Основы геодезии и топографии
.pdfp1/p2 = m22/m12 , |
( 3.43 ) |
. .
. |
|
|
|
|
|
|||
|
|
n |
|
, |
|
|||
|
|
m, |
|
|||||
|
|
|
|
|
||||
|
|
M = m/Ön. |
||||||
|
, |
|
||||||
|
|
|
|
|
|
|||
|
|
|
P/1 = m2/M2. |
|
|
|
|
|
2 m2/n , |
|
|
|
|
||||
|
|
|
= n. |
|
|
( 3.44 ) |
||
,
, ,
. |
|
|
|
|
|||
! |
|
|
|
|
|
||
1, 2, |
… , |
n 1, |
2, |
… , |
n |
" |
|
|
|
|
|
||||
Lo = ( 1p1 + 2p2 npn)/(p1 + p2 |
+ … |
+ pn) = [ p]/[p]. |
|||||
# |
$ |
|
|
|
|||
" . |
|
|
|||||
n : |
|
||||||
1 1, |
12 , … , 1p |
c |
1, |
||||
21 , 22 , … |
, 2p |
2, |
|||||
|
. . . |
|
|
|
|
|
|
n1, n2, … |
, np |
n . |
|||||
%
Lo=( 11+ 12+… + 1p+ 21+ 22+ 2p+… + n1+ n2+… + np)/(p1+p2+… +pn).
71
& $
1 =( 11 + 12 + … + 1 )/ 1,2 =( 21+ 22 + … + 2 )/ 2 ,
…
n =( n1+ n2 +… + np)/pn .
$
Lo = ( 1p1 + 2p2 + … + npn )/(p1+ p2 +… + pn)=[ p]/[p]. ( 3.45 )
!
, , % $ ,
L [p]. ,
, "
, . .
= [p]. |
( 3.46 ) |
–
, $
, .
,
" " . ' |
|
|
, |
|
; |
"
.
"
|
m |
|
" |
|
|||
|
|
. ' |
|||||
|
|
|
|
|
|||
, |
$ |
p/1 = 2/m2, 2 = |
|||||
pm2. |
|
|
|
|
|
|
|
& 1 , 2, … |
, |
n , " |
|||||
1 , 2, … |
, n |
m1, m 2, … , m n, |
|||||
$ :
2 = p1m12
72
2 = p2m22
. . . . .
2 = pnmn2
$ %
n, |
______ |
|
|
= [pm2]/n. |
( 3.47 ) |
|
||
|
___ |
|
|
= / [p]. |
( 3.48 ) |
( ,
1, 2 … , n |
1, 2, … |
, n , " " |
" |
" |
|
||
|
_________ |
|
|
|
= [p 2]/(n-1). |
( 3.49 ) |
|
|
(3.49) |
|
|
|
|
||
|
___________ |
|
|
|
Mo = [p 2]/[p](n-1). |
( 3.50 ) |
|
:
_______ |
|
= [pd2]/2n , |
( 3.51 ) |
, |
|
__________ |
|
= [p 2]/2(n-1), |
( 3.52 ) |
|
|
. |
|
73
3.11.
3.11.1.
( )
)
, *,
*. +
* %, "
*, ,
, ,
, . C" " "
( ) mt $
_________________ |
|
mt = m12 + m22 + ... + mn2 , |
( 3.53 ) |
m1 = m2 = ... = mn = 0.1 (
$ ); n = 16 mt = 0.4 .,
4 * 1 : 10 000.
3.11.2.
& $
$ , $
1 |
1, 2 |
2 |
|
mx1 |
my1, mx2 |
mx2. ' |
|
$ |
|
|
|
s2 = (x2 |
– x1)2 + (y2 |
– y1)2. |
( 3.54 ) |
! (3.54)
:
s2ms2=(x2– x1)2(mx1)2+(x2– x1)2(mx2)2+(y2– y1)2(my1)2+(y2– y1)2(my2)2. ( 3.55 )
, % $
74
mx my "
, . .
mx1 = my1 = mk1 , |
mx2 = my2 = mk2 , |
( 3.56 ) |
mk1 mk2 − 1 2 |
||
. |
|
|
$ " (3.53) |
|
|
_ |
_ |
|
mt = mk 2 |
mk = mt/ 2 . |
( 3.57 ) |
$ (3.56) (3.55), |
|
|
s2ms2 = (mk1)2[(x2 -x1)2+(y2 -y1)2] + (mk2)2[(x2 -x1)2+(y2 -y1)2], |
( 3.58 ) |
|
|
|
|
ms2 = (mk1)2 + (mk2)2, |
( 3.59 ) |
|
(3.57) |
|
|
ms2 = 0.5[(mt1)2 + (mt2)2]. |
( 3.60 ) |
|
mt1 = mt2 = mt |
|
( 3.61) |
ms = mt, |
||
. . $
$ .
$ ( ) -
mt
|
________ |
|
|
ms0 = mt2 + m 2. |
( 3.62 ) |
mt = 0.4 m = 0.08 ms0 = 0.41 , . .
$ ,
" .
75
3.11.3. ,
'
( ) $
% .
