Основы геодезии и топографии
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+ .' ,!.+!$ / & , $ $# (+ . $# + .'0 0
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% ' '- "' 5
1. !'! "' 5 )
u = kx + C, |
( 3.10 ) |
. / – * + '' '! ' !+ "* '$!, u – ,- '' '! '"' 5 , k # - ( $ )''- , '-.
, +!, ' $, (3.10) ,* $ * + '' ( . $!, $# $ ''! ' :, ( " * $ ' '! ' "' 5
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U = kX + C. |
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( 3.11 ) |
- $!) +!, ' $, (3.11) (3.10) |
" $-,!) (3.2), '! . * |
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!, * $# * 3." 6 & !* "' 5 !+ "* '$! |
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u = k x. |
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( 3.12 ) |
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( .' 1 +!, ' $, +! . * ( " ''- "**- '! n:
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[ u2]/n = k2[ /2]/n.. |
( 3.13 ) |
! ! + .')) ,!.+!$ !) 6 & ! (+ . ) $ ) ( +*" (3.8), $ [ u2]/n [ x2]/n (+ . $!, )0 $ & ,!.+!$- + .' 1,!.+!$ 1 6 & "' 5 !+ "* '$!. & '! , 1 + mu mx $, $ $, '' , * 3 * '!( !$#:
mu = kmx. |
(3.14 ) |
2. !'! "' 5 ) |
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u = k1 x + k2 y + C. |
( 3.15 ) |
8+ $ '-1 '! ' )1 !+ "* '$ , . ) 2$ "' 5 * * |
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U = k1X + k2Y + C. |
( 3.16 ) |
- $!) '- +!, ' $,! (3.16) $, $ $,"0 / 1 ' , +!, ' $,! (3.15), ( " *:
u = k1 x + k2 y. |
( 3.17 ) |
* ) n * + ' , * 3' '!( !$# n $! 1 +!, ' $,. , .),- (+!,- ! $ 2$ 1 +!, ' $, , ,!.+!$, 3 , 1 +! . , '! n, ( " *:
[ u2]/n = k12[ x2]/n + k22[ y2]/n + 2k1k2[ x y]/n. |
( 3.18 ) |
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u2]/n, [ x2]/n, [ y2]/n ), )0 $ ) ,!.+!$!* + .' |
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6 & , ! , '! [ x y]/n (+ ", ' ! |
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* + ' $+ * $ ) '" 0 . $!) * + ' |
& #6 *, |
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$&+ * ( *! $ ( .' ' , +!, ' $, (3.18), ( ,
& '! , ,1 .)/ |
, 2$ +!, ' $, + .' ,!.+!$ |
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6 & &" ,!* m, m1 |
m2 , * 3 * '!( !$# |
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mu²= k1²m ²+ k2²m ². |
( 3.19 ) |
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3. !'! "' 5 ) |
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u = k1x1 + k2x2 + k3x3 + … + knxn + C. |
( 3.20 ) |
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8 $"(!) |
!'! ' (+ .-."/ *", |
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* 3." 6 & !* "' 5 |
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+ .' * ,!.+!$ * 6 & !* |
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mu²= k1²m1²+ k2²m2²+ … |
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+ kn²mn². |
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( 3.21 ) |
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u = x1 ± x2 ± x3 ± |
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± xn + C, |
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( 3.22 ) |
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mu²= m1²+ m2²+ m3²+ … |
+ mn². |
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( 3.23 ) |
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* + ' ) +!,' $ '-, $ $# m1 = m2 = m3 = … |
= mn = |
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m, $ |
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mu = mÖn . |
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( 3.24 ) |
"' 5 ) &/ , .! |
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u =ƒ (x1, x2, … |
xn ). |
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( 3.25 ) |
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8 ,-+!3 ' ) , +!, ' $, (3.25) |
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x1, |
x2,… xn |
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U + Du = ¦(X1 + Dx1, X2 + Dx2, … |
, Xn + Dxn ). |
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( 3.26 ) |
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! 3 , (+!,"0 ! $# ( .' +!, ' $,! , +). |
+! |
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$&+ , ( *! $ '- ,$ + $ ( ' ,-6 , ( " *: |
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U + u =¦(X1, X2 … , Xn)+(¶¦¤¶x1)Dx1+(¶¦¤¶x2)Dx2+ … +(¶¦¤¶xn)Dxn. |
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( 3.27 ) |
$-,!), $ U = ¦(X1, X2, … Xn), '!1 . *: |
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Du = (¶¦¤¶x1)Dx1 + (¶¦¤¶x2)Dx2 + … |
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+ (¶¦¤¶xn)Dxn,, |
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( 3.28 ) |
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$. . 6 & ! "' 5 , &/ * , . +!,'! ( ' *" . + '5 ! " 2$ "' 5 .
