. . . .

. . . .

. .

. .

. .

. .

. .

. .

BRAND GUIDE /

- . . - , - , - , . 00 - .

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

O ()

O ()

2

, - - , , -, , , - .. ., 450 300 .

, - - - - , , .. ., 15 ( ) 23 (- ) , 13 - .

-- - , , , , - .. ., - -, 50 27 .

- - , - , , , 27 ( ), 36 (- -) 110 .

- , - 2012 , 15 -- 100 .

- , - , - - , 15 18 -

. ., . .,

. ., . ., . .

2015

. . ., - . ., . ., . ., . . - , , - . .: , 2015, 592 .

- . - , . , - . , -- , : , -. , -- . - - .

- , .

- , - . - , , - .

, .

ISBN 978-5-93728-146-3

. . . . . .

.

,

27.11.2014 . 6088 1/16+. . . 74,8.. . . 81,4. 500 .

294

107258, , . . 17, . 2.

.: +7(499) 168-21-28E-mail: buchnauch@mail.ru; a2211@bk.ru

3

(. navigatio, navigo ) - . , () . : - , , - , . , , . , 2/3 , - . , :

; ; - .

( ) ( ) - .

, - , - .

- - .

- - . , .

: - ;

- ;

( , ) , .

, , , - , , .

, , . -

, - , , .

, , - , :

;

, ;

.

, - : , - , . , - - . , - , . . . , - .

- : . . ( - VIII); . . ( VIII 13); . . : 3, 7, 811, 1418, 22, 23; . . : 8, 1518; . . VIII ( 1921) 6.

, : - , .

- , - , - , . . . . . , . . , . . , . . , . . , - , . . . , . . . - . . .

. . , . . , . . .

1

. 5238

6

- , , 30 .

: , - .

, - , - . , , .

- 30 , - , - . , : ; , - . - () ( ) - ( ). - : -, , .

- . - .

- , - . , , , . .

, - - : - , , , , . - , . , - , -- .

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

, - , , .

8 1

1

1.1. - ,

, - . - .

XVIIXVIII - -, . (16431727), - , . - = 1/230. - , . 1 , , 1/214 1/314 [19].

(17131765), , , - . [19]

g = ge(1 + sin2),

: ge ; ; , = 2,5aU 2/ge ; a ; U .

- . XIX - - , . -

9 1.

, . :

;b ;c = a b ;

=a ba

;

e a b

a=

2 2

2 ;

=e a b

b

2 2

2 .

, - XYZ, , :

X Ya

Zb

2 2

2

2

21 0

++ = . (1.1)

, :

a b b

eb e=

=

= +

11

12

2

; b a a e a

ea ee

= = =+

=

( ) ;1 1

1

2

2

=

= = +

a ba

ee

1 1 11

1

2

2

; 2 2 2 e e ; (1.2)

e a b

a

e

e2

2

2

2

2 2

2

1

=

= =

+ ( ) ; =

=

=

e a b

b

e

e2

2 2

2 2

2

2

2

1 1

( )

( )

.

1.1 -, XIXXX .

-, , . , , , . , - , : 25 38 . , - . .

1.

10

1.1

( )

a, 1/

1800 6 375 553 334,00

1819 6 376 896 302,78

1830 6 377 276 300,81

1830 6 376 542 299,33

, , 1841 6 377 397 299,15

1844 6 377 096 302,5

1861 6 378 547 283,0

. 1866 6 378 206 294,98

1880 6 378 249 293,47

1893 6 377 717 299,7

. 1906 6 378 200 298,3

1909 6 378 388 297,0

(-42, -95) , , . 1940 6 378 245 298,3

GRS-80 1980 6 378 137 298,257222101

WGS-84 1984 6 378 137 298,257223563

-90, -90.02, -90.11 1990, 2007, 2012 6 378 136 298,257839303

1996 6 378 136.49 298,25642

2000 6 378 136.6 298,25642

-2011 2012 6 378 136.5 298,2564151

-42 1942 ., 760 1946 . . -95 , 28.07.2000 . 568 1.07.2002 .GRS-80 , , 1979 . WGS-84 (World Geodesic System, 1984) - , -. -90 - 1990 , -90.02 - 1990 , -90.11 -90. (IERS International Earth Rotation and Reference Systems Service).

-2011 - , 28.12.2012 . 1463 .

11

1.

. 1.1 ,

- (18191903), - . -, , , . , 1873 . (18081882) . - , - , [19]. , (. 1.1). () - [19]. - (1.1) - , . , , - : , (), . .

( ), a , - . - ( ) 0 - , . - 0 , -

: ii

2 = min . , , ( ),

. 1.1. , -

(

), a , . - ( ) 0 - , . 0 , - :

mini

i 2 . , , (.. ), . , - , -. , --.

-, 1940 . .. (1878-1948), - :

0 = 0 0 = 594618,71 0,16; L0 = 0 0sec0 = 301938,55 + 3,54; 0 = 0 0tg0 = 1214036,13 + 2,66 ; 0 = 0.

0, L0 , 0 , 0, 0 - ( ), 0 , 0 0 - , - . ( - .) - (-) , , .

, -, .

h

1.

12

. , - , -. , --.

-, 1940 . . . - (18781948), :

0 = 0 0 = 594618,71 0,16; L0 = 0 0 sec0 = 301938,55 + 3,54; 0 = 0 0 tg0 = 1214036,13 + 2,66 ;0 = 0,

: 0, L0 ; 0 ;

0, 0 ( );

0 ; 0 0 , -

.

. (-) , - , .

, -, - .

1.2. -

. , , . . - - . -, - . - X, Y, Z () B, L . - .

: -

X, OY, OZ (. 1.2). , : , OZ. OX .

13

1.

. 1.2

OXYZ . , , , - . .

- . 1. () IAG (International Association of

Geodesy) GRS-80 ITRS (International Terrestrial Reference System ), - ITRF (International Terrestrial Reference Frame). ITRS / ITRF . - IRS/ITRF () IERS (International Earth Rotation and Reference Systems Service), . , 4000 ITRF, - , . IERS - , , : ITRF-89, ITRF-94 . . , ITRS , - ITRF . - (), (), (), -

Z

YX

LO

B

1.

14

DORIS. ITRF2000. ITRF2005 2007 ITRF2008 2010 .

2. - WGS-84 (World Geodesic System, 1984), . WGS-84 (Terrestrial Reference Frame TRF), - 17 , . WGS-84 IERS. WGS-84 (http://earth-info.nga.mil/GandG/publications/tr8350.2/tr8350_2.html). () - WGS-84. TRF WGS-84. 1994 WGS-84 (G730), 1997 WGS-84 (G873), 2002 WGS-84 (G1150). - TRF WGS-84(G1150) ITRF2000 1 .

3. 1990 (-90). Z , 19001905 . 28 2000 -90 . 20 2007 - - , - 1990 , -90.02. - -90.02 IERS/ITRF2000 . 28 2012 . - -90 -90.11. -90 :

, ;

; ; WGS-84, ITRS - .

-90 -90.02 WGS-84 , -90.11 .

() - , - ( 51794-2008) [14] ( -90.02).

B

15

1.

B

X

Z A

X

ZY m Y

Z Y

Z X

Y X

= +

+

+

+

( )1

1

1

1

+

xyz

, (1.3)

: Dx, Dy, Dz ; x, y, z ; m .

B (-90, -90.02, WGS-84) 51794-2008 [14].

- . ().

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

L ;

.

( , , ( ), . .) - 90 +90, 180 +180. - B L - .

, , , , . - : , , .

1.

16

-, . -, , , - . , , , , , , .

, , . , -. , , - . , , . - , - . 90 +90, 180 +180. - .

, -, . :

B = , L = sec,

: , .

35, 40. 45.

-, ( ) ( ), ( ). - , - . , - , , - . - , , - ( ) ( ). , - , [37].

17

1.

. . () - , . .

1946 . - 1942 (-42). 28 2000 1995 (-95). - -95 -90 - -42. -95 - -. 1977 . 28 2012 - 2011 (-2011). - :

(a) 6378136,5 ; (a) 1/298,2564151.

, -90 (-90.02, -90.11) . - -42, -95 -2011. - WGS-84. -2011, - -90.11 WGS-84. -2011. -2011 .

1989 1997 EUREF. EUREF - - . ( ), . EUREF WGS-84 .

35 ITRF, 1989.0 ETRF-89 (European Terrestrial Reference Frame 1989 .). ETRF-89, , , -. ETRF-89 - . 1991 ETRF - EUREF-89. 3...4 .

B L . - , - B L .

1.

18

. 1.3 ,

NAD-83, - , , . - GRS80. SIRGAS, GDA94.

-

, . - OXYZ: , OZ, X () . = 0, Y = 0.

1.3 , . . , . M , - RM, OZ. 1.3 :

- ;

- -;

- .

1.3 .

b (1.2):

.)1( 2 tgeXZ (1.7) Z (1.4) b (1.2):

1)1(

)1(22

2222

2

2

eatgeX

aX ,

22 sin1cos

e

aX . (1.8)

Z (1.8) (1.7) .sin

sin1)1(22

2

eeZ (1.9)

h h ,

. 1.4,

.sin,cos

hZZhXX

(1.10)

90 +

r

Z

X

O

P

R

M

M'

. 1.3. ,

19

1.

, - u.

1.3 X Z

Xa

Zb

2

2

2

21+ = , (1.4)

,

Xa b

ddX2 2

0+ =

.

, Z dZ / d , - , Z = f (X ), . 1.3 :

ddX= + = tg ctg( )90 , (1.5)

:

Xa

Zb2 2

0+ =( ctg ) Z X ba

= 2

2tg. (1.6)

b (1.2):

Z X e= ( ) .1 2 tg (1.7)

Z (1.4) b (1.2):

Xa

X ea e

2

2

2 2 2 2

2 2

1

11+

=( )

( )

tg ,

X ae

=

cos

sin

1 2 2. (1.8)

Z (1.8) (1.7)

Z a ee

=

( )

sinsin .