& $ 1 2
tg = (y2 – y1)/(x2 – x1). |
( 3.63 ) |
$ (3.63)
m 2 = (y2 – y1)2(mx1)2/(x2 – x1)4 + (y2 – y1)2(mx2)2/(x2 - x1)4 +
+ (my1)2/(x2 – x1)2 + (my2)2/(x2 – x1)2. |
|
( 3.64 ) |
|
, mx1 = mx2 = mk1 mx2 = my2 =mk2, |
|
||
m 2/cos4 = [(x2 – |
x1)2 + (y2 – y1)2](mk1)2/(x2 – x1)4 + |
|
|
+ [(x2 – |
x1)2+(y2 - y1)2](mk2)2/(x2 – |
x1)4, |
( 3.65 ) |
$ (3.54) (3.57) |
|
|
|
m 2/cos4 = s2( mt1)2/2(x2 – x1)4 + s2(mt2)2(x2 – |
x1)4. |
( 3.66 ) |
|
(3.66) cos = (x2 – x1)/s, |
|
||
mt1 = mt2 = mt |
m 2 =[(mt1)2 + (mt2)2]/2s2. |
|
( 3.67 ) |
|
|
( 3.68 ) |
|
|
m = mt /s. |
|
|
! m $ . !
m 2 = 0.5[(mt1)2 + (mt2)2](3438’/s)2, m = 3438’mt/s. |
( 3.70 ) |
76
) (3.70) ,
$ .
, ( . 27),
= 21 – 23 = arctg[(y1 – y2)/(x1 – x2)] – arctg[(y3 – y2)/(x3 – x2)].( 3.71 )
(3.71),
,
$ (3.54) (3.57) mx = my = mk
m 2 = (mt1)2/2(s21)2 |
+(mt3)2/2(s23)2 + |
|
|
+ [(1/s21)2 + (1/s23)2 |
– 2cos /s21s23](mt2)2/2. |
|
|
. 38 |
|
|
|
|
mt1 = mt2 = mt3 = mt |
|
|
m 2 = mt2[1/(s21)2 +1/(s23 )2 – |
cos /s21s23]. |
( 3.72 ) |
|
s21 = s23 = s, = 90˚, |
_ |
|
|
=180˚ |
m =90˚ = mt 2/s, |
( 3.73 ) |
|
|
_ |
|
|
|
m =180˚ = mt 3/s, |
( 3.74 ) |
|
, (3.70).
3.11.4.,
+ $ " ,
" % . & ,
$ ( )
$
xi yi mx
my .
( ) $
77
|
n |
|
|
|
|
2P = xi(yi+1 – yi-1). |
|
( 3.75 ) |
|
|
i=1 |
|
|
|
% $ xi |
yi, |
|||
|
|
|
|
|
n |
n |
n |
|
|
2dP = (y1+1 – yi-1)dxi + xidyi+1 – xidyi-1. |
( 3.76 ) |
|||
i=1 |
i=1 |
i=1 |
|
|
|
n |
n |
n |
n |
- , n- xiy+1 = xi-1yi |
xiyi-1 = xi+1yi , |
|||
|
i=1 |
i=1 |
i=1 |
i=1 |
$ (3.75) $ |
|
|
|
|
n |
n |
|
|
|
2dP = (yi+1 – yi-1)dxi + (xi-1 – |
xi+1)dyi |
|
||
i=1 |
i=1 |
|
|
|
, ,
(3.56) (3.57)
n
mP2 = [(xi-1 – xi+1)2 + (yi+1 – yi-1)2](mti)2/8.
i=1
! − ,
$ n 2, 1 3, 2 4 . . #
Di , |
, |
$ |
si-1 |
si $ i-1 i+1 |
|
i: |
|
|
(xi-1 – xi+1)2 + (yi+1 – yi-1)2 = (si-1)2 + si2 – |
2si-1sicos i . |
|
' |
n |
|
|
mP2 = Di (mti)2/8. |
( 3.77 ) |
|
i=1 |
|
. (3.77) $
" . / " ,
.
,
$ s1 = s2 = ... = sn =s, 1 = 2 = ... = n =
$ , mt1 = mt2 =... = mtn = mt;
78
___ |
|
mP = ssin( /2)mt n/2 . |
( 3.78 ) |
&
, 1: , mti = mt
|
___________ |
|
|
mP = mt P(1 + K2)/2 , |
( 3.79 ) |
, n =4 = 1 |
|
|
|
__ |
|
|
mP = mt P. |
( 3.79 ) |
3.11.5.
−
$ , h a,
P = ah/2.
$
lnP = lna + lnh – ln2
,
(mP/P)2 = (ma/a)2 + (mh/h)2. |
( 3.80 ) |
' $
,
, .
$ ,
ma/a = mh/h = 1/N,
_
mP =P 2/N.
, . . s1 = s2 =
s3 = s4 =s, ms1 = ms2 = ms3 = ms4 = ms ,
79
_ |
|
mP = mss = ms P mP/P = ms/s. |
( 3.81 ) |
, "
, ,
$ .
,
$ ,
. + , ,
$ $ (3.81).
3.11.6.
/ ( , )
" ,
. + , ,
, $ .'
.
!
, −
« » « »
:
L = L2 – L1 , R = R2 – R1; |
|
|
= ( L + R)/2 = 0.5(L2 – L1 + R2 – |
R1), |
( 3.82 ) |
L1, L2, R1, R2 − «
» (L) « » (R) 1 2
.
mL1 = mL2 =mR1 =mR2 = m , m − .
(3.80)
m = m ,
80