.MK-ID 97141514

  • _____________________________________________________________________________3

    . 2202-61/5 07.02.2014 .

    ,

    " "- .

    2011, .

    .

    " "-

    -

    . -

    -

  • 4

    , () , . , , , , , . II , 2003 . . , , . II , , .

    , , . , . , , , . -. . -. , . , , ..., , . .- () , . -. , . -. - . , .

  • _____________________________________________________________________________5

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . 4 . . . . . . . . . . . . . . . . . . 6 ................. ........................................................... 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 . . . . . . . . . 14

    . . . . . . . . . . . . . . 15 () ............................................ 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 ....................... 19 .................................................................... 20 ............. 22

    - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 .... 22

    , . . . . . . . . . . . . . . . . 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 ................................................................. 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ........................................................... 33 .............................................................................................................. 38 . . . . . . . . . . . . 39

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 . . . . . . . . . . . . . . . . . . 45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    ............................ 52 . . . . . . . . . . . . . 52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

  • 6

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 . . . . . . . . . . . . . . . . . . 62 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 . . . . . . . . . . . . . . . 66 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 . . . . . . . . . . . 71 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

    . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 . . . . . . . . . . . . . . . . . . . . 76

    . . . . . . . . . . . . . . . . . . . . 82 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 ..................................................................... 88 .......................................................................................................... 90 , ............................... 92 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    . . . . . . . . . 94 . . . . . . . . 95 . . . . . . . . . 96 , . . . . . . . . . . . . . . . . . . . . . 97

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

    . . . . . . . . . . . . . . . . . . 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 . . . 106 ................................................................................................... 109

  • _____________________________________________________________________________7

    .................................................................................................... 116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

    , GNSS GPS . 129 . . 130 ... 13 0 TOPCON........................... 13 4 WILD TC-1000 , GNSS I GPS............ 1 40 GPS LEICA VIVA GS08. . . . . . . . . . . . . . . . . 145 . . . . . . . . . . . . . . 149 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

    () . . . . . 150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 . . . . . . . . . . . . . . . . . . . . . . . . . . 152 . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

    . . . . . . . . . . . . . . . . . 155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 . . . . . . . . . . . . . . . . . . . 158 . . . . . . . . . . . . . 160 . . . . 276

    . . . . . . . . . . . . . . . . . 163 . . . . . . . . . . . . 163 . . . 165 . . . . . . . 167

    GPS - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 . . . . . . . . . . . . . . . . . . . . . . 175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 (), . . . . . . . . . . 179

  • 8

    (). . . . . . . . . . . . . . . . . . . . . . . 180 . . . . . . . . . . . . . . . . . . . . . . . . 182 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

    . . . . . . . . . . . . 189 . 192

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 .. . . . . . . . . . 194 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 .. . . . . . . . . . . . . . . . . . . . . . . . . . . 196 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

    .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 , . ................................................................................................ 204

    ......................... 206 ................................................................................. 206

    . . . . . . . . . . . . . . . . . . . 208 ...... 209 ......... 210 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

    . . . . . . . . . . . . . . . . 212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

    . . . . . . . . . . . . . . . . 213 . . . . . . . . . . . . . . . . . . . . . . . . . . 216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 . . . . . . . . . . . . . . . . 219

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 . . . . . . . . . . . . 227

    ...................................................................................... 227 ........................................................................................ 230 ............ 231

  • _____________________________________________________________________________9

    . . . . . . . . . . . . . . . . . . . . . . 234 ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 ............................ . . 234 .................................................................. 234 ........................................................ 235 ........................................................... 236 ...................................................................... 237 ................................................................... 237 .............................................. . . . . . . . . . 239 .............................. 240 ......................................... 241 ............................................... 244 - ................................................ 247 ............... 248 .................................... 249 ............................................................................... 250 ............................................ 251 ........................... 251 ..................................................................... 252 .................................................. 253 ................................. 253

    . .................................. 257 ....................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

    ..................... . . . . . . . . 258 .......................................... 258

    . . . . . . . . . . . . . . . . . 259

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

  • 10

    -

    -

    ,

  • _____________________________________________________________________________11

    () (), . , . , 1300 II , , , . , , .

    , , . harpedonaptes (). , .

    , . , , . , .

    (), .

    , . . , , , . , , . , .

    : , , -, ,

  • 12

    . , , , , , .

    , , , :

    (.. ) (.. ) . ,

    . , () , . , , , () . , () , , , .

    , , . () , ( ). : , , , , .

    , , , , .

    , , , , . , , - .

  • _____________________________________________________________________________13

    . ( 900 - 800 . ...). , . , (384 -322 ....) , () 400 000 (1 = 192 m), .

    (276-196 ....). . .. , . , ( ) . , , ( ). , , , .

    1528 . . , (Q=10.011km) 1% . .

    , , . 1615 1617 ., , . , .

    1669 . . (Q) 10 008,9 km, ().

    . 1672 .,

  • 14

    . , . , (1643-1727 .) . , . , , .

    , . 1718 . , . , 1735 . . , ( ) 1744 . , . ( ), 1736 1737 ., , , . , , , (. 1.1).

    (.1.1)

    , 18- , , , . 40 000 000- . .

  • _____________________________________________________________________________15

    (. 2.1)

    , , , 1872 . . , , . , , , (. 2.1 ).

    , . 50 m, 150 m. . , , , . .(. 3.1)

    (. 3.1)

  • 16

    , , . 1 . 1:

    ElipsoidGolema poluoska

    a (m)Mala poluoska

    b (m)Spleskanost

    Dalamber (1800) 6 375 653 6 356 564 1 : 334.0

    Besel (1841) 6 377 397 6 356 079 1 : 299.2

    Hajford (1909) 6 378 388 6 356 911 1 : 297.0

    Krasovski (1942) 6 378 245 6 356 863 1 : 298.3

    WGS (1984) 6 378 137 6 356 752 1 : 298.3

    , . , . . , 1984 ., WGS 84, , , .

    , , , .

    . ( ) , () .

    (,) , . () ( ) . 0 90 , 0 - 90 , , (. 4.1).

  • _____________________________________________________________________________17

    (. 4.1)

    () () ( ) . . 0 180, .

    , (. 5.1).

    (.5.1)

    (V) . . (N) . .

    () (.5.1). .

  • 18

    T1 ( S ) (.6.1). , , T T T1 , S , . 00 360 0 .

    , T1 S .

    (.6.1)

    . A. 0 360 .

    , - .. , . () P1P2 = 2b (.7.1).

  • _____________________________________________________________________________19

    (.7.1)

    , : P1- , P2- . (), , , ( E 1E E2). , , , (U1U U2 ). , , (P1E1P2E2).

    (.8.1), , :

    () - , 2a = 12 - , 2b = P1P2 - ,

    OFOF 21 == -

    (.8.1)

  • 20

    ( a b ) . , , . , :

    a

    e = ...(1.1)

    , :

    b

    e = ...(1.2)

    . (.8.1) :

    222 ba =

    ( 1.1) (1. 2) :

    222

    2

    abae = ...(1.3)

    2

    222

    bbae = ...(1.4)

    :

    aba

    = ...(1.5)

    , , . , :

    bba

    = ...(1.6)

    . :

    b

    ac2

    = ...(1.7)

    .

  • _____________________________________________________________________________21

    , ( a b), ( a ).

    M, , . , . , (.9.1).