1

1

2

2 2 (1.9)

H h , - 1.4,

X X H

Z Z H

= +

= +

cos ,

sin .

(1.10)

1.

20

. 1.4

1.4 , X

X PM= cos .

(1.8),

PM ae

= 1 2 2sin

.

N ae

= 1 2 2sin

, PM N= . (1.11)

N. (1.8), (1.9) (1.10) h

X N H Z N e H= + = + ( ) cos , ( ) sin . 12

- , X Y - . - 51794-2001

X N H Y N H Z e N H= + = + = + ( ) cos cos , ( ) cos sin , sin . ( )12 , (1.12)

: N ae

= 1 2 2sin

.

j l X, Y, Z. (1.12) :

. 1.4. -

. 1.4 , X cos PMX .

(1.8),

22 sin1

eaPM .

22 sin1

eaN , NPM . (1.11)

N. (1.8), (1.9) (1.10) h

sin)1(cos)(

2

heNZhNX

, X Y - . - 51794-2001

.sin1,sincos)(,coscos)(

2

hN)e(hNhN

ZYX

(1.12)

P

0

Z

X

h

r

21

1.

= arctgYX

, 0;

= sign (Y) / 2, = 0. (1.13)

(1.12)

X Y N H2 2 2+ = +( )2 cos . (1.14) (1.12)

tge2

=+

+

Z NX Y

sin.

2 2 (1.15)

- . - (1.15), .

, arctgZ

X Y2 2+. : ,

, 0,0001. H (1.14):

H X Y N= + 2 2

cos (1.16a)

X Y2 2+ cosj, (1.12) sin j :

H X Y Z a e= + + 2 2 2 21cos sin sin . (1.16)

Z = 0, j = 0 H .

X Y2 2+ = 0, j = sign(Z) /2 H .

. (1.6)

Z X ba

= 2

2tg.

1.3 , tg =ZX

, , - :

tg tg tg tg ' ' .= = ( ) ba e2

2

21 (1.17)

. ,

tg tg

=

sin( )

cos cos, :

sin( ) sin cos . = e2

= ,

= + e2 sin (cos sin ) , :

1.

22

=

e

e

2

2 21

sin cos

sin.

:

=e

e

2

2 21

sin cos

cos

= +

ee

2

2 21

sin cos

cos

= +

ee

2

2 21

sin cos

sin

.

2 ,

sin sin .2 2 (1.18)

1...2.

=

h

Ha

1 .

0 10 1, - .

. 1.3 :

X RM a2 2 2+ =( ) X ab

Z a22

22 2+ = , RM a

bZ = .

1.3 , RM a = sin , Z b= sin .

X a= cos , ZX

ba

= tg . -

tg =ZX

, :

tg tg = ba

. (1.19)

(1.17) (1.19):

tg tg = ba

. (1.20)

1 =ba

tg tg

=

sin( )

cos cos, sin( ) sin cos = .

= , = + = + cos sin( ) cos sin cos2

=

cos sin

cos12

.

23

1.

=

cos sin

sin.

12

:

2

22

2sin sin . (1.21)

, , (X, Y, Z ) (B, L, H ) . () - (), , -95 -42, . - . - X, Y, Z, (1.3) -. . . (19091991) - .

B= B + DB, L= L + DL, H= H + DH. (1.22)

51794 [14]. .

. - .

- . UTM (Universal Transverse Mercator) .

, , - , () . - . - 60 . - , 6. . () 3 , 9 . . n- (6n3) (. . 1.5). UTM 3 180 .. - 177 .. , - 31- UTM. , ( 1:1 000 000) 180 ..

-. x ; y ( UTM y). 500 000 , 10 000 000 .

1.

24

. 1.5

, y - . , x = 6 650 457, y = 4 307 128, , 6 650 457 . - y 4 , 500 000 , 192 872 .

() - . 6.3 51794-2008. , -42 -95, - . - . - , .

, 1.6. -, .

1.3. , -

. . , - , -. , - . ( ) (. 1.7).

n- (6n-3) (. . 1.5). UTM 3 180 .. 177 .. , - 31- UTM. , - ( 1:1000 000) - 180 ..

-. x ; y - ( UTM y). - 500000 , 10000000 .

, y - . , x =6 650 457, y = 4 307 128, , 6 650 457 .. y 4 , - 500 000 , - -192 872 .

. 1.5. -

()

. 6.3 51794-2008. - , 42 -95, - . . -, .

, - 1.6. -

180

50

3 6 9 12 15

P3 X

Y

P2P1

25

1.

. 1.6

. 1.7

RN - R Z = Z (X ):

R ZZ

=+

( )

.1

2

3

2 (1.23)

(1.5) : =Z ddX

12

sin

, : ( )cos

sin sin1 1

123

2

2

2

3

2

3+ = +

=Z

.

. 1.7

RN - R Z = Z(X):

23

21 )(R . (1.23)

(1.5) :

dXdZ 2sin

1 ,

:

3

23

2

223

2

sin1

sincos1)1(

.

dXd (1.8):

.)sin1(

)1(sin

23

22

2

de

eadX

(1.23) : 3

2 2 222

3 332 2 2 22 2

1 1 1 1

1 1N

( ) a sin ( e ) a( e )R M sin .sin ( e sin ) ( e sin )

(1.24)

R , , , -. , - (. 1.3), .cos Rr

N

M

/2

, -.

. 1.6.

1.3. ,

-. . , , - . - , -. - ( ) (. 1.7).

x1, y1, H1

-2 x2, y2, H2g(H2g)

( )

B1, L1, H1

X1, Y1, Z1

-1 x1, y1, H1 (H1g)

H = H+

X2, Y2, Z2

B2, L2, H2

x2, y2, H2

H = H +

1.

26

ddXj

(1.8):

dX a e

ed=

sin ( )

( sin )

.

1

1

2

2 2

3

2

(1.23) :

R M a e

eN =

+

=

( ) ( )

( )

1 1 1

1

2

3

2

3

2

2

2 2

3

2

sin

sinsin

sin

==

a e

e

( )

( sin )

.1

1

2

2 2

3

2 (1.24)

R , -, , . - , (. 1.4),

r RE= cos .

(1.8), r X= ,

r ae

=

cos

sin,

1 2 2 :

R r ae

E = = cos sin

. 1 2 2

(1.25)

(1.11) N. , N . . M. RE = N, RN = M.

(. 1.4). , = N :

OP = N sin Z = N e2 sin . (1.26)

, (. 1.5), :

K ddSA

=

,

: dS ; d

dS.

dS d. , dN = dS cos A, d = dS sin A. dN d dN d dN = dN / RN, d = d / R. - d

d = dN cosA + d sinA = dN / RN cosA + d / R sinA = dS cos2A / RN + dS sin2A / R ,

27

1.

KA RA , , - :

KR

AR

ARA A N E

= = +1

2 2cos sin

. (1.27)

- , [37]:

R R R a ee

ae e

c E N= =

( )=

+ ( )1

1 1 1

2

2 2 2 2 2sin cos . (1.28)

1.4. -

. , - . - . - - . , , , , . , . - , (2- ) m, :

m w w w G F + +( ) = + , (1.29)

: G m ;

F ( ); w , w , w ,

.

, -, . , , - w G . , - G , :

m w G = . (1.30)

:

m w F = . (1.31)

1.

28

(1.31) - . , -, , - , . , , - , . - -, OXYZ. - , , , .

,

= +U , (1.32)

: U ( -);

(), U U >> .

( GRS-80) U = 7 292 1151011 /, - ( U = 15,0410669 o/, 3600 ). , . - (105106) %.

-. 26 000 , U , 23,5. - () - 50,2 . - -. U 18,6 , 10 [3, 49]. , (, . .), , -. , - = +U , . - . - 1899 . -, 1961 . , 1988 . (International Earth Rota-tion Service IERS).

29

1.

(). - (Xp Yp) - OXYZ, , , -, 19001905 ., 1984 . . .

. ( ) , - :

s = 2 / 2 / U. (1.33) 0,00164 100 [23],

. , , -. 24 , 60 -, 60 .

- . - , :

= 2 / ( ) 2 / (U ), (1.34): -

.

, -, , . -, , - . , , - , . , ,

= 2 /(U ), (1.35)

. 24 , 60 , 60 . , . , s , -, , s = 23 56 4,1 . (, , ) UT, . - UT0. UT0 UT1.

() - (Xp Yp) UT1. -

1.

- - . : - , .

- , , - 9 192 631 770 , - Cs133. . - , () . 1012...1014, - 1015...1016 [49]. UT, - UTC (Universal Time Coordinated ). UTC.

- , : SU , US . UTC UTC(SU) 1 .

, , - , - OZ ( -) . , - - O XY Z , O Z , . .

31

2

- , , -. , :

, ; , .

(-) , - .

2.1.

- : l, j, , , , , Y, Z (. 2.1).

. 2.1 N

Z

X Y

O

N

1

H

X

1.

32

, - ( 1 , H = 0), (1.8), (1.9). 1 (. 2.1) :

X = X1 + H cos cos; Y = Y1 + H cos sin; Z = Z1 + H sin, (2.1)

, , l, j, , Y, Z :

X a H Y a H

Z a e H

= +

= +

=

+

cos cos ; cos sin ;

( )12

= sin ; sin . 1

2 2e (2.2)

2.2. , [5]. -

1 , (. 2.2):

, 1; 1 .

, 1 , , - . , - 1 :

r r= +1 , (2.3)

r , , n

,

= H nn| |

. (2.4)

OXYZ. - ,

n , F(X1, Y1, Z1) = 0, (,

n 1 X1, Y1, Z1):

n FX

Xax

=

=2 1

2; n F

YYay

=

=2 1

2; n F

ZZbz

=

=2 1

2,

(1.1):

n n n na

Xa

Ya

Zb

ab a

Zx y z

2 2 2 2

2

1

2

2

1

2

2

1

2

2

2

2 2

1

24 4

1= + + = + +

= + bb

ee22

21

.