8 + 1 . $ +!, ' $,! (3.25) +!, ' $," (3.28) '! -,!0 $ (+ , . ' * "' 5 (3.25) ' ' *" , ." ( ' !+ !5 )"' 5 ).
! ! 2 5 '$- ¶¦¤¶11, ¶¦¤¶12, … , ¶¦¤¶1n , +!, ' $, (3.28)
– ( $ )''- , '-, $ , 2$ * |
" ! !, * $# * 3." |
6 & !* $! 3 , $ , +!, ' $, |
(3.21), ! . ,!$ #' , . ) |
+ .' 1 ,!.+!$ 1 6 & &". $ (+!, . ,! +*" ! (3.22),
, $ + ,* $ k1, k2, … , kn . # &"."$ ¶¦¤¶11, ¶¦¤¶12, … , ¶¦¤¶1n,
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mu²= (¶¦¤¶x1)²m1² + (¶¦¤¶x2)²m2²+ … (¶¦¤¶xn)²mn². ( 3.29 )
3.6. )** 52% %"#& 35%* (7#65% % #+! "#& 35(0 3 #)/
+ * $ "0 + . '" * 3' (+ . $!, $# , , . :
L = [7]/n = (1/n)71 + (1/n)72 + … + (1/n)7n..
$ 0 .! , .' , $ '! ), ) $ ) ' ' "' 5 $. #'-1* + ' . 8+ * ')) ' +*" " (2.21), * 3' '!( !$#:
M²= (1/n)²m1²+ (1/n)²m2²+ … + (1/n)²mn².
! ! , * + ' ) +!,' $ '-, $ $# m1 = m 2 = … |
= mn, $ |
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M2 = n(1/n)2m2 = m2/n |
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M = m/Ön.. |
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( 3.30 ) |
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+ .')) |
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+ )$' 6 ( (+!, , " ! |
+!,' $ '-1 |
* + ' |
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+ . ' |
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$. #'-* |
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+ " #$!$ * * + ' ), $. . |
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C = L – 7. |
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( 3.31 ) |
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[C] = nL – [7]. |
( 3.32 ) |
8 . $!, , ,* $ L ,-+!3 ' L=[l]/n , '! . *, $
[C] = 0. |
( 3.33 ) |
9$ +!, ' $, ,-+!3! $ , $, , + )$' 6 1 ( (+!, +!,' $ '-1 * + ' . ' "3 $ '$+ * (+!, #' $
,- ' ) !+ * $ + . '-
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& #6 ' $, " ! , $ ' '! ' * + '
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6 - L . |
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= 7 i - X, |
Ci |
= L - 7i . |
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i + Ci = L – X .
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& '! , 6 & " !+ * $ + . '-, +!,'"0 L - X, +
d = Di + Ci, * 3' !( !$# |
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Di = d - Ci.. |
(3.34 ) |
) +).! n * + ' &". $ $ # 3 +!, ' $, , .! (3.34).
, . * , ,!.+!$ ,- (+!,- ! $ 2$ 1 +!, ' $,
3 *:
[D2] = nd2 - 2d[C] + [C2].
8 , $," , + )$' 6 1 ( (+!, [C] = 0, 8 2$ *", +! . , , '- ( .' +!, ' $,! '! n, ( " *
[D2]¤n = d2 + [C2]/n
m2 = d2 + [C2]/n.
2$ * +!, ' $, ., ' , $'- , '- m d. $ ''"0
6 & " !+ * $ |
+ . '- d |
* 3' |
!* ' $# |
$, $ $,"0 / + .' |
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6 & 2$ |
+ . '- |
M2 = m2/n, ( ( " * |
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m2 = m2/n + [C2]/n.. |
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6 , 2$ +!, ' $, $' $ #' m, ( " * |
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m = Ö[C2]/(n – 1) . |
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( 3.35 ) |
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8 . $!, , ,-+!3 ' . ) m +!, ' $,! (3.35) , +*" " " = |
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m/Ön , '! . * |
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M = Ö[C2]/n(n-1). |
( 3.36 ) |
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! * &+! *, +*" ! (2.35) ( , ) $ ,- $# + .'0 0,!.+!$ "0 6 & " ( , + )$' 6 * ( (+!, !* ( 6 & !*)
+).! * + ' .' $ 3 , '-.