    (.9.1)

    (P1MP2)

    M, . , M. (A1MA2). (B1MB2). . M . , (= 0), 90 (= 90 ).

    (M) :

  • 22

    32

    2

    1

    1

    )sine(

    )e(aM2

    = ....(1.8)

    (N) :

    221 sine

    aN

    = ...(1.9)

    ( ) , ( ) .

    ()

    , , (.10.1).

    (.10.1)

    , , , . , . , (M N) :

  • _____________________________________________________________________________23

    MNR = ...(1.10) 100km

    R= 6377 km.

    (), (.11.1).

    (.11.1)

    , , , . , , . . , ABCD A 1 B1 C1 D1 (.12.1).

    (.12.1)

  • 24

    , (.13.1).

    (.13.1)

    , , , . , , , . , , (.13.1).

    , . , , , , , .

    , . , .

  • _____________________________________________________________________________25

    ( - ), , , . , , .

    ( .14.1)

    , ( ) (.14.1). ( -) , , (.15.1).

    (.15.1)

  • 26

    . , , : 1. ; 2. ; 3. .

    , , : - ( ); - ( ); - ( ) . , .

    , .. , .

    . ( ) , . .

    , : - ( ) ; - ( ); - .

    () , - (.16.1)

  • _____________________________________________________________________________27

    (.16.1)

    (.17.1)

    ( ) , (.17.1).

    , 0 90 ( .18.1).

  • 28

    (.18.1)

    . -, , , .

    . . , :

    - ; - ; - ; - ; - ; - ;

    - .

  • _____________________________________________________________________________29

    -

    -

    . , - - 3 (.19.1).

    (.19.1)

    ,

    7- , 21 . . , 21- x-, , y-, . x y O (.20.1).

    (.20.1)

    , x , y .

  • 30

    y-, , x- 500 000 m. , y- 500 000 m, 500 000 m. y 7, 7- .

    ,

    , . , , . , 18- , .

    - , - . , , , 90% 10% . , (Mtre de archives), . 30 , , , 1872 . , . 6, , . (Bireau international des poids et mesures ) , . 30 , , , , .

    (90% 10% ) . , ( a - b ) 1 m. (.21.1)

  • _____________________________________________________________________________31

    (.21.1)

    SI , 1

    650 763,73 ( Kr 86 ) , 2P10 5d5.

    , , : 1/299 792 458 .

    : 1 dkm () = 10 m = 101 m. 1 hm () = 100 m = 102 m. 1 km () = 1000 m = 103 m. 1 Mm () = 106 m. 1 Gm () = 109m. 1 Tm () = 1012m. ,

    , .

    : 1 dm () = 0,1 m = 10-1 m. 1 cm () = 0,01 m = 10-2 m. 1 mm () = 0,001 m = 10-3 m 1 () = 0,001mm = 10-6 m 1 m () = 0,001 = 10-9 m

    , , . .

  • 32

    , . , , , , , , . :

    1 (, ) = 2,634 cm

    :

    1 in (inch , - ) = 2,54 cm 1 ft (foot , - ) = 12 in = 30,48 cm 1 yd (yard , ) = 3 ft =91,439 cm 1 mile ( - ) = 1639,34 m 1

    0,6858 m.

    :

    1 m2 ( ) = (1 1) m2

    1 a () = (10 10) m2 = 100 m2 1 dka () = (10 100) 2 =1000 m2 =10 a 1 ha () = (100 100) m2 = 10 000 m2 = 100 1 km2 ( )= (100 x 1000)2 =1000 000 m2 = 100 ha : 1 = 1600 . = 918,67 m2

    () , () , . () . . 10 ( ) . :

    10 = 60' ( ) 1' = 60" ( )

  • _____________________________________________________________________________33

    1g( ), . , , , .. 1g = 100 c = 100` ( ) 1 c = 100 cc = 100`` ( )

    :

    gogo :: 400360= ...(1.11)

    ( 1.11 ) :

    == ggo . 900109

    oog . 111

    910

    == ... (1.12)

    , .

    , , , . . , ( ) , (d) ( ) (.22.1).

    (.22.1)

    (.22.1), :

  • 34

    =d ...(1.13)

    :

    go 4003602 == ...(1.14)

    :

    go 200180==

    ... (1.15)

    () :

    3572957795157180 .. ooo

    o ==

    ''.)'(' 3438746771343760180 ==

    ...(1.16)

    ''206265806'.'206264')'6060180('' ==

    :

    7636619772463200 .. ggg

    g ==

    ccc

    c .)( 63661977246366100200 ==

    ...(1.17)

    cccccc

    cc .)( 6366207724636619100100200 ==

    ()

    :

    =

    == o

    or

    ... (1.18)

    () :

    === rrro ...(1.19)

    1. :

    498662 .g=

  • _____________________________________________________________________________35

    ggo . == 90109

    49866290 ..o =

    2487456 .oo =

    '''oo 551456 =

    2. :

    '''o 3749136 =

    oog . == 1119

    10

    82694136 .oo =

    82694136111 ..g =

    0299152 .gg =

    cccgg 9902152 =

    3. () , : r = 1, 3564:

    35641.r =

    roo =

    35641357 ..oo =

    7217277 .oo =

    '''o 184377 = 4. 0 :

    'o 35225 =

    ''

    r =

  • 36

    ''

    r 343813535

    =

    936883.r =

    5. , :

    rgg =

    936883763 ..gg =

    779256250 .g= cccg 9377250 =

    (),

    , . (1 :M) (1/M), M . 1 : M .

    D , d , :

    MDd 1

    = ... (1.20)

    O (1.20) :

    MDd = ...(1.21)

    : MdD = ...(1.22)

    (1.21) 1 : M, M. (1.22) (D), 1 :, d ( ) M.

    : 1: 250, 1: 500, 1: 1 000, 1: 2 000, 1: 5 000 1: 10 000.

  • _____________________________________________________________________________37

    , , . , 1: 500 1:1000. , .

    , : 1 : 25 000, 1 : 50 000, 1: 100 000 1: 200 000. 1: 200 000 .

    , , . , , , , , .

    6. d = 13, 7 mm 1: 2 500 :

    d= 13, 7 mm = 0, 00137 m M = 2 500

    D=d . M= 0, 0137. 2 500 = 34, 25 m

    D= 34, 25 m

    7. d 1 : 5 000 137, 53 m ?

    m.D 53137=

    5000=M

    027505000

    53137 ..MDd ===

    mm..d 527100002750 ==

    mm.d 527=

  • 38

    :

    - ? - ? - ? - ? - ? - ? - ? - ? - , ? - ? - - ? - ? - ? - 1 Mm ( ) ? - : 10 y () 100 m ? - m2 ? - m2 ? - ? - ? - : = 370 = 42g ? - 600 ? - 1 m ? - ? - 1 : 2500

    320 m ?

  • _____________________________________________________________________________39

    -

  • 40

    . , , . , (.1.2).

    (.1.2)

    , . , , . , , . , :

    1. , 20 km; 2. , 15 - 25 km; 2. , 9 - 18 km; 3. , 5 - 13 km; 3. , 3 - 7 km; 4. , 1 - 4 km.

    1. 27 8 , 35 (.2.2).

  • _____________________________________________________________________________41

    (.2.2)

    1. . , . , : .. , , , (.3.2).

    (.3.2)

    , , .. (.3.2). , , (.4.2).

  • 42

    (.4.2)

    , ( 1b ) ( 111 ,, ), (.5.2).