, n

aZ m= + 2 1 12 2 ,

: m eb e

2

2

2 21

= ( )

.

33

2.

(2.3)(2.4) -

r1 r X1, Y1, Z1 X, Y, Z , :

xH

aZ m

Xa

Ha Z m

X= +

= +

2

1

2

11

2 2

1

2

1

2 21;

yH

aZ m

Ya

Ha Z m

Y= +

= +

2

1

2

11

2 2

1

2

1

2 21;

zH

aZ m

Zb

Ha e Z m

Z= +

= +

2

1

2

1 11

2 2

1

2 2

1

2 21

( );

X X Ha Z m

X X Ha Z m

= + +

= + +

1

1

2 21 1

1

2 21

11

;

Y Y Ha Z m

Y Y Ha Z m

= + +

= + +

1

1

2 21 1

1

2 21

11

;

Z Z Ha Z m

Z Z Ha Z m

= + +

= + +

1

1

2 21 1

1

2 21

11

.

X1, Y1, Z1 X, Y, Z:

X Xa Z m

H a Z m1

1

2 2

1

2 2

1

1=

+

+ + ; Y Y

a Z m

H a Z m1

1

2 2

1

2 2

1

1=

+

+ + ; Z Z

a e Z m

H a e Z m1

2

1

2 2

2

1

2 2

1 1

1 1=

+

+ +

( )

( )

, , X1, Y1, Z1 (1.1), - :

. 2.2

r1r

nA

1A

X

Z

O

A1

r1

1.

34

Xa

Ya

Zb

2

2

2

2

2

21

+ + = , (2.5)

: aH a Z m

Z m2

1

2 22

1

2 2

1

1=

+ + ( )+

; bH a( e ) Z m

( e ) Z m2

2

1

2 22

2

1

2 2

1 1

1 1=

+ + ( ) + ( )

.

. - , Z1 1 . 1

n - . :

sin = = +

nn

Za e Z m

z

1

2

1

2 21 1( )

. (2.6)

(2.6) , m2, Z2:

Z a ee1

2

2 2 2

2 2

1

1=

( ) sin

sin

,

, , 1 12 2+ Z m :

11

11

2 2

2 2+ =

Z m

e sin .

, , (2.5),

a a H - e2 2 22

1= + ( )sin ; b b H - e- e

2

2 2

2

2

1

1= +

sin . (2.7)

, , - .

, , - , -, , . , ( ) ( -) . R1 ( RE), () R2 ( RN).

R , (. 2.3) :

35

2.

R R Ho = + cos ,

: R a- e

No

=

=

cos

sincos

12 2

-

, , - .

:

RR R

H a- e

H R H N Ho

o1

2 21

1= = + = + = + = +

cos cos sin. (2.8)

R No1 - , , -.

2.3 1 1.

, - (), . :

r r R no e2 2= ,

: r2 - , ;

r - ;

ne ;

Ro2 .

, , - .

. 2.3

C

r

r2

R2

R

R

A1

X1

A

Z

O

R1

1.

36

, - , . , - , , - :

R R H a - e

- eHo2 2

2

2 2 3

1

1

= + =( )

+( )

sin . (2.9)

- , X = X() Z = Z():

R

X Z

Z X Z X

/

2

2 23 2

2

2

2=

+

2

.

( ), Y = 0 , , , (2.9).

2.3 , -. , R2 R1.

R , , :

12

2

2

1R R R

= +cos sin

. (2.10)

R

R2 R R1,

.

2.3. 1.3, -

, (a+h), . X, Y, Z :

X2 + Y2 + Z2 = (a+h)2. (2.11)

OZ , = b/a, X = ,

37

2.

Y = Y, Z =Z.,,(2.11):

X Y Z a h2 22

2

2+ + = +( )

. (2.12)

:

ah=a+h; ()h==b/a; h2=(h2bh2)/ah2=1h2=12=2; b a e a e h e b h eh h= = + = + 1 1 1 12 2 2 2 .

,,. h .-h-[11].h-:

X Ya

Zbh h

2 2

2

2

21 0

++ = . (2.13)

h- -h.-[11].,1.3,(1.17),(1.19)(1.20),h-.

, , h-.-.h-.,.

h-. . ,h- , .2.4,h-,.2.41,-h-,1h-.112,h-12.11h-(-1),-h-11.,111,(1.17).,1211.,(-)h-,

1.

38

. 2.4 h-

. , h-, - h, .

2.4 , 12 12 h- 1 1. 12 ( -12), - 2 h-, 2 - 12: 2 = 12. , -2 2, -1 1.

- , , [11], - , h = (h ) , .

h- a b h bh:

R ae

a he

R he

h o1

2 2 2 21

2 21 1 1

=

=+

= +

sin sin sin ; (2.14)

R a e

e

a h e

eR hh o2

2

2 2

3

2

2

2 2

3

2

2

1

1

1

1

=

=

+

= +

( )

( sin )

( ) ( )

( sin )

(( )

( sin )

1

1

2

2 2

3

2

e

e . (2.15)

Ro1 Ro2 - , , h-.

1 11

12 2

X

Z

1 1 11 2 12

39

2.

R h- , , (2.10).

h-:

R R R a h ee

a he e

c = =+ ( )

=+

+ ( )1 22

2 2 2 2 2

1

1 1 1

( )

sin cos . (2.16)

, l, j, , Y, Z, h-. . (2.2) - (2.13) h-, ah :

a a a H ee

Hh2 22 2

2

22

1

1= + +

cos, (2.17)

: = 1 2 2e sin .

a a ah hh2 2 22= + + , (2.18)

h = h(H) h - H. (2.17) (H / )2:

1

1

1

1

1

1

2 2

2

2

2

2 2

2 2 2

=

=

ee

H H

ee e

cos

cos

sin

=

H ee e

H24 2 2

2 2 2

2

1 1

cos sin

( )( sin ).

, . , ,

h He

= 1 2 2sin

(2.19)

,

H h e= 1 2 2sin . (2.20)

2.4.

- N (. 2.1). , . N - . . 123, 1, 2, 3 , N, .

1.

40

i1i2i3, i1, i2, i3 X, Y, Z .

, - - . , , - . , , .

. ( , Y, Z - l, j, ) (d, dY, dZ dl, dj, d ). dS, :

dS2 = dr2 = (dl 1 + dj 2 + d 3)2 = (dX i1 + dY i2 + dZ i3)2.

, - , :

dS m d m d m dH m m d dm m

2

11

2 2

22

2 2

33

2 2

12 21

13 31

= + + + + ++ +

( )( ) dd dH m m dH d + + ( ) ,23 32

: m X Y Z a H112 2 2

=

+

+

= +

cos

=

+

+

=

+

2

22

2 2 2 21

;

( )m X Y Z a e H 3

=

+

+

=

=

2

33

2 2 2

12 21

1

;

;m XH

YH

ZH

m m ==

+

+

=

= =

+

X X Y Y Z Z

m m X XH

Y

0

13 31

;

YYH

Z ZH

m m X XH

Y YH

Z ZH

+

=

= =

+

+

=

0

023 32

;

.

(2.21)

= 1e2 sin2 . (2.21) (2.2).

(2.21) (2.8)(2.9) , , - dS2 :

41

2.

dS R d R d dH2 12 2

2

2 2 21= ( ) + ( ) + cos (2.22)

, dl, dj, d , mij (i = 1 3, j = 1 3):

dS2 = dP dP , (2.23): dP = [dl dj dH] -. (2.21) :

M =( )R

R1

2

2

2

0 0

0 0

0 0 1

cos

. (2.24)

, - dS2 . , . - , , 3- : l, j, . , - . () : det(M) 0, - :

det cosM = ( ) R R1 2 22 1 0 , , ( ), pi= 2 . .

-

, l, j, h. - h . dS2 .

, ( , Y, Z ) - l, j, h, (1.8), (1.9), a - (a + h):

X a h Y a h

Z a h e

=+

=

+

=+

cos cos ; cos sin ;

( ) ( )12

= sin ; sin . 1

2 2e

(2.25)

1.

42

, dS2 :

dS m d m d m dh m m d dm m

2

11

2 2

22

2 2

33

2 2

12 21

13 31

= + + + + ++ +

( )( ) dd dh m m dh d + + ( ) ,23 32

: m X Y Z a h112 2 2

=

+

+

=

+

cos

=

+

+

=

+

2

22

2 2 2 21

;

( ) ( )m X Y Z a h e 3

=

+

+

=

2

33

2 2 2 2

1

;

sinm XH

YH

ZH

e 22 22

12 21

13 3

1

0

( );

;

= =

+

+

=

=

e

m m X X Y Y Z Z

m m 11

23 32

0=

+

+

=

= =

+

X Xh

Y Yh

Z Zh

m m X Xh

Y

;

+

= +

Yh

Z Zh

a h e e

( ) ( )

( sin cos ).1

2

2

4

(2.26)

= 1e2 sin2 . h :

Mh

R

R R e

R e e

=

( )

1

2

2

2

2

2

2

2 2 2

0 0

0

0 1

cos

sin cos

sin cos sin

(( )12 e

2

, (2.27)

: e2 h- ; = 1e2 sin2 ; R1 R2 h-, -

(2.14) (2.15).

-, , , - h . h-, h- , . - , , l, j h, . ( h ), , .

43

2.

( pi=

2)

l, ( ) . N , . - (). (l, j, ), , , - .