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3.9. 1 - )# 3 ) 0 52% %"#& 35(0 (7#65# ( )('(
#$! )#*1( %$)(3"*! 2(0)/, %2)("(&)/,#$! )#0
!3.!) , '! * + '! .,!3.- , * + ' )
+!,' $ '- |
('!(+ * +, * + ' ) |
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+'-* |
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c$ + '!* + (+ * $+ * ' , + ,!' ), |
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(+ . $# ( +! ' $)*, ( " ''-* . ) |
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8" $# * $ ) +). ., '-1 +!,' $ '-1 * + ' |
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7.1, 72' |
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7.1, |
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7.n, |
7'n. |
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d1 |
= 7.1 - 7'1 |
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d 2 |
= 7.2 |
- 7'2 |
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d 3 |
= 7.3 - 7'3 |
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d n = 7.n - 7'n . |
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8+ & 6 & ' * * + ' 2$ +! ' $ . 3'- +!,')$# ) |
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$ + $ '! ' ). . |
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+! ' $ di |
(+ . $!, )0 $ & |
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+!,' '" 0 . |
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6 & ! |
2$ 1 +! ' $ |
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+! ' $ * 3." * + ' )* $ ''-* '! ' )* ) &"."$ |
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d1 = d1 - 0 = d1, |
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d2 = d2 - 0 = d2, |
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… |
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dn = dn - 0 = dn , |
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$. . 6 & +! ' $ di +!,'- !* * +! ' $)*. |
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+ .')) ,!.+!$ !) 6 & ! .' |
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md = 9[ d2]/n ,
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! $! ! |
di = di, $ ( " * |
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md =9[d2]/n . |
( 3.37 ) |
! '$# di $# "' 5 ) .,"1 +!,'$'-1 * + ' |
di = |
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7.i - 7'i , |
( 2$*" |
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md =m9n, |
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. m – + .')) ,!.+!$!) 6 & ! .' * + ' ). $0 .!
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m = md/92.
8 .$!, , , 2$ +!, '$, ,*$ md |
,-+!3 ' (3.35), |
( " * |
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m =9[d2]/2n. |
( 3.38 ) |
+*" ! (3.38) .!$ , * 3'$# |
,- $# + .'0 0 |
,!.+!$"0 6 & " .' * + ' ) ( +! '$)* ., '-1 * + ' , , +! '$)1 '$ $*!$1 6 &. (+$,' * " ! ."$ (+ .$# 0 $# +! '$ .
$*!$"0 6 & " , +! '$ * 3' .$!$' $'
(+ .$# (+ & #6 * * + ' ( +*" |
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D = [d]/n , |
( 3.39 ) |
$. . ! + .' !+ *$ '! ' +! '$ . |
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0 , $*!$"0 6 & " +! '$ di, '! . * |
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Ei = di - D. |
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'- Ei ."$ $!$# , + )$' 6 * |
6 & !* '! |
' ,!' $ , $ ' ( " '- , + " #$!$ &+! ,!' )
+! '$ * + ''-1 '! ' di |
+ .' !+*$ |
'! ' ) D. 8 .$!, , , +*" " |
m = 9[C2]/(n-1) ,*$ C |
, '- Q, ( " * |
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md = 9 [E2]/(n-1) |
m = md/92 = 9[E2]/2(n-1) . |
( 3.40 ) |
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8+!, #'$# ,-' ) '! ' |
D E '$+ +"0 $ ( |
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+*" |
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[E] = 0, |
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( 3.41 ) |
$3.$, '' +*" [C] = 0. |
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3.10. 8 )5% "(&)(3"# $9-:"%"(2 ) %2)("(&)/,
#$! )#0
) (+ . ' ) '! & '!. 3' '! ' ) +).! ' +!,'$'-1 * + ' 5 ' 1 $'$ ,, .)$ $! '!-,! *- , ! * + ' , ( !-,!0 / $( '# '!. 3'$ ,-( ' ''-1 * + ' . * + ' ) (+ . )$) +*"
P = k/m2 . |
( 3.42 ) |
+*" (3.42) $# - (+ , #' , $+
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! * &+! *, , * + ' ) $# , '!, &+!$'
(+ ( +5 '! #'!) ,!.+!$" + .' ,!.+!$ 6 & 2$
* + ' ).
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* 3' ",$# "* '#6 $# , .' $ 3 |
+!. |
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+ .')) ,!.+!$!) |
6 & ! m1 , 1, |
! * + ' 0 72 − |
$,$ $, '' m2 p2. |
) 2$1 * + ' |
p1 =k/m12 p2= k/m22 , $".! |
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