    (.5.2)

    , . 1 :

    msin

    asin

    csin

    b===

    1

    1

    1

    1

    1

    1

    ...(2.1)

    (2.1) 1a 1c ,..

    111

    11

    sinmsinsin

    ba == ...(2.2)

    :

    111

    11

    sinmsinsin

    bc == ...(2.3)

    , . 2b , , 1b , (.5.2).

  • _____________________________________________________________________________43

    , S (.6.2).

    (.6.2)

    (, , , , , ) , , 100 400 km (.7.2).

    (.7.2)

    2. 1. . 2. , 2. (.8.2).

    (.8.2)

  • 44

    3. 2. , , . 4. , .

    1, 2. 3. . . 3. 4. . , , .

    , , GPS , EUREF 1996, GPS (Global Positioning System) , , GPS .

    2004 . GPS , 1 : 25 000. GPS , ..

    (. 9.2)

    2010 ., GPS- , GNSS ( Global Navigation Satellite Sustem ) , (MAKPOS) (. 9.2). GPS - .

  • _____________________________________________________________________________45

    , 1. 1,20 1,40 m. (.10.2).

    (.10.2) (.11.2)

    0,25 x 0,25 x 0,75 m, , 3 - 5 cm . () (.11.2).

    , (.12.2), .

    (.12.2)

  • 46

    , . : , , , . (.13.2).

    (.13.2)

    1 .. . , 1 , , , , . , . , 1.

    , : , .

    (.14.2)

  • _____________________________________________________________________________47

    , .. . (.14.2)

    , , 900.

    . , (.15.2)

    (.15.2)

    .

    (.16.2)

    , (.16.2).

  • 48

    (.16.2) B :

    += AB ...(2.4)

    , , : () , , .

    () BA , A

    B, , x- ( x- ) . (.17.2).

    (.17.2)

    BA A B

    AB (.17.2).

    BA AB :

    180= BAAB ...(2.5)

    , (.17.2) :

    ( )( ) x

    yxxyy

    tgAB

    ABBA

    =

    = ...(2.6)

  • _____________________________________________________________________________49

    , BA :

    ( )( )AB

    ABBA xx

    yyarctg

    xyarctg

    =

    = ...(2.7)

    (.16.2), B, :

    ( ) ( )2222 ABABAB xxyyxyD +=+= ...(2.8) :

    BA

    BA

    ABcos

    xsin

    yD

    =

    = ...(2.9)

    :

    ( ) ( )BA

    ABBA

    ABAB

    cos

    xx

    sin

    yyD

    =

    = . ..(2.10)

    BA 45, :

    ( )

    yxyxtg BA

    +=+ 45 ...(2.11)

    (.18.2)

  • 50

    (.18.2) , 0 360. y x. .

    I

    (0o-90o) II

    (90o-180o) III

    (180o-270o) IV

    (180o-360o) AB yyy = + +

    AB xxx = + +

    xy

    .

    :

    I ++

    = x)(y)(tg =BA

    II +

    = x)(y)(ctg 90+= BA

    III

    = x)(y)(tg 180+= BA

    IV +

    = x)(y)(ctg 270+= BA

    ( ) .

    xy

    ,

    (90, 180

    270

    ) ,

    y x .

  • _____________________________________________________________________________51

    1. : 8976

    54

    23 i ,,

    :

    y x 2 21 660,32 14 515,78 3 21 536,46 14 483,16 4 21 100,71 15 218,51 5 22 801,43 14 819,16 6 21 790,78 13 635,21 7 20 944,14 13 424,70 8 21 269,97 14 467,91 9 22 142,96 14 444,01

    . y x 32+450=1200 14' 44'' : 2 21 660,32 14 515,78

    : 3 21 536,46 32 = 750 14' 44'' 14 483,16

    y = +123,86 x = +32,62 x+y=+156,48 x-y = -91,24 D = 128,08 . y x

    45+450=148012'52'' : 5 22 801,43 14 819,16 : 4 21 100,71 45 = 1030 12' 52''

    15 218,51 y = +1 700,72 x = - 399,35 x+y = +1 301,37 x-y = - 2 100,07 D = 1 746,98 . y x

    67+450=301002'13'' : 7 20 944,14 13 424,70 : 6 21 790,78 67 = 2560 02' 13''

    13 635,21 y = - 846,64 x = - 210,51 x+y = - 1 057,15 x-y = + 636,13 D = 872,42 . y x

    98+450=316034'06'' : 8 21 269,97 14 467,91 : 9 22 142,96 98 = 2710 34' 06''

    14 444,01 y = - 872,99 x = + 23,90 x+y = - 849,09 x-y = + 896,89 D = 873,32

  • 52

    . , BA ABD , (.19.2), y x :

    BAAB

    BAAB

    cosDx

    sinDy

    =

    = ...(2.12)

    (.19.2).

    (.19.2)

    ( AA x,y ) A y x, B( BB x,y ). (.19.2) :

    xxx

    yyy

    AB

    AB

    +=

    +=

    ...(2.13)

    2. 120

    12080 =169

    o1733 D=769,45m 80

  • _____________________________________________________________________________53

    ( 80y = 21100,71; 80x = 15218,51).

    ( )

    m,x

    m,y

    m,,,xxx

    m,,,yyy

    ,cos,cosDx

    ,)'''sin(sinDy

    ,xm,D

    ,y

    769,45

    9114467

    9721269

    9114467607505115218

    9721269261697121100

    60750337116745769

    261693317167

    511521845769

    71211003371167

    120

    120

    80120

    80120

    12080

    12080

    80

    8012080

    =

    =

    ==+=

    =+=+=

    ===

    +===

    ==

    ==

    ( 00 x,y ) , ( BBAA x,y,x,y ) (AB), .. ( BA , ) (.20.2).

  • 54

    (.20.2)

    AB. (.21.2).

    (.21.2)

    A B, CD ( yx ) 1 2 ( ), PB , PC, P1, P2 PD. (B) , :

    BC pp =0 ...(2.14)

  • _____________________________________________________________________________55

    B, C D, , :

    AB

    ABBA x

    yarctg

    =

    AC

    ACCA x

    yarctg

    = ...(2.15)

    AD

    ADDA x

    yarctg

    =

    (1) o . o ..

    BAo =1

    02 =CAo ...(2.16)

    33 =DAo

    , :

    3321 oooo ++= ...(2.17)

    o , :

    oB =

    0 += oC

    11 += o ...(2.18)

    22 += o

    3 += oD , ( ), . ( DCB ,, ) B, C D

    .,, DACA

    BA .

    v o , . , iv :

  • 56

    BBAv =1

    CCAv =2 ...(2.19)

    DDAv =3

    ,. . ][v =[( )]= 0 ...(2.20)

    o , . , , . v :

    6 2. 15 3. 25 4.

    , . , ( io ) :

    BBA po =1

    CCA po =2 ...(2.21)

    DDA po =3

    , i , :

    BB po +=

    CC po +=

    11 po += ...(2.22)

    22 po +=

    DD po += , , . 5. : - , - , , - ( iiio = ),

  • _____________________________________________________________________________57

    - ( o ),

    (noo i ][= ),

    - , : ii o +=

    - , (2.18), :

    on += ][][ -

    iiiv = , :

    0][ =v . 3. :

    Y [ ]m X [ ]m 10 13 952,73 19 486,84 13 17 779,66 18 903,69 15 15 289,86 16 752,50 32 13 776,62 17 487,66 52 15 050,18 19 361,79 87 17 386,05 16 758,22

    ' " ' "

    15 52 0 00 00 32 13 0 00 00 87 73 35 43 6 23 30 21 32 279 40 02 15 45 23 40 6 306 12 58 10 294 31 04 10 317 41 45 52 339 58 58

  • 58

    =5215 16 14' 44" =8715 89 50' 37"

    =3215 295 54' 41" =1015 333 56' 27" ???