N : , a1, b1, g1, a2, b2, g2, a3, b3, g3, ai, bi, gi (i = 1, 2, 3) - i. ai, bi, gi (i = 1, 2, 3) -

C =

1 1 1

2 2 2

3 3 3

,

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

dS2 , : ai, bi, gi, , , :

dS 2 = dP dP ,:

=

+

M

R K K

K R

1

2 3

2

3

2

3

2

2 3 3

3

2

3

2

2

3 3

3

2

3

2

2

1

2

1 10

1

++

=

+( )

3

2

3

2

3

2

2 2

1

2

2

1 2

10

0 0 1

K K R a e R R; ;3

= 1e2 sin2 ; R1 R2 ; dP = [da db dH] -.

1.

44

, , (2.24) ( Mh (2.27)) - : , - , a, b.

N , a, b, g () . a, b, g 2 2 2 1+ + = . (a b, a g, b g) - . , a= const ( b= const, g= const) . - , , - , . (), - . , -, (1 a2 b2) .

, . - , - , . l j : . -, , .

m11 m22 -

( ) j ( h) . , , - , . - dS2. , .

(l0, j0, 0) - :

m11*= m11(l0, j0, 0); m22*= m22(l0, j0, 0); m33*= m33(l0, j0, 0), (2.28)

0, - , - l, j, . ds2 , - , :

2.

d d d dH

R R

H

2 2 2 2 2 2 2

2

1

22

2

= + +

= ( ) = = (;

cos ;* * *const )) = =2 2 1const; .H

(2.29)

-, E, YN, ZH :

E = R1* cosj* l; YN = R2* j; ZH = (2.30)

, - E, YN, ZH . ds2 :

d dX dY dZE N H2 2 2 2= + + . (2.31)

- E, YN, ZH. l, j, - 1(l1, j1, 1). - 1 1 , - E, YN, ZH.

46

. 3.1

3

( , ) , . , - , . A D (. 3.1):

) ( ) - ( -);

) ( );) ( ),

( ) .

. , -.

, : , -, . .

, , - , , D. , D .

( ) , . S , i- j- , - , ij i- j- i-

3. ( , ) , . , , . A D (. 3.1): ) ( ) ( ); ) ( ); ) ( ), ( ) .

D A

1

. 3.1.

) . , . ) , : , , . . ), , , , D. ), D ).

( )) , . S , i- j- , , ij i- j- i- (. 3.2). ( ) , , () .

A

D

A

D

) ) )

3. ( , ) , . , , . A D (. 3.1): ) ( ) ( ); ) ( ); ) ( ), ( ) .

D A

1

. 3.1.

) . , . ) , : , , . . ), , , , D. ), D ).

( )) , . S , i- j- , , ij i- j- i- (. 3.2). ( ) , , () .

A

D

A

D

) ) )

3. ( , ) , . , , . A D (. 3.1): ) ( ) ( ); ) ( ); ) ( ), ( ) .

D A

1

. 3.1.

) . , . ) , : , , . . ), , , , D. ), D ).

( )) , . S , i- j- , , ij i- j- i- (. 3.2). ( ) , , () .

A

D

A

D

) ) )

47

3.

. 3.2

(. 3.2). - ( ) , , () .

- 12 (. 3.2). , () , 12 S . () S - 12 21. , . : , , h-, .

3.1.

. , , , ; D, - D, D, D (. 3.3).

YZ - YZ, OZ OZ (. 3.3).

, D YZ - :

2

1

A21

A12

P2

P1

S

1.

48

X N H Y N H

Z ND D D D D A D D D D D A

D

= +( ) = +( ) =

cos cos( ); cos sin( );

DD D D( ) sin ;12 +( ) e H (3.1)

: N a eD D= 1 2 2sin .

AXYZ , Y , AX - (. 3.4).

YZ , Y, Z , , S. :

-, ; - ;

.

- D. YZ D :

= = =X S A A Z S A A Y S AD D D D D D D D D D Dsin cos , sin sin , cos . (3.2)

AXYZ , YZ. :

n, Y

(NA + HA), N a eA A= 1 2 2sin ;

. 3.3

Z ZA

P

Y

A

D

Z

X

Y

D

D

YA

A

49

3.

. 3.4

Z (90 A) , Y ;

(1.26) Z n = 2 NA sin.

D - AYAZA :

X Y N H X Y ZZ Y N H XD D A A A D A D D

D D A A A

= + + =

= + + +

( )cos sin ; ;

( )sin

DD A A Acos sin . e N2 (3.3)

. YZ

X N H Y N e HA A A A A A A A A= + = = + ( )cos ; ; [( ( ) )] sin . 0 1 2

D X D, Y D, Z D, D YZ (3.3).

3

. 3.3.

, D YZ .

( )( )( ) ;sin)1(

);sin(cos

);cos(cos

2DDDD

ADDDDD

ADDDDD

HeNZ

HNY

HNX

j

lljllj

+=

+=

+=

(3.1)

.sin1 22 DD eaN j= AXYZ , Y , X (. 3.4).

. 3.4.

Y

A

Z ZA

lDl

Z

X

Y

P

jD

D

YA

jA

O

nD

AA

X

Y

A

HAZA

NA

S

HD

ND90-jA

nA90-jD

; D D ; D , D; n, nD D; , D; A .

D

D

D

D

D

1.

50

(3.3) X D, Y D, Z D. - cosjA, sin jA . sin jA , cosjA .

= + + = + +

X X Z e NY X Z eD D A D A A A A

D D A D A

sin cos cos sin ;

cos sin

2

2NN N H Z YA A A A D Dsin ( ); .2 + = (3.4)

, . YZ (3.1). (3.4). (3.2)

A ZX

AX Z

YS X Y ZD D

DD

D D

DD D D Darctg arctg=

= +

= + + ; ; .

2 2

2 2 2 (3.5)

D YZ X D, Y D, Z D, .

, - . - XA, YA, ZA XD, YD, ZD,

S X X Y Y Z ZD D A D A D A= ( ) + ( ) + ( )2 2 2 .

, S, , A

, , - .

, D. - AXYZ (. 3.4). D (3.2). YZ - (3.3). , (3.1).

, : S, , A (3.2) X D, Y D, Z D; , X D, Y D, Z D (3.3) - XD, YD, ZD;

XD, YD, ZD (3.1) D, D, D.

(3.1) :

tg D AD

D

( ) = YX

, :

D AD

D

arctg= +YX

, D 0; (3.6)

D = A + sign(YD) /2, D = 0.

51

3.

(3.1) cos( D A ), sin( D A ) :

X ( ) Y ( ) N HD D A D D A D D D + = +( ) cos sin cos . (3.7)

(3.1) , :

tg( ) ( )

DD D A D D A

D D

D D

= +

+ +

ZX Y

N eN H

D

cos sin

sin

( ) cos

2

DD

.

(3.7)

tg( ) ( )

DD D D

D D A D D A

=

+ +

Z N eX Y

2sin

cos sin. (3.8)

D (3.8) ,

D

D

D D A D D A

arctg( ) ( )

0 = + Z

X Ycos sin.

- . D. -

(3.7) cosD, (3.1) sinD.

H X Y Z N eD D D A D D A D D D D( ) ( )= + [ ] + + ( ) cos sin cos sin sin 2 D D N . -

. [32]:

D

Dacrtg=+

Z b eP e a

2 3

2 3

sin

cos, H P ND

DD=

cos, (3.9)

: P X Y a Zb P

= + =D D

Darctg2 2 , .

3.2. -

. - (), . -, , ( . 3.5 , . 3.4 ), . , 90 , . , , . , , . , - , .

: j l (. 3.5).

1.

52

. 3.5 , i i+1

( 3.6 ), -, . - , . - - (. 3.6). , /2 /2.

() - . .

, - . , , -. , , - . , , . - 6 371 116 6 371 000 [37]. - , - ( ). - ():

N

i+1

i

L

Bi B0i

( = 0)

Bi+1

/2

53

3.

. 3.6

R R Rc E N= , (3.10)

: RE RN , , , . - - :

X Y Ze

aH

H2 2

2

2

2

1+ +

= , : Z a X Y eH H2 2 2 2 21= ( ) ( ).

:

2 2 2 2 2 2 2 2 21= + + = + +X Y Z a e e X YH H H( ) ( ) .

a eH H, 2 . , X Y2 2 2 2+ = cos , , :

2

2 2

2 2

2

2 2

1

1 1=

=+

a ee

ae

H H

H

H

H

( )

cos sin.

:

=+

ae

H

H12 2

sin. (3.11)

N

7

. 3.6.

() . .

, . , , . , , . , , . 6371116 6371000 [2]. , ( ). ():

NEc RRR , (3.10) RE RN , - , , .

22

222

1 HHa

eZYX ,

: )e()YXa(Z HH

22222 1 . :

)YX(e)e(aZYX HHH222222222 1 .

2HH e,a - . ,

/2

=0

N

1.

54

i YL

L . - 3.5 3.7 0i i Bi :

sin

sin

sin

sin

=

i

L

( )i,

sin sin( ) sin sin = i L i . :

cos cos cos sin sin cos2 2

= + pi pi ( ) ( )L L i , cos sin cos = L i .

:

cos cos cos sin cos sin2 2

pi piL = + ( ( ) ( ),) i sin cos cos = ( )i L .

, :

sin sin sin sin

sin cos cos

=

=

( )( )

i

i

L

L

i ;

;; .cos sin cos = L i (3.12)

, j l .

. 3.7

N

9

. 3.7.

i

i+1 ( ii+1) L i i+1 . 3.5 3.7 i i+1 N

,)()

2

Li

L

sinsin

sin(sinD 1i

i1

).i11i sin(cossinsinD iLL

),))()()( 2222 i11ii1ii cos(-sin(sincoscoscosD iL ).i11ii1ii cos(coscossinsincosD iL i N , i 90. ii+1

L L) ) L .i 1 2 2cos( ) cos( cosD sin( sinD cos

Ni+1

1 1 i i i i i( ) ( ) [ )]i 1 1 i2 2cos( ) cos ' cos - ' sin ' sin - ' cos - (

. ,

).i11ii1ii cos(cossinsincoscossinD iLL ,

).),

),

i11ii1ii

i11ii1ii

i11i

cos(coscossinsincosDcos(cossin-sincoscossinD

sin(cossinsinD

iL

iLL

iLL

(3.13)

i+1

i

/2-i+1 /2-i

DLL

B0i

i+1

i-

i+1-i

i+1-i

Bi Bi+1

55

3.

-

i i+1 ( i i+1) yL i+1 i . 3.5 3.7 i i+1 N

sin

sin

sin

sin

2 1Di

iL

i L(

pi

+

+

=

1 )

( ),

sin sin cos sin(1 1D i i iL L = + + ).

cos cos cos sin sin2 2 1 2 2 1

D i i i iL = + + +( ) ( ) ( ) )pi pi pi pi ( ccos 1( i i+ ),

cos sin sin cos cos cos1 1D i i i i i iL = + + + + 1 ( ). i N , i 90.

ii+1 :

cos( ) cos( ) cos sin( ) sin cos1 2 2CP D Di

+ = + pi pi L L L .

DNi+1 :

cos cos cos sin sin cos1 2 1 2 1( )CPi i i

+ + += + pi pi' ' 'i i( ) ( ) [pipi +( i i1 )] .

,

sin cos cos sin sin cos cos1 1 1D i i i i iL L = + + + i ( ).

, :

sin sin cos sin

sin cos cos sin

1 1DD

i i iL L

L L

= =

+ +

(

i

);

ii i i i i

iD+ + +

+

= +

1 1 1

1

sin cos cos

cos sin sin co

(

i

);

L ss cos cos1 1 + + i i i i( ). (3.13)

, DL L,y .

ym

-. , yL (i+1)- , (3.13). (i+1)- N : ( /2, /2 ).

, -

, ( ).

1.

56

. N N (. 3.6 3.8), , :

sin(

sin(

sin(

sin(

pi pi

pipi

2

3

2

2

2

+

=

)

)

)

),

i

cos cos cos2

= + pisin( ). :

cos cos cos sin sin cos2 2 2

3

2( ( ) ( )pi pi pi pi = + + ) ( ) .

sin cos sin sin cos cos = + pi ( ).2 DN

cos KM cos cos sin sin cos2 2 2 2

(

= + ( ) ( ) ( ) ) (pi pi pi pi pi 2 ),

D

cos KM cos cos2

= ( ) pi ,

sin cos sin cos cos cos2

= + + pi sin ( ) .

. 3.8

11

. 3.8.

,

).(

),(

),(

lljjjj

lljjjj

llj

p

p

p

++=FL

+=F

+=FL

2

2

2

coscoscossinsincos sin

coscossinsincos sin

sincoscos cos

(3.14)

, .

(3.14). (3.14) :

).()( LF= 2sincos coscos pllj DN N

)()(sin)()( 2222 cossincoscos-cos pppp jjj LF+F=

).( += 2coscossinsincossin pjjj (3.5) jcos :

)()(sin 2

lljjlljjjj

jpp +++

=FL

22 cossincoscoscoscossincoscossin

.)]([

)(

jlljjjjj

lljp

p

++

=+

coscossincossincoscossinsin

coscos

2

2

p/2-F

ym

3/2p-l+l

L-p/2

p/2-j

j

M

N

p/2-j

l-l-p/2

F

N

57

3.

, :

cos cos cos sin

sin cos sin sin

2

= +

=

pi( );

+

= +

cos cos

sin cos sin sin cos cos

2

pi( );

+ picos 2( ).

(3.14)

, .

(3.14).

(3.14) :

cos cos cos cos = ( ) .

N N

cos cos cos sin cos2 2 2 2

( ) ( ) sin ( ) ( )pi pi pi pi = +

sin cos sin sin cos sin = + .

(3.14) cos :

sin cos cos

sin cos cos cos2

=

+ +

pi

sin ( ) ccos sin cos 2 + pi2

( )

cos cos

sin sin cos cos sin co

2 + =

pi( )

[

ss sin cos cos2( )] .pi + +

(3.14) sinF ,

cos sin sin sin sin cos cos = + ( ) .

, :

cos cos cos cos

sin cos sin sin cos

= = +

( ) ;

ssincos sin sin sin cos cos sin

;

( ) . = +

(3.15)

, l .

-

, , - . .

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

1.

58

-, , . - () , (Y) (YL) - 0,3 . - - : j1 j2 . : j. , - . j l - . : L F, (, j), (, j) (L, F) - .

, , - [11, 37], j l . , -. . , - , .

, , -, , , , .

-: L F, (, j) (L, F) . L F l, j. L, F DL, , , . - [37]. 200300 , , - .

- , .

1. .2. , :

- ;

59

3.

, .

3. , ( ) : ;

;

, - , - .

4. - , - .

, , , , , , , , (. 3.9). , - .

: -1 -2, , : l-1, j-1 l-2, j-2.

, . - l, j, :

) -1 -2 - ( (1.17))

j-1 = (j-1), j-2 = (j-2);

) j l - ( (3.12) , = + /2):

{l; j} = ({l-1; j-1} ; {l-2 ; j-2}).

{L; } - . ( (3.15) = + /2):

{l; j} = ({L; } ; {l; j}).

, 0, L -.

, , - , -

1.

60

. 3.9

- . , l* j* , :

{l*; j*} = ({l-1; j-1}; {l-2; j-2}).

{L; } -.

-- {L; } {L; }, - . , , :

{L; } ({l*; j*}) {l; j} {l; j} 1({l; j}) {L; }

(3.16)

{L; } ({l; j}) {l; j} 1 {l; j} 1({l*; j*}) {L; }.

- , III- . - , -.

3.3. ,

XIX . - , - [12, 19, 23, 37]. (, [37]).

14

{L; } ({l*; j*}) {l; j} {l; j} -1({l; j}) {L; } (3.16)

, III- . , -.

3.3.

, XIX- . , [1, 2, 5, 9]. (, [2]). , , , , . .

.

APB' (.3.10), AP B'P ( , ) dS , .

. 3.10.

L

L

90

dS

dA90dA

'

dLP

A

A

N

P

61

3.

. 3.10

, , , - , . - .

.

APB (. 3.10), -

AP BP ( , ) dS , .

dS A A. B BC. - A B dB dL , B dA . AB :

R dB dS AN = cos ; r dL R B dL dS AE = = cos sin , (3.17)

: r ; B, L .

- B L , . , u.

(3.17) , - [37]:

= + + dAdS

A A1 2cos sin , (3.18)

: 1 2 L = const () B = const ().

14

{L; } ({l*; j*}) {l; j} {l; j} -1({l; j}) {L; } (3.16)

, III- . , -.

3.3.

, XIX- . , [1, 2, 5, 9]. (, [2]). , , , , . .

.

APB' (.3.10), AP B'P ( , ) dS , .

. 3.10.

L

L

90

dS

dA90dA

'

dLP

A

A

1.

62

, - : = 0. (3.18), - :

dAdS

A A+ + = 1 2 0cos sin . (3.19)

M , (2.24) = 0, 1 2 :

1221 1

0=

=

=det cosM

mL R B R

RLE NN ;

2111 1 1=

=

=

det cos

( cos )

cos

sin

MmB R B R

R BB R B R

a BN NE

E

E +

ae B B2 23

sin cos.

= 1 2 2e Bsin . 2 - (1.24) (1.25) -

: 2 = = tgBR

BrE

sin.

1 2 (3.19), :

dAdS

Br

A =sin sin 0 . (3.20)

, , . :

cos pipi

( ) = +

A dA A dL A dL Bcos cos sin sin cos

2.

, dAdL

B= sin , - (3.17) (3.20).

(3.17) (3.20) - :

dBdS

AR

dLdS

AR B

dAdS

BR

AN

= = = cos

;sin

cos; sin .

E E

tg (3.21)

- .

-, : - :

r sin A = const. (3.22)

, A, (. 3.10 3.11).

63

3.

A r, C r dr+ , RN dB, -:

dr = RN dB sin B. (3.23)

(3.17) : cos A R dBdSN

= sin A r dLdS

= .

r dA, dr , :

cos sinA r dA A dr R dBdS

r dA r dLdS

drN + = + .

(3.20) (3.23), :

cos sinsin sin

sinA r dA A dr R dB r B Ar

r dLdS

R dB BN N + = ,

cos sin sin sin sin sinA r dA A dr R dB B A A R dB BN N + = = 0 .

, - : r Asin = const . .

1.3 , - u a : r a u= cos , (3.22) :

a u Acos sin = const, (3.24) : - - . (3.24) , u = 0, - o , : a A =sin o const (3.24) :

cos sin sinu A A= =o const. (3.25)

(3.22), (3.24) (3.25) - .

. 3.11

.AsinRtgB

dSdA

;BcosR

AsindSdL

;R

AcosdSdB

E

E

N

(3.21)

. ,

: :

rsinA = const. (3.22) , A, (.3.10 3.11).

P

A

C drr

rB

BNR

. 3.11.

A r , C drr , RNdB, :

dr = RNdBsinB. (3.23) (3.17)

dSdBRAcos N dS

dLrAsin . rdA, dr , :

drdSdLrrdA

dSdBRdrAsinrdAAcos N .

(3.20) (3.23), : BsindBR

dSdLr

rAsinBsinrdBRdrAsinrdAAcos NN ,

0 BsindBRAsinAsinBsindBRdrAsinrdAAcos NN .

, : . . constAr sin . 1.3 , u a :

uar cos , (3.22)

constAsinucosa , (3.24) : . (3.24) , u = 0, o, : (3.24) constAsina o

16

1.

64

(3.21)

, - , , - [37]:

1) APB (. 3.12), -, , ;

2) 111 (. 3.12) - ;

3) - .

() - ( 1), (3.21), :

dd

dd

dd

= = = cos ;sin

cos; sin ,tg (3.26)

: j, l ; a ; s . . -

(3.26) .