    . . . iiio = ii o += f ' " ' " ' " ' " "

    15 52 16 14 44 0 00 00 16 14 44 16 14 45 -1

    87 89 50 37 73 35 43 14 54 89 50 28 +9

    32 295 54 41 279 40 02 14 39 295 54 47 -6

    6 306 12 58 322 27 43

    10 333 56 27 317 41 45 14 42 333 56 30 -3

    10 28 70 31 04 59 23

    13 45

    24 13

    =1332 70 31' 10" =1032 5 02' 03"

    =1532 115 54' 41" =5232 50 30' 03"

    . .. iiio = ii o += f ' " ' " ' " ' " "

    32 13 70 31 10 0 00 00 70 31 10 70 31 04 +6

    6 23 30 21 94 01 25

    15 115 54 41 45 23 40 31 01 115 54 44 -3

    10 5 02 03 294 31 04 30 59 5 02 08 -5

    52 50 30 03 339 58 58 31 05 50 30 02 +1

    24 03 70 31 04 59 23

    35 20

    59 23

  • _____________________________________________________________________________59

    . .

    (.22.2)

    T( 00 x,y ), , , , Ta Tb T (.22.2).

    , T, Ta, Tb.

    Ta Tb ba d.

    :

    ab

    abba xx

    yytg

    = ...(2.23)

    :

  • 60

    ba

    abba

    ab

    cos

    xx

    sin

    yyd

    =

    = ...(2.24)

    ba , a b , (T,Ta,Tb) , :

    ab

    babb

    baaa

    =

    =

    =

    ...(2.25)

    ,

    180 , ..: 180=++ ba ...(2.26)

    , :

    msin

    dsin

    dsin

    da

    b

    b

    a ===

    ...(2.27)

    (2.27) :

    ab

    ba

    sinmd

    sinmd

    =

    = ...(2.28)

    ad bd

    a b , :

    bbbaaa sindysindy == ;

    bbbaaa oscdxoscdx == ; ...(2.29)

    , 0y 0x , . , :

    bbaa yyyyy +=+=0

    bbaa xxxxx +=+=0 ...(2.30)

    :

    - (2.23) (2.24) ba d.

  • _____________________________________________________________________________61

    - (2.25) a , b (2.26) ,

    - sin

    d (m)

    (T,Ta,Tb),

    - (2.28) ad bd ,

    - (2.29) ,

    - (2.30) T.

    4. o61 o51 o23. .

    Y X

    =a 84 41' 52" bT 32 16 045. 69 71 284. 96

    =b 159 42' 05" aT 15 15 398. 05 69 571. 16

    + 647. 64 +1 713. 80

    =ba 20 42' 05" =d 1 832. 09 =m 1 896. 69

    == baaa 63 59' 47" =ad 1244. 34

    == babb 41 00' 00" =bd 1704. 68

    == ab 75 00' 13"

    180 00' 00"

    == aaa sindy +1 239. 02 == bbb sindy + 591. 38

    == aaa cosdx + 114. 99 == bbb cosdx -1 598. 81

    =+= aa yyy0 16 637. 07 =+= bb yyy0 16 637. 07

    =+= aa xxx0 69 686. 15 =+= bb xxx0 69 686. 15

  • 62

    , , , .

    (.23.2)

    a b , (.23.2), :

    a

    a

    a

    aa x

    yxxyytg

    =

    = ...(2.31)

    b

    b

    b

    bb x

    yxxyytg

    =

    = ...(2.32)

    (2.31) (2.32) :

    aaa tg)xx(yy = ...(2.33)

    bbb tg)xx(yy = ...(2.34) (2.33) (2.34) :

    bbaaab tg)xx(tg)xx(yy = ...(2.35) :

    bbaabaab tgxtgx)tgtg(xyy += ...(2.36) :

    ba

    bbaaabtgtg

    tgxtgx)yy(x

    += ...(2.37)

  • _____________________________________________________________________________63

    (2.37) x , ax , bx , :

    ba

    bababa tgtg

    tg)xx()yy(xx

    = ...(2.38)

    ba

    aababb tgtg

    tg)xx()yy(xx

    = ...(2.39)

    : Atg)xx()yy( babab =

    Btg)xx()yy( aabab = ...(2.40)

    Ctgtg ba = :

    CAxa = C

    Bxb = ...(2.41)

    (2.31) (2.32) :

    aaa tgxy =

    bbb tgxy = ...(2.42) , ,

    , ..

    bbaa yyyyy +=+=

    bbaa xxxxx +=+= ...(2.43)

    , , , :

    - ( ab xx ) ( ab yy ),

    - A, B C, (2.40),

    - ax bx , (2.41),

    - ay by , (2.42), ,

    - ( 0y , 0x ) (2.43).

  • 64

    5. o6 o15 o32. .

    T : 6 aT : 15 bT : 32

    =by 13 776. 62 =bx 17 487. 66 =a ' "

    322 27 43

    =ay 15 289. 86 =ax 16 752. 50 =b 94 01 25

    ab yy - 1 513. 24 = ab xx + 735. 16

    == babab tg)xx()yy(A + 8938. 14

    =atg

    =btg

    - 0.7683828

    - 14.2164739

    == aabab tg)xx()yy(B - 948. 36 == ba tgtgC + 13.4480911

    == aaa tgxy - 510. 70 ==

    CAxa + 664.64

    == bbb tgxy + 1 002.54 ==

    CBxb - 70.52

    =+=+= bbaa yyyyy0 14 779.16 =+=+= bbaa xxxxx0 17 417.14

  • _____________________________________________________________________________65

    :

    - ? - ? - ? ? - ? ? - , ? - ?

    ? - ? - ? -

    ? - ? - () ( )

    , ? -

    ? -

    ?

  • 66

  • _____________________________________________________________________________67

    . , (.1.3).

    (.1.3)

    , ',

    , (.1.3) , , , .

    . ( - 00) , 1 2(.2.3), ..

    = 2 - 1 ...(3.1)

    (.2.3)

  • 68

    , , (.3.3).

    (.3.3)

    1 2 (.4.3), :

    = ( 2 + 3600 ) - 1 ...(3.2)

    (.4.3)

    , , , , .

    (.5.3)

  • _____________________________________________________________________________69

    , (), (TA), , (TB). - . (1) , (A) , (2) (B) (.5.3).

    , .

    - , .

    (.6.3)

  • 70

    : , , , , , , . (.6.3).

    ,

    ,

    , . . . , , , (.7.3).

    (.7.3)

    . , (.7.3).

    , , .

    . , , , , .

    :

    - () (V- V); - () (H - H); - () (C - C)

  • _____________________________________________________________________________71

    - (L - L).

    (.8.3).

    (.8.3)

    , ,

    .. . () , , , . , . . , , . , , , (.9.3).

    (.9.3)

  • 72

    . ( ) (.10.3) , (.10.3). . .

    (.10.3) (.10.3)

    , , . () . , (.11.3).

    (.11.3)

    30, 20, 10, 2, 3 6 (.11.3).

  • _____________________________________________________________________________73

    , . Rp (.10.3) . . . , , . Lv (.10.3) , .

    (.20.3)

    , , . , , . , , . (.20.3)

    , . , .

  • 74

    (.21.3). , . , .

    (.21.3)

    , . , . , , . , , 2/3 .

    , .

    .

    (.22.3).

    , , : (CM) (L-L), (.22.3). , , , (.22.3).

  • _____________________________________________________________________________75

    (.22.3)

    , , . 2,26mm., 1 , . 2mm, . , , , (.23.3).

    (.23.3)

    . , .. (.24.3 ).

    (.24.3)

  • 76

    . , , , , .

    .. . ,

    , . :

    Rb

    = ...(3.9)

    : b - ;

    R - . (3.9) ,

    R, ( ), , . , , .

    , , R , (L-L) . , 1 () , ( ), ( ), (.25.3).

    (.25.3)

  • _____________________________________________________________________________77

    , - ( ), , .

    , . , , , . , : .

    , . , , . , :

    1. () , . , ,

    2. , , .

    () (.30.3).

    (.30.3)

    () , :

    - (),

  • 78

    - () - , .

    , , : , (.31.3).

    (.31.3)

    , , . , , ( ) , , (.32.3).

    (.32.3)

    , , . , , (.33.3). , . , . , (.33.3).

  • _____________________________________________________________________________79

    (.33.3)

    .

    , . , , , (.34.3)

    (.34.3)

    , , , . , . . .

  • 80

    . , , . .. . ( ), . .. , . ( ) , , .

    ,

    . , , (.35.3) (.35.3 ), .

    (.35.3) (.35.3)

    . , , . : , , (.36.3).

    (.36.3)

  • _____________________________________________________________________________81

    , . : , , . , , , , .

    . , .. . , , .

    , , , . , , , . , . . , , . , , , 2 - 3 mm . , , . .. .

    , . , . , , . 2-3 m ,

  • 82

    , . , , 2 5 mm.

    (.41.3)

    , . 4 - 6 . , (.41.3).

    () () . , , () . .

    . , . , , , . , . , , , , .

    , , :

    - , , ;

    - . , ;

  • _____________________________________________________________________________83

    - .

    - () , . , . , . 1-2 mm.

    . . , . . , , . .

    . , , . , .

    , , . , . . , .

  • 84

    ,

    . , , . , (V-V), (H-H), (C-C) (L-L) , :

    1 : (L-L) (V-V);

    2 : (V-V) ;

    3 : (H-H) (C-C) ;

    4 : (H-H) (V-V).

    . , , (1) (2), , (.42.3).

    (.42.3)

    , . , , . , , . , .

  • _____________________________________________________________________________85

    . ( ) (1 2), (k) (.43.3). .

    (.43.3)

    , . , (3). , (.44.3).

    (.44.3)

    , .

  • 86

    , (1 2). , , (.45.3).

    (.45.3)

    (1 2), , (3) . , . .

    ( ) (21 ),

    , (1 2)

    (21 ) (k1).

    . ,

    (22 )

    (3), (22 ) (k2)

    . .

    , (H-H) (C-C) , :

    ) )

  • _____________________________________________________________________________87

    . , A . (I) (.46.3) A (L1) . , (I-I) (H-H) (.47.3).

    (.46.3) (.47.3)

    (II) (.46.3) , (L2), (II-II). , , (H-H).

    (.47.3) L1 (I -I) L2 (II-II) (2C), :

    ( ) ( )CLL 2180 102 = ...(3.23) 2-3

    , , . (2C) , . L2 (-C). . A . A (KH-KH). , , A , .

  • 88

    , , L1 I 29226 42", L2 II 112 28 30" : L1 L2 180 2Np , . . 108, . , :

    ''4262292)180''3082112(2 += C

    ''1082 +=C (-) II :

    112 28 30- 54 = 112 2736

    , 112 2736. , (KH-KH) , (112 2736), A (.48.3).

    (.48.3)

    (H-H) (V-V), . , .

    () . A (.49.3).

  • _____________________________________________________________________________89

    (.49.3)

    , , . , , , I II A, . (II-I) , . , , .

    , , .

    (.50.3)

  • 90

    , , K1 K2 , (.50.3). K2 , K1 , , .

    . , : , , , , , , . , , .

    (.51.3)

    , (.51.3). , , .

    321 ,, 4 ,

    321 ,, 4 :

  • _____________________________________________________________________________91

    11 =

    122 =

    233 = ...(3.24)

    344 =

    . , , , , .

    . giro, , . , , , . , , , .

    , , , . ( 043210 PPPPPP ). , , , , .. ( 012340 PPPPPP ). , , , (.52.3).

    , . , . , . , C (.52.3), :

  • 92

    I

    II

    C P0 0' 0'' P1 1' 1'' P2 2' 2'' P3 3' 3'' P4 4' 4'' P0 5' 5''

    (.52.3)

    , ..

    2ii

    i

    +

    = ...(3.25)

    i i . , , . , , .

  • _____________________________________________________________________________93

    I

    II

    C P0 0'+ 0"+ 12 0+13/2 P1 1'+ 2 1"+ 11 1+13/2 P2 2'+ 3 2"+ 10 2+13/2 P3 3'+ 4 3"+ 9 3+13/2 P4 4'+ 5 3"+ 8 4+13/2 P0 0'+ 6 0"+7 5+13/2

    , (

    2

    13),

    . , 1 :

    0101011 213

    213 =

    +

    +== PP ...(3.26)

    , (2C), :

    I)II(c = 1802 ...(3.27) (3.27) II

    , I . (2C)

    , , . , , .

    , , . , . n :

    nA

    180= ...(3.28)

  • 94

    .

    , . , 1 12 , 2 - 10 , 2 - 8 , 3 - 6 , 3 - 4 4 - 3 .

    , , 2 () , 3 () .

  • _____________________________________________________________________________95

    :

    - ? - ? - ? - ? - ? - ? - ? - ? - ? - ? - ? ? - ,

    ? - ? - ? -

    ? -

    ? -

    ? - "" ? -

    ? -

    ? -

    ? -

    ? -

    , ?

  • 96

    ,

  • _____________________________________________________________________________97

    , , .

    , 4- . , ( ), . :

    ( p z )

    ( n,...,, 21 ),

    ( ns,...,s,s 21 ), . (.1.4)

    .

    (.1.4)

    , , (x,y) (P) (Z) , (p) (z) .

    , , .. . , ,

  • 98

    . .

    , , .. . , , , . (.2.4) .

    (.2.4)

    , , I . (II , III IV ), , . (.3.4) .

    (.3.4)

  • _____________________________________________________________________________99

    , .. (.4.4).

    . , , , .

    (.4.4)

    , ( 100 ) , , .. . , , (.5.4). .

    (.5.4)

  • 100

    , , .

    () . , , , .

    , . , , . ( ) , , , .

    , . . , , , , , , . , , , .

    , , , , , . , , . .. , , , , , , . , 1, .

    , , . :

  • _____________________________________________________________________________101

    - 1800 , , ;

    - . , 2: 1;

    - 2, 5 km , 2,0 km 1,5 km . 4,0 km.

    - , .

    , :

    - ;

    -, , . , , , , .

    - . , , , , , .

    - .

    - , , , , .

  • 102

    ,

    , . .

    . , : ( , , ) ( ). , , .

    , , (.6.4).

    (.6.4)

    , , (.7.4). .. .

  • _____________________________________________________________________________103

    (.7.4)

    , , .

    , . , , , .. , . 27.

    (.8.4)

  • 104

    , , . , , . , . 27 , .

    ,

    , , . , , .

    . , , .

    ( , , ), , . , , , , . , , , .

    , . . , : (.9.4).

  • _____________________________________________________________________________105

    (.9.4) ,

    . , , , . , . , . .

    , , , . . . , .

  • 106

    , , .

    (.10.4)

    (.10.4) D D r, ..

    rDD = ...(4.14)

    , r . r.

    (.11.4)

    BBC (.11.4), :

    2tghr = ...(4.15)

    ABB , :

    Dharcsin= ...(4.16)

    r :

  • _____________________________________________________________________________107

    21

    21

    =

    DhDD ...(4.17)

    :

    DhrDD2

    2== ...(4.18)

    (r) (4.15). , (4.14), D. , 1 cm.

    , :

    22 hDD = ...(4.18) D, (.21.4)

    : cosDD = ...(4.18)

    (4.16) .

    ,

    , , . , .

    , . , , . 3 , (.12.4).

  • 108

    (.12.4)

    , , , , : a, b c. :

    ( ) += 180 ...(4.23) :

    msin

    csin

    bsin

    a===

    ...(4.24)

    :

    msin

    a=

    ...(4.24')

    (b c). (4.24) :

    sinmc

    sinmb

    =

    = ...(4.25)

    (.13.4) (.14.4)

    , (.13.4), :

    cosccosba += ...(4.26)

  • _____________________________________________________________________________109

    (, ) a (.14.4). , , . , 180 ..

    180++ ...(4.27)

    f :

    ( ) ++= 180f ...(4.28)

    , :

    3fvvv === ...(4.29)

    , , :

    3

    3

    3

    fv

    fv

    fv

    +=+=

    +=+=

    +=+=

    ...(4.30)

    , : 180=++ ...(4.31)

    , (4.24) b c, ..

    sinsin

    asinmb == ...(4.32)

    sinsin

    asinmc == ...(4.33)

    , , (4.26).

    .

  • 110

    (.15.4)

    (.15.4) b c . (4.24). m :

    msin

    c=

    ...(4.34)

    :

    msin

    b=

    ...(4.35)

    , :

    mbarcsin= ...(4.36)

    , 180, ..

    ( ) += 180 ...(4.37) a, :

    sinma = ...(4.38) (4.26).

    (.16.4)

    , (.16.4), ..

  • _____________________________________________________________________________111

    cosbccba += 2222 ...(4.39)

    :

    cosbccba += 222 ...(4.39')

    , :

    acbcacos

    2

    222 += ...(4.40)

    abcbacos

    2

    222 += ...(4.41)

    : 180=++

    ,

    . () b c (.16.4), , :

    2

    2

    +

    =+

    tg

    tg

    cbcb ...(4.42)

    ( + )/2, :

    290

    2

    =+ ...(4.43)

    ( + )/2 (4.43), ( )/2, :

    22 ctg

    cbcbtg

    +

    = ...(4.44)

    :

    ( )( )

    +

    =

    2

    2 ctg

    cbcbarctg ...(4.44)

  • 112

    ( + )/2 ( )/2, ..

    22

    22

    +=

    +

    +=

    ...(4.45)

    , :

    sinc

    sinb

    sina

    ==

    a, , ..

    sinsin

    csinsin

    ba == ...(4.46)

    ( )/2 . b>c , b

  • _____________________________________________________________________________113

    2. b c ABC = 48o3400, = 64o2600 = 67o0100 a = 138,28 m h = 4,12 m.

    ( ) 0010180 =++ 023001 ===fvf

    000067

    045264

    043348

    =

    =

    =

    ( ) 0000180 =++

    45645184

    2813822

    ,sin

    am

    ,haa

    ==

    ==

    m,sinmc

    m,sinmb

    81169

    39166

    ==

    ==

    3. ABC b = 166,39 m c = 169,81 m = 48o3400.

    02 001067

    02 006264

    02 004348

    =

    =

    =

  • 114

    004348

    81169

    39166

    =

    =

    =

    m,c

    m,b

    m,,a

    ,,,a

    cosbccba

    3013812419126

    972373944642883563227685

    222

    ==

    +=

    +=

    235264

    4464696995620275

    2

    222

    =

    =

    +=

    ,,cos

    acbcacos

    230067

    4744602329217976

    2

    222

    =

    =

    +=

    ,,cos

    abcbacos

    230067

    235264

    30138

    =

    =

    =

    m,a

    4. a1 ( 15) , .

    Y X 15 46 332,26 52 338,26 16 46 826,28 52 422,31

    .

    2 1 0o 00' 00" 15 48 56 20 3 111 42 50 2 60 36 00 15 102 12 40 16 173 44 20 32108

    49111

    22336108

    88254111

    21

    221

    22

    222

    11

    22

    ,hcc

    ,haa

    ,h,c

    ,h,a

    '

    '

    '

    '

    ==

    ==

    ==

    ==

  • _____________________________________________________________________________115

    2,3,15

    1,2,15

    2,3,15

    056375

    0132104

    036462

    046341

    2

    2

    2

    =

    =

    =

    =

    m,sinmc

    m,sinmb

    ,sin

    am

    35102

    4376

    099115

    22

    21

    2

    2

    ==

    ==

    ==

    37929802

    22

    7518432108

    89314376

    11

    1

    11

    1111

    111

    111

    ,tg

    ctgcbcbtg

    ,cb,c

    ,cb,b

    =

    =+

    =

    =+=

    ==

    m,a

    ,sinsin

    csinsin

    ba

    8481

    8481

    1

    11

    11

    1

    11

    =

    ===

    ( )

    041371

    210111

    808197

    180

    21011119371930

    12501

    3

    3

    3

    333

    333

    23

    3

    2

    3

    3

    223

    =

    =

    =

    +=

    ====

    =+=

    ,sinacsin

    sinc

    sina

    m,xya

    ( )

    8454100

    360 132

    =

    ++=

  • 116

    , . , , , , , .

    , , 1:10 000. , , , , . , , , (.17.4).

    (.17.4)

    , , . , , , , . , , , , , . , , .

  • _____________________________________________________________________________117

    , , : - , , - - .

    :

    1801 += 'nnn ...(4.47)

    (4.47) , , ,

    (+180) (-180

    ).

    .

    (.18.4)

    32 , , 2 2, (.18.4), :

    180221

    32 += ...(4.48)

    32

    21 ( ) 2 , (.19.4), :

    180221

    32 ++= ...(4.49)

  • 118

    (.19.4)

    (4.47), (4.48) (4.49), (.18.4) (.19.4) , 180. , ( ) 180, 180 , .

    180, 360

    , 360

    . , (.20.4).

    (.20.4)

    (.32.4), :

    1

    1

    +

    +

    =

    =

    iiii

    iiii

    cosdx

    sindy

    ...(4.50)

  • _____________________________________________________________________________119

    ( 21.4).

    (.21.4)

    , , : ( BBAA x,y,x,y ) ,

    ( AC ) (DB ) , ,

    A, B, C D. (A, 1, 2, 3, B) (d1, d2, d3 d4), .

    (4.47), (4.48) (4.49), (.33.4), :

    1805180

    1804180

    1803180

    1802180

    180

    3213

    3213323

    21221

    32

    1112

    1

    1

    +++++=+=

    ++++=+=

    +++=+=

    ++=+=

    +=

    BAACB

    BDB

    AAC

    B

    AAC

    AACA

    AACA

    ...(4.51)

    (4.51) . , :

    1801

    += =

    nn

    ii

    AC

    DB ...(4.52)

  • 120

    (4.52) . , .

    (f) , (4.52), ..

    ( )

    +=

    =

    n

    ii

    AC

    DB nf

    1180

    ...(4.53)

    , :

    n,' 01= ...(4.54)

    n,' 51=

    f>, , f

  • _____________________________________________________________________________121

    BB

    A

    AA

    cosdxsindy

    cosdxsindy

    cosdxsindy

    cosdxsindy

    34344

    323

    3233

    22

    2122

    11

    111

    ==

    ==

    ==

    ==

    4

    3

    2

    1

    ...(4.57)

    , . , , , ..

    [ ] [ ] ABAB xxxyyy ; ...(4.58) fy fx :

    ( ) [ ]

    ( ) [ ]xxxf

    yyyf

    ABx

    ABy

    =

    = ...(4.59)

    , (fd). :

    22xyd fff += ...(4.60)

    (.22.4)

    (.22.4). , , B, B' , , fd. , (4.60), :

  • 122

    22 lfd += ...(4.61)

    , , l . , l () . , . , .

    . , 50 m, :

    [ ] [ ]

    [ ] [ ]

    [ ] [ ] 0500004000600

    0500003000450

    0500002000350

    ,d,d,

    ,d,d,

    ,d,d,

    III

    II

    I

    ++=

    ++=

    ++=

    ...(4.62)

    :

    [ ] [ ] 030000012000100 ,d,d, ++= ...(4.62') fd>,

    , . , , .

    fd

  • _____________________________________________________________________________123

    444444

    333333

    222222

    111111

    x'

    y'

    x'

    y'

    x'

    y'

    x'

    y'

    vxxvyy

    vxxvyy

    vxxvyy

    vxxvyy

    +=+=

    +=+=

    +=+=

    +=+=

    ...(4.64)

    , , .:

    ''B

    ''

    ''''

    '''

    'A

    ''A

    xxxyyy

    xxxyyy

    xxxyyy

    xxxyyy

    43434

    323323

    212212

    1111

    +=+=

    +=+=

    +=+=

    +=+=

    ...(4.65)

    5 : 1 2 .

    . Agl.vrednost [o] [] []

    6 32 210 00 00 1 19 51 40

    1 6 300 05 00 2 80 37 40

    2 7 71 47 15 1 180 02 00

    KOORDINATI T~. Y X o 6 14 779,16 17 417,14 o 7 15 511,46 17 492,00 o 32 13 776,62 17 487,66 o 87 17 386,05 16 758,22

  • 124

    - d () h (. .)

    (m) (m)

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

    6 - 1 327,26 6,18

    1 - 2 226,93 5,64

    2 - 7 276,79 6,22

    :

    .19 1 2

    =632 940 01 25 =6 169

    0 51 40

    =877 1110 22 38 =1 140 32 40

    =2 251 45 15

    d 1 = 327,20 =7 175 11 23 d 2 = 226,86

    d 3 = 276,72

    7 87 79 24 23 2 264 13 00

  • _____________________________________________________________________________125

  • 126

    , , , (.23.4).

    (.23.4)

    , , , . , i , :

    ( ) =

    =n

    iinf

    11802

    ...(4.66)

    'i , :

    ( ) 'nfn

    ii

    =

    +=1

    1802 ...(4.67)

    .

    , yf xf ,.:

    [ ] [ ] 00 == '' x;y ...(4.68)

    :

  • _____________________________________________________________________________127

    [ ]

    [ ]xf

    yf

    x

    y

    =

    = ...(4.69)

    , .

    ,

    . (.24.4) , , , . , , , .

    (.24.4)

    , (4.47), (4.48) (4.49), , (4.64), (4.65).

    6 : 3 4 ,

    Y X

    1 10 005, 84 20 276, 49 2 = 1820 15 00

    2 10 210, 00 20 132, 20 3 = 1060 38 40

  • 128

    S1 = 132,56

    S2 = 110,26

    :

    21 = 182

    0 15 00

    ''03'5753180

    ''23'18127180

    332

    43

    221

    32

    o

    o

    =+=

    =+=

    mSxSymSxSy

    89.64cos m15.98sin

    34.80cos 105.44msin4322

    4322

    3211

    3211

    ====

    ====

  • _____________________________________________________________________________129

    75.1162089.6486.05120

    404.59 1089.15315.44 10

    86.0512034.8020.13220

    315.44 10105.44210.00 10

    234

    234

    123

    123

    =+=+=

    =+=+=

    ==+=

    =+=+=

    xxxyyyxxxyyy

    , , (), , (), (.39.4).

    (.25.4) AA x,y , BB x,y -

    , i , id , . , , , . , (.25.4), :

  • 130

    111 - +=+= OBBA

    222 - +=+= OBBA ...(4.71)

    ................................................

    nnBBAn O +=+= -

    :

    OBBA = - ...(4.72)

    .

    ( i ), (4.71), ( id ) , ( ii x,y ) :

    1111 yysindyy AA +=+=

    2222 yysindyy AA +=+= ...(4.73) ......................................................

    nAnnAn yysindyy +=+=

    :

    1111 xxcosdxx AA +=+=

    2222 xxcosdxx AA +=+= ...(4.74) ......................................................

    nAnnAn xxcosdxx +=+=

  • _____________________________________________________________________________131

    . , .

    , , . , , .

    (.26.4)

    . , (.26.4), , .

    , 1, a 1 . , 1 c 1 .

    1, :

    )(180 += - ...(4.101)

  • 132

    :

    msin

    csin

    bsin

    a===

    ...(4.102)

    b c , : mb = sin ...(4.103)

    mc = sin ...(4.104)

    1 . , D , . .

    22 )xx()yy(D ABAB -- += ...(4.105)

    1 c( )D )( , . , :

    sinD

    sinc

    =

    :

    sinDcsin =

    :

    )sinDcarcsin( = ...(4.106)

    :

    )(180 += - ...(4.107)

    , 1 ,

    360 ..

    ( ) += -1 360 ...(4.108) , (4.107), (4.108) (1), .

    (.42.4), 1, a , 1 .

  • _____________________________________________________________________________133

    (.27.4)

    . 1, . .

    )(180 += - ...(4.109)

    :

    msin

    csin

    bsin

    a===

    ...(4.110)

    b c :

    mb = sin ...(4.111)

    mc = sin ...(4.112)

    1, D A , ..

    22 )xx()yy(D ABAB -- += ...(4.113)

    :

    sinD

    sinb

    =

    :

    )sinDbarcsin( = ...(4.114)

    , , :

    )(180 += - ...(4.115)

  • 134

    (.28.4), 1, 1P , 2P , , .

    (.28.4)

    , 1, 1 2 1P , 2 2P , 1a 2a . 1b , , .

    , , 21APP , :

    )180 22 += 2(- ...(4.116)

    :

    msin

    csin

    bsina

    ===22

    22

    2

    ...(4.117)

    2b c , :

    mb =2 2sin ...(4.118)

    mc = 2sin ...(4.119)

    1AP1 1(a )c )( 1 . , 1 1 :

    2- 1

    90=+2

    11 ...(4.120)

  • _____________________________________________________________________________135

    1 1 ,

    : 221

    1

    111 gcotcacatg

    +=

    -- ...(4.121)

    1 1 , ..:

    221111

    1 -+= ...(4.122)

    221111

    1 -+= ...(4.123)

    1b , :

    11

    11

    11

    sinsin

    csinsin

    ab == ...(4.124)

    , 2ABP , , D , . .

    22 )xx()yy(D ABAB -- += ...(4.125)

    :

    sinD

    sinb

    =2 ...(4.126)

    :

    )sinDbarcsin( 2= ...(4.127)

    3 2ABP 180 ,

    : )180 += (-2 ...(4.128) (.43.4), 1

    360 , :

    )(- 1 +=360 ...(4.129)

    , 21 3 360 , . .

    )(360 321 ++= - . ...(4.130)

  • 136

    :

    - ? - ? - ? - ? - ? - ? - ? - ? -

    ? - ?

    ! - ? -

    ? - ? - ? - ? -

    ? -

    ? -

    ? -

    ? -

    ? -

    ? -

    ?

  • _____________________________________________________________________________137

    , GNSS GPS

    - TOPCON WILD TC 1100 , GNSS GPS GPS Leica VIVA GS08

  • 138

    , , . .

    , :

    - , ,

    - , ..

    - , . , ,

    , , .

    . (v) (t) ..

    tvS = ...(5. 1)

    , : seckm/c 300 , , , , (.1.5).

    (.1.5)

  • _____________________________________________________________________________139

    , : tcS =2

    2tcS = ...(5.2)

    , . , , (.2.5).

    (.2.5)

    t, , , , t, , .

    , , :

    - - ()

    . ,

    : - , , - , , - , . ,

    , , .

    , , , . , ,

  • 140

    () , , , , . , .

    . , , .

    , . - , . , , .

    , , ( .3..5)

    (. 3.5)

    , .

    , , , , (.4.5).

  • _____________________________________________________________________________141

    (. 4.5)

    , . , , . , , , , :

    ( ) ( )[ ] mmS kmSm = 6103232 ...(5.3)

  • 142

    TOPCON105N

    Topcon 105N

    (IPX 6), (.5.5)

    (.5.5)

    Topcon 105N

    : -

    , ;

    - ;

    - ,

    - , .

    POWER ZEROSET. (PSM) (PPM). , .

  • _____________________________________________________________________________143

    , (.6.5)

    (.6.5) , Battery empty .

    , (F1) (F2).

    F4 (ENTER).

    Topcon 105N :

    - 5. -

    : (2 mm + 2ppm), 2000 m. 45 mm

    30X, 1,3 m. 9 (

    ) , 24 000 . Topcon 105N ( ), 24 (.7.5).

  • 144

    (.7.5)

    : POWER - (ON/OFF)

    ; * - ,

    , ;

    XYZ- ; DIST- ; ANG - ; MENU - ; ESC -

    ; ENT - ; F1,F2, F3 F4 - . Topcon 105N

    .. , . (F1), (F2),(F3) (F4). , Topcon 105N , . , , , .. , , .

    , , 8 500 4000 .

  • _____________________________________________________________________________145

    WILD TC- 1100

    , , . WILD TC - 1100 (.8.5).

    (.8.5)

    , WILD TC-1100 , . , , , (, , ) . , .

    , . . .

    WILD TC - 1100 "1 , "3 .

    ppmmm 22 ppmmm 25 .

    . , , .

  • 146

    3 km, . . ( 30 km ). , (- 200C) (+500C). 30X.

    ( ) , . , . . , , , , .

    WILD TC 1100, LEICA - WILD TC 2002. , , , . , :

    - ",10 - ",50 - ppmmm 11 - ppmmm 13 WILD TC 2002

    WILD TC 1100. , , , , WILD TC 1100.

  • _____________________________________________________________________________147

    ,GNNS GPS

    GNSS (Global Navigation Satellite System), GPS (Global Positioning System) . , : NAVSTAR, GLONASS GALILEO. .

    NAVSTAR GPS, GPS .

    GPS , : , , .

    24 , 20 000 km , 6 , 12 . (.9.5).

    (.9.5) GPS

    , : - L1, , 1575,42 MHz 19 sm - L2, , 1227,60MHz

  • 148

    24 sm. L2 , (PPS).

    (L1 L2) : C/A ( ), P , (PPS - Precise Pozitioting Service) (.10.5).

    (.10.5)

    , 3 (.11.5).

    (.11.5)

    ( Colorado Springs) ), (Hawaii, Ascension Island, Diego Garcia Kwajalein ) (.12.5).

  • _____________________________________________________________________________149

    (.12.5)

    , , , , .

    .

    GPS GPS , .

    GPS , , , () () (.13.5).

    (.13.5)

  • 150

    . (.14.5).

    (.14.5)

    , . , , .

    GPS C/A , . GPS , , . : , (WGS 84), GPS .

    .

  • _____________________________________________________________________________151

    GPS Leica VIVA GS08

    Leica VIVA GS08 GPS , , .

    Leica Geosystems GPS . , Net-Roverot , (.15.5).

    (.15.5)

    SMARTWORX VIVA LT , , .

    Leica VIVA GS08 . RTCM 3.1, ( - ) . 3.5 G .

    GNNS NAVSTAR . GNNS . VIVA GS08 Netrover, GNNS GS08 SmartAntena (.16.5).

  • 152

    (.16.5) (.17.5)

    GS08 SmartAntenna- , . (Bluetooth) ( ).

    . Leica VIVA GS 08 ( GEB211 7,4 V/2.2Ah Lithiumlon) 7 .

    CS 10, LeicaVIVA GS08 . (.17.5).

    CS 10 : - ( VGA), - (1 GB NAND), - (512 MB SDRAM CF/SD), - - (Bluetooth) .

    Leica VIVA GS08 . , Leica, .

  • _____________________________________________________________________________153

    , GNNS GPS :

    - ?

    - !

    - ?

    - () ?

    - ? - ? -

    WILD TC -1100? -

    WILD TC -1100?

    - GPS- ? - GPS- ?

  • 154

    ()

    GPS -

  • _____________________________________________________________________________155

    , , ( ), , , .

    , : , , GSP , .

    ()

    ()

    . . , , d(.1.6).

    (.1.6)

    , (.2.6).

  • 156

    (.2.6)

    A, B S

    . (S) A B, , (.2.6).

    , , . , (72) , (73) , . , (.3.6). , , .

    (.3.6)

  • _____________________________________________________________________________157

    . , . , . , , .

    . , , .

    . , , . 1 999, 1. , , , .

    150 - 200 , , , .. . , . 4', 3'.

    . , , .

  • 158

    , . , 130 m, , 150 m, () , 180 m. , (.4.6).

    (.4.6)

    (.5.6), (.6.6) .

  • _____________________________________________________________________________159

    (.5.6)

    (.6.6)

    , . , 5 - 10 sm, 1 - 2. , .

  • 160

    . , (.7.6)

    (.7.6)

    (x)

    (y) (-), (.8.6).

    (.8.6)

    , ,

  • _____________________________________________________________________________161

    (x). (y) , .

    . , ,

    90. ,

    . .

    , ( 100 300 m) . , ,