. (3.21) (3.26), B, L, A S , , j, l, a, s , , 1...4:

dBd

f dLd

f dAd

f dSd

f = = = =1 2 3 4; ; ; . (3.27)

. 3.12

constAsinAsinucos o . (3.25) (3.22), (3.24) (3.25) .

(3.21)

, , , [2]:

1. APB (. 3.12), , , .

2. 111 (. 3.12) .

3. .

17

. 3.12

() ( 1), (3.21), :

,sintgdd

;cossin

dd

;cosdd

(3.26)

: , ; ; . . (3.26) . . (3.21) (3.26),

L

S

12.A12.'21.

B1(2, 2)

A1(1, 1)

P1

constAsinAsinucos o . (3.25) (3.22), (3.24) (3.25) .

(3.21)

, , , [2]:

1. APB (. 3.12), , , .

2. 111 (. 3.12) .

3. .

17

. 3.12

() ( 1), (3.21), :

,sintgdd

;cossin

dd

;cosdd

(3.26)

: , ; ; . . (3.26) . . (3.21) (3.26),

L

S

12.A12.'21.

A(B1, L1)

A1.2B(B2, L2)

P

65

3.

, -

[37]. - - . , . [37], .

:1) ()

( 3.12: S, 11 );

2) ;

3) .

, 1, 2, 3, 4, (3.27), , (3.21) (3.26). (3.21) dB, dL, dA dS, (3.26) d, d, d d, (3.27), :

dBd

AR

dSd

dLd

AR B

dSd

dAd

N

= =

=

cos

cos;

sin

cos

cos

sin;

1

E

ttg

tg

BR

A dSdE

sinsin

.1

(3.28)

, = a, f3 = 1. , j = , (1.20): tg tgu B e= 1 2 .

cos x x

=+

1

12tg

,

:

cos

cos sin

cos.

uB e B

e ue

=

=

1

1

1

12 2

2 2

2

(3.28) :

dAd R e

dSd

=

=1

11

2

E

,

:

dSd

R e a ee B

a e u= =

= E 1

1

11

2

2

2 2

2 2

sincos .

1.

66

(3.28) :

dLd

uB R

dSd

uB

e e uE

= = = cos

cos

cos

coscos .

11 1

2 2 2

S L :

S a e u d= 1 2 21

2

cos

; (3.29)

L e u d= 1 2 21

2

cos

. (3.30)

1 2 ( . 3.12 1) ( . 3.12 1) .

(3.29) (3.30)

-. (, ) , , , - . .

[37]. 111 (. 3.13), - , 3.13: - 1 11 - 1.

. 3.13

dcoseaS

2

1

221 ; (3.29)

deL

2

1

22 cos1 . (3.30)

1 2 ( . 3.12 1) ( . 3.12 1) .

(3.29) (3.30)

. (, ) , , , . .

[2]. 111 (. 3.13), , . 3.13: 1 11 1.

P1

19

. 3.13.

111 11 m, 11 (/2 1). ( 12) m 1 111 cosm, sinm, tg1:

1u 2.1A

1

1

1

1

sinsin

)2/cos()2/cos(cos

uum ;

)u(tg)(tgAcos .

1

121 2/

2/

cosm :

1

2.11

1

2.11

coscoscos

)2/sin(cos)2/sin(cos

AuAum

;

112112 2sin ucosAsin)u/(Asinmsin ; (3.31)

D1

A12

A21 B1(u2, 2)

2-u1 2-u2

1

A0 A1(u1, 1)

A1 B1

1

67

3.

111 - 11 m, 11 (/2 s1).

u1 A1 2. ( a12) - m s1 111 - cosm, sinm, tg1:

coscos( )

cos( )

sin

sinm u u=

=

pipi /

/

2

2

1

1

1

1

; cos( )

( ).A u1 2

1

1

2

2=

tg /

tg /

pi pi

cos m :

cossin( ) cos

sin( )

cos cos

cos

. .m u A u A=

=pi

pi /

/

2

2

1 1 2

1

1 1 2

1

;

sin sin ( / ) sin cosm A u A u= = 12 1 12 12sin pi ; tgtg

11

1 2

=uAcos

..

(3.31)

1 , 1 1, , D1C1 , D1C1 /2. , s1 , D11. D1C1 P1 D1C1 D1 P1 /2 -, m A0 ( D1 ). - (3.31) (3.25),

sin m = sin Ao . (3.32)

(3.32) m A0. 1 , .

111 :

sin cos sin cos sinu m A= +( ) = +( ) 1 0 1 , cos cos sin2 2 0 2 11u A= +( ) . cos2 u (3.29),

dS a e e A d= + +1 2 2 2 0 2 1cos sin ( )

dS a e ee

A d= +

+( )1 11

2

2

2

2

0

2

1cos sin .

(1.2) =

e ee

2

2

21

ba

e= 1 2 , - (3.29) :

dS b e A d= + +( )1 2 2 0 2 1cos sin .,

k e A2 2 2 0= cos (3.33)

dS b k d= + +( )1 2 2 1sin . (3.34)

1.

68

- :

1 11

2

1

8 16

2 2

1

2 2

1

4 4

1

6

6

1+ +( ) = + +( ) +( ) + +k k kk

sin sin sin sin ( ) ... .

, - , k 4 . :

sin cos sin cos co2

1 1

4

1 1

1

2

1

22

3

8

1

22

1

8 +( ) = +( ) +( ) = +( ) +; ss ,4 1 +( )

1 11

4

3

64

1

4

1

162

2 2

1

2 4

2 4

1

+ +( ) = +

+

+ +

k k k

k k

sin

cos

++( ) +( ) k4

164

4cos .

(3.35)

:

K k k K k k K kA B C= + = =14

3

64 4 16 128

2

4

2 4 4

; ; , (3.36)

(3.34)(3.36) (3.29):

S b K K K d= +( ) +( ) + A CB cos cos ...2 2 41 10

.

, :

S K b K b K b= + +A C B sin cos( ) sin cos( )2 2 4 21 1 . (3.37)

S :

=

+ + + +S

K bKK

KKA A

C

A

Bsin cos( ) sin cos( )2 2 4 21 1 . (3.38)

(3.30) d d.

sinsin

cos

sin

cos = =

mu

Au0 , -

111, (3.26):

d Aud = sin

cos

0

2.

(3.30) d, -:

dL d e u e u Aud= + +

2

2

4

4 0

22 8cos cos ...

sin

cos.

cos cos sin2 2 02

11u A= +( ) u - s sin2 1 +( ) , (3.30) :

69

3.

L e A e e A e A= + + +

20

2 2

2

0

2

2

0 1

02

14 8 8

2sin

cos cos cos ( ) d

, , :

L e A e e A e A A= +

2

0

2 2

2

0

4

0

2

0

21

4 8 16

sincos

sin cossin coss( )2 1 + .

K e e A e K e A = +

=

1

2 8 16 16

2 2

2

0

2

4

2

0cos ; cos , (3.39)

L A K K= + +( ) sin sin cos0 12( ) . (3.40)

(3.38) (3.40) . S (3.38) .

(3.37) (3.40) . Dl . - Dl = DL, Dl . - - .

, , 12 -

L1 B1 , - L2 B2 , S (. 3.12). .

1. :

tg tgu e B1 2 11= .

sinsin

sincos

cos

sin.u B e

e B, u B

e B1

1

2

2 2

1

1

1

2 2

1

1

1 1=

=

(3.41)

X = a cosu, Z = b sinu.

2. , (3.31), (3.32) - A12 u1:

1.

70

sin cos sinA u A0 1 12= ; ctg 1 =cos cos

sin

u Au

1 12

1

;

sin22ctg

ctg1

12

=

+1 1; cos2

ctg

ctg1

=+

2

1

2

1

1

1.

(3.42)

sinu1 = 0, sin2s1 = 0 cos2s1= 1.3. KA, K, K, K, K (3.36) (3.39).4. S (3.38), -

. =

SK bA

. :

=

+ +S

K bKKA A

Bsin cos( )2 1 .

:

=

+ + + +S

K bKK

KKA

B

A

C

A

sin cos( ) sin cos( )2 2 4 21 1 .

:

cos( ) cos cos sin sin2 2 21 1 1 + = ; cos( ) cos ( )4 2 2 2 11

2

1 + = + . (3.43)

5. (. . 3.13). 111

sinsin sin

cos

=

Au

1 2

2

. .

111 11, - 111 11, Dl:

cos

=cos cos sin sin coscos

u u Au

1 1 12

2

.

=

arctg

sin sin

cos cos sin sin cos

Au u A

12

1 1 12

.

11B1

sin sin cos cos sin cosu u u A2 1 1 12= + ,

cos u2 cos. (1.20),

, :

B ue

ue u

u u2

2

2

2

2

2

1

1 1=

=

=

+arctg

tgarctg arctg

sin

cos

(sin cos cos 11 122

1 1 121

sin cos )

(cos cos sin sin cos )

Ae u u A

cos.

71

3.

6. dl :

= + +( )sin sin cosA K K0 12( ) , (3.40):

L2 = L1 + Dl dl.

.

, -

. , :

K A A KA = + = 6356863 02 10708 849 13 474 5354 9 82 0 2 0, ( , , cos )cos ( , ,; B 9978

2 238 0 006 33523299

2

0

2

0

2

0

2

0

cos )cos

( , cos )cos ,

A A

K A A K

;

;C = + = ( ) =

28189 70 10

14094 47

2

0

2

0

10

2

0

2

cos cos

( cos )cos

A A

K A

;

AA0 1010 . :

0 1 11

2 2= +( ) K S K KA CBcos sin .

. (3.42) -, (3.43):

= ++

+ +[ ]0 1 0 1 02 5 2sin cos( ) ( )K K KBA C.

:

= + + [ ]{ }K K AB sin sin sin2 21 0 1 0( ) . :

sin sin cos cos sin cosu u u A2 1 1 12= + , Bu

e u2

2

2 2

21 1=

arctg

sin

sin.

, :

=

A u AA u21

12

12

arctgu

1

1 1

cos sin

cos cos cos sin sin .

.

() . sin u1, cos u1, sin A0 , k2 , cos2A0 -

, :

k e A2 = 2 2 0cos ; 1 arctg= sin

cos cos

uA u

1

12 1

.

1.

72

:

K k k kA = + + + 1 25664 12 5

2

2 2( ) ,

K k k kB = + + 2

2 2

512128 64 37( ) .

: =S

b KA,

b . :

2 2 21

41 2 21

2 m m mK KS

= + = + +( )

=, sin cos cos cos , B B bbKA+ .

:

B u u Au

2

1 1 12

11=

+

+ arctg

sin A2 0

sin cos cos sin cos

( ) (sin sin c

oos cos cos );

sin sin

cos cos sin sin cos

u AA

u u

1 12

2

12

1 1

=

arctg

AA12;

K A ( A )C = +

16

4 4 32

0

2

0cos cos ,

: a (a-b)/a,

L L K A K Km2 1 0 21 2 1 2 2= + + + + ( ) sin sin cos cos cosC C C m( ) { } ; =

A A

A210

12

arctg 1 1

sin

cos cos cos sin sinu u .

.

, L1, B1 L2, B2 -

, S - : 12 L1, B1 - 21 L2, B2. .

L = L2 L1 . (3.41) sin u1, cos u1, sin u2, cos u2.

- L, (3.40). . , - (3.40), , A0 1 , , , S A12. (3.40) - . .

73

3.

, l = L, -

. (3.13):

sin sinA

sin cosA12

12

= =

cos sin ;

cos sin sin cos

uu u u u2

2

1 1 22

2 2

= +

cos ;

cos sin sin cos cos cos .

u u u u1 1

(3.44)

p, q, n. :

A pq12

= arctg ; =+

arctgp qn

2 2

. (3.45)

(3.42) sinA0 cos2A0 :

sin cos cosA A u0 12 1= ; cos sin2 0 2 01A A= . (3.46)

(3.40):

= L + sin A0K,

K (3.39). A0 (3.44)(3.46),

1 (3.31):

11

1 12

= arctgsin

cos cos

uu A

.

(3.40):

= + + +( )L A K Ksin sin cos0 12( ) . u1, u2 L

(), - . , 21 (3.13) 21 12 :

A uu u u u21

1=

arctg cos1 2 2 1

sin( )cos

sin cos sin cos

.

(3.37):

S K b K b K b= + +A C sin cos( ) sin cos( )2 2 4 21 1 .

.

(3.41)

. l = L+. .

= 0. (3.44)(3.46) A12, cos2A0 - (3.44)

= +

arctgp A q A

nsin cos

.12 12

1.

74

K K . - :

K A A K = ( ) = 33523299 28189 70 10 28189 942 0 2 0 10cos cos ( c; oos )cossin sin cos cos

2

0

2

0

10

2

2

0

10A A

X u u A K K X

= =

;

2 ;1 ssin( ) . , dl 0,5 108

(, 0,01 30 ). :

K A A K

K

A

B

; ;= + =

=

6356863 02 10708 849 13 474 4 487

1

2

0

2

0, ( , , cos )cos ,C

00708 938 17 956 2

47

2

0

4

0

2, , cos (cos )cos

( c

=

= + +

A Y A X

S K K X

; ;

B

A oos )cossin( )cos

sin cos

2

0

2

0

10

21

110A A A u

u u =

; arctg

cos2 1

ssin cos

.u u1 2

.

() (3.41)

. l = L+. .

= 0. (3.44), (3.45) n, p, q:

cos = n; sin = +p q2 2 ; = arctg sin( cos ). (3.47)

:

sin cos sinA p u /0 1= ; cos sin2 0 2 01A A= ;

cos /cosm2

2 2 1 2 0 = cos sin sinu u A ; KC = +

16

4 4 32

0

2

0cos ( cos )A A ,

: a ;

= + + +( )( ) sin sin cos cos cos1 2 1 2 20 2K A K Km mC C C { } . (3.48) , dl 0,5 108. :

k e A2 2 2 0= cos ; Kk k kA = + + + 1 256

64 12 5

2

2 2( ) ; K k k kB = + +

2

2 2

512128 64 37( ) ;

= + +( )

=

K K

S bK

m mB B ;sin cos cos cos

(

21

41 2 2

2

A ); (3.49)

A pq 12

= arctg ; A uu u u u21

1=

arctg

cos1 2 2 1

sin( )cos

sin cos sin cos.

.

75

3.

, -

. [32].

(3.41)

. l = L. (3.44), (3.47) s, sin s, cos s. , a a, - S:

M = +

sin

cos1; N = +

sin

cos1;

U u u= +( )sin sin1 2 2 ; V u u= ( )sin sin1 2 2 ; (3.50)

= +( )0 25, MU NV ; S a= +( ) . 3000 5 .

- l = L. -

s (3.44), p, q, n - , . - S:

M = +

3

1

sin

cos; N = +

3

1

sin

cos;

U B B= +( )sin sin1 2 2 ; V B B= ( )sin sin1 2 2 ; (3.51)

= +( )0 25, MU NV ; S a= +( ) .

3 000 30 .

- . 12 , , -

. : - - b1, 360 b2. (. . 3.13):

sin 1

=

sin sinsin

B2 ; sin 2

=

sin sinsin

B1 ;

cos 1

=

sin sin cos

sin sin

B BB

2 1

1

; cos 2

=

sin sin cos

cos sin

B BB

1 2

2

.

(3.52)

1.

76

:

A B B12 2 1 1 1 2 2 2 2=

cos sin cos cos sin cossin

; (3.53)

A A12 1 1 12= ( ) +arctg sin cos ,: .

3000 2 .

arctg( ) , , . - - atan2(x1, x2) , - x1/x2, : .

3.4. -

() , h-. a (a+h) h-, .

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

(3.17), (3.19) ; , (3.19) (3.20), , - (3.21), R1 R2 - .

, -

B, L, A, S j, l, a, s 1 4 (3.27), . - (. 1.3). -

77

3.

, . a , (. 3.14).

u [5].

|32|, , :

O O R B ae B

H B

a H e Be

3 2 12 2

2 2

2

1

1

1

= =

+

=

=+

cossin

cos

sin

sin22 2 2

1BB a

e BB =

cos

sincos

, 32

O O O O u a u3 2 3= = cos( ) cos( ) .

, :

a u ae B

BM =

cos( )sin

cos1

2 2. (3.54)

Z :

Z OO a ee B

H B= =

+

2

2

2 2

1

1

( )

sinsin .

, |2| |2|

k ba

= :

OO O O k O O u k O O u ba

b u2 2 3 3= = = =

sin( ) sin( ) sin( ) .

. 3.14

(3.17), (3.19) ; , (3.19) (3.20), , (3.21), R1 R2 .

, .. B, L, A, S , , , 1 4 (3.27), . (. 1.3). , . a , (. 3.14). u [8]. |32|, , :

BBe

aB

BeBeHaBH

BeaBROO cos

sin1cos

sin1sin1cos

sin1cos

2222

22

22123

.

28

. 3.14.

, 32 )ucos(a)ucos( '323 .

, : Bcos

Bsinea)ucos(a

221

. (3.54)

Z : BsinH

Bsine)e(aOOZ

22

2

21

1.

, |2| |2|

abk :

+

C

B

Z

O O2

O

O

r

1.

78

, :

b u a ee B

H B ( ) =

+

sin

( )

sinsin

1

1

2

2 2. (3.55)

:

a ee B

H Ba e H e B

ee

+

=

+

( )

sinsin

sin

s

1

1

11

1

1

2

2 2

2

2 2

2

2iin

sinsin

sin .

2

2

2 2

21

11

B

e B be B

e B

=

(3.55) :

b u b ee B

B =

sin( )

sinsin

1

1

2

2 2. (3.56)

(3.54) (3.56), :

sin( )

cos( )( )

sinsin :

cos

sin

uu

u ee B

B Be B

B e= =

= tg tg1

1 11

2

2 2 2 2

22 .

,

tg tg( )u B e= 1 2 . (3.57)

(3.57) B u - .

:

1) ;2) ;3) j .

, 1, 2, 3, 4 , - (3.21) (3.26). :

dBd

AR

dSd

dLd

AR B

dSd

dAd

N E

= =

=

cos

cos;

sin

cos

cos

sin;

1

ttg

tg

BR

A dSdE

sinsin

.1

(3.58)

, = a, f3 = 1. , j = , tg tg = B e1 2 . , , -

S L -:

79

3.

S a e u d H e= + ( ) 1 12 2 2 2 11

2

cos

; (3.59)

L e u d= 1 2 21

2

cos

. (3.60)

1 2 - 1- 2- .

- , .

-, - 2 , , - (3.37) (3.40) [5]:

S H e a eK K K +

+( ) +( )1 1

2 2 4 2

2 2

1 1

A CB sin cos sin cos , (3.61)

L A K K

+ +( ) sin sin cos0 12( ) , (3.62)

k, KA, KB, KC,K , K (3.33), (3.36), (3.39). ,

: - , , - . , S (3.29) (3.37) (3.59) (3.61), L (3.30) (3.40) (3.60) (3.62).

. , = const , 12 P1 L1 B1 , - L2 B2 P2, P1 S (. 3.2).

[5].1. u1, 1

, (3.41).2. 0 0

s1 01. . B1 = p/2 ( 1 ),

, . s1 l :

1.

80

B1 = p/2 ( ), u1 = p/2, A0 = 0, A12 = p, A21 = 0, s1 = p/2, l = 0; B1 = p/2 ( ), u1 = p/2, A0 = p, A12 = 0, A21 = p, s1 = p/2, l = 0.

B1 p/2 ( ), 0 s1 (3.42).

0 .

3. , , , a, b, - . - (3.33), (3.36), (3.39).

4. s - 1 2 S s1. .

S H e a e K= + ( ) 1 12 2 A s s. s - (3.61)

S H e a e K a e

K KC

= +

+( ) + 1 1 1

2 2

2 2 2

1

A

B sin cos sin c

oos .4 21 +( )( )

5. s2 02

s2 = s1 + s (3.63)

, 2 2. , 1, 2 . - 12 ( 21), s1 ( s2) - l1 ( l2) 1 ( 2) 0 p/2:

cos(A12) = 0 sin(A12) = 1, s1 = p/2, l1 = p/2; cos(A12) = 0 sin(A12) = 1, s1 = p/2, l1 = p/2; cos(A12) 0, l1 = arctg(sin(A0) tg(s1)); s2 = s1 + s = p/2, l2 = p/2; s2 = s1 + s = p/2, l2 = p/2; p/2 < s2 < p/2, l2 = arctg(sin(A0) tg(s2)); s2 > p/2, l2 = arctg(sin(A0) tg(s2)) + p; s2 < p/2, l2 = arctg(sin(A0) tg(s2)) p.

l = l2 l1, 2 , u2 , :

tguu u A

u u21 1 12

1 1

= + ( )

sin cos cos sin cos cos

cos cos sin

ssin cos

. A12

6. 2 (3.41).7. L2 (3.40):

L L A K K2 1 0 12= + + +( ) sin sin cos( ) .

81

3.

8. 21 :

tg( )( )

( ) ( )A u

A u211

12 1

=

+ pi

sin cos sin

sin sin sin cos c

oos

.A12

(3.64)

- .

,

. , = const 1 2, L1, B1, L2, B2 , - S 12 1. - .

1. L = L2 L1, u1 u2 , (3.41).

2. 12 , . , , = L , :

cos sin sin cos cos cos ; sin cos ;

sin

= + = u u u u

A

1 2 1 2

2

12

1

= = = sincos

sin; sin sin cos sin cos

sin

u A u A u2 0 0 1 12 1

pi

=

cos

sin;

sin sincos

sin.

u

A u

2

21

1( )

, , s - , , sin . . - a, b :

= sin A K0 .

:

I = +

:

cos sin sin cos cos cos ; sin cos ;

sin

I I I Iu u u u

A

= + = 1 2 1 22

12

1

III

II

I

I

I

u A u

A

=

=

=

sin cos

sin; sin

sin cos

sin;

sin

pi

2

21

1

0

( )

ccos cos sin

sin;

sin sin cos

sin sin

u u u uII

II

I1 2

1

2 1 =

ctguu1

.

1.

82

I I I I I I IK A K A= + +( )sin sin sin cos .0 0 12 (3.65)

:

= + = + + = + II I I I I ( ). (3.66)

u1, u2 - l , , -. , 12 21 :

cos cos sin sin cos cos cos ; sin sin = + =II IIu u u u A A1 2 1 2 12 12ssin cos

sin;

cos; sin sin

pi

=

u

Au

A AII

2

1 1

12

1

21ctg ctgtg

( ) ( 2211II u )pi

=

sin cossin

.

, - 12 21, . 12 1 :

B1 = 0, cos cossin

sinA A u12 0 2= =

;

B1 0, coscos cos cos cos

sin sinA u u

u121 2

1

=

.

21 :

cos cos cos sin sin sin( )A A A u21 12 12 1pi = .

12 21 .

3. (3.61):

S H e a e K K K= + +( ) +( ) 1 1 2 2 4 22 2 1 1 A CB sin cos sin cos .

- .

, - , , s1 0/0. - . ( /2, /2), :

L > 0, 12 = p/2; 21 = p/2; L < 0, 12 = p/2; 21 = p/2;

S L a H= + ( ) . (3.67)

L [ , ].

83

3.

3.5.

, , . - , (. 3.1). : , , ( ) , - D. , D (. 3.1), , SAD . , SAD , - CD AC. , D -. , - SAD .

.

.

, . . , .

n , (1.1):

n FX

Xax

=

=2

2; n F

YYay

=

=2

2; n F

ZZ

a ez=

=

2

12 2( )

. (3.68)

l = AC , , -

:

( ) ( ) ( )X X Xa

Y Y Ya

Z Z ZbCA C

C

A

C

A C

C + + =2 2 2

02 2 2

. (3.69)

b2 = a2 (1 e2) . , (1.1) ,

:

X Xa

Y Ya

Z ZbC

A

C

A

C

A

2 2 21 0+ + = . (3.70)

, (, ) , , . , , , . - YZ 1Y1Z1 (. 3.15), : Z1 - (3.70), 1 Y1

1.

84

. , 1 - .

[5, 12]. (3.70) ,

, Y, Z:

X Xa

Y Ya

Z Zb

A A A

2 2 21+ + = . (3.71)

L, :

L Xa

Ya

Zb a

X Y Z

e=

+

+

= + +

( )A A A

A A

A

2

2

2

2

2

2

2 2

2

22

1

1

==Laa . (3.72)

La:

L X Y Z

ea = + +

( )A A

A2 2

2

22

1

. (3.73)

(3.71) h:

X XL

Y YL

Z ZL e

aLa a a a

A A A+ +

=( )1

02

2

, (3.74)

: aLa

2

;

XLa

A , YLa

A , Z

L eaA

( )12

nh

YZ.

1Y1Z1 :

Z aL

Za

1

2

1= = =*

const . (3.75)

Y Z 1Y1Z1 - 3.2.

3.2 YZ 1Y1Z1 ( )

M X1 Y1 Z1

X cos( )^

XX1 cos( )^

XY1 cos( )^

XZ1

Y cos( )^

YX1 cos( )^

YY1 cos( )^

YZ1

Z cos( )^

ZX1 cos( )^

ZY1 cos( )^

ZZ1

85

3.

,

ng , :

cos( )^

XZ XLa

1 =A ; cos( )

^

YZ YLa

1 =A ; cos( )

( )

^

ZZ ZL ea

1 21

=A . (3.76)

M, (. 3.15).

1 Y, ( )^

ZX12

=pi

cos( )^

ZX1 0= .

( )^

ZZ1

[0, ], sin( )^

ZZ1

0 :

sin( ) cos ( )^ ^

ZZ ZZL

X Ya

1

2

1

2 21

1= = +

A A.

( )^

ZZ1

[0, ], - ( )

^

ZY1

[0, /2], cos( )^

ZY1

0 sin( )^

ZY1

0 . , cos( )

^

ZY1

M :

cos( ) cos ( )( )

.^ ^

ZY ZZ ZL e L

X Ya a

1

2

1

2

2 2 2

2 21 1

1

1= =

= +A

A A

3.15

sin sin / cos( )

ZY ZZ ZZ ZL ea

1 1 1 22

1

=

=

=

pi A .

. 3.15 Y Z 1Y1Z1

. 3.16 ZY1Y ZZ1Y

35

. 3.15. YZ 1Y1Z1

1 Y, 1 2^

( ZX ) . 1 0^

cos( ZX )

^ 1( )ZZ [0 ], :

^

1sin( ) 0ZZ

^ ^2 2 2

1 1os ( ) A Aa

1sin( ) 1 cZZ ZZ X YL

)

.

^ 1(ZZ [0 ], ^

1( )ZY

[0 /2], .. . , ^ 1( ) 0ZY cos^

1sin( ) 0ZY ^

1cos( )ZY :

2^ ^2 2

1 1 2 2 2

1cos( ) 1 cos ( ) 1 .(1 )

AA A

aa

Z 2ZY ZZ XLL e

Y

. 3.15

)e(LZZZcosZZ/sinZYsin

a

A2111 1

2

. ZY1Y ZZ1Y (. 3.16), :

1 1 1 1

^ ^ ^ ^ ^ ^cos( Z Y ) cos( Z Z ) cos( Z Y ) s in( Z Z ) sin ( Z Y ) cos s in( Z Z ) cos

. (3.77)

1 1 1 1

^ ^ ^ ^ ^ ^cos( YY ) cos(Y Z ) cos( ZY ) s in( Y Z ) sin( ZY ) cos( ) s in(Y Z ) cos .

. 3.16. ZY1Y ZZ1Y

Y

ZY1

Y1 Z

Z1

X1 X

Y

Z1

35

. 3.15. YZ 1Y1Z1

1 Y, 1 2^

( ZX ) . 1 0^

cos( ZX )

^ 1( )ZZ [0 ], :

^

1sin( ) 0ZZ

^ ^2 2 2

1 1os ( ) A Aa

1sin( ) 1 cZZ ZZ X YL

)

.

^ 1(ZZ [0 ], ^

1( )ZY

[0 /2], .. . , ^ 1( ) 0ZY cos^

1sin( ) 0ZY ^

1cos( )ZY :

2^ ^2 2

1 1 2 2 2

1cos( ) 1 cos ( ) 1 .(1 )

AA A

aa

Z 2ZY ZZ XLL e

Y

. 3.15

)e(LZZZcosZZ/sinZYsin

a

A2111 1

2

. ZY1Y ZZ1Y (. 3.16), :

1 1 1 1

^ ^ ^ ^ ^ ^cos( Z Y ) cos( Z Z ) cos( Z Y ) s in( Z Z ) sin ( Z Y ) cos s in( Z Z ) cos

. (3.77)

1 1 1 1

^ ^ ^ ^ ^ ^cos( YY ) cos(Y Z ) cos( ZY ) s in( Y Z ) sin( ZY ) cos( ) s in(Y Z ) cos .

. 3.16. ZY1Y ZZ1Y

Y

ZY1

Y1 Z

Z1

X1 X

Y

Z1

1.

86

ZY1Y ZZ1Y (. 3.16), :

cos( ) cos( ) cos( ) ( ) sin( ) cos (^ ^ ^ ^ ^

Z Y Z Z ZY Z Z ZY Z1 1 1= + =sin sin 11

1 1 1

Z

YY Y Z ZY Y Z ZY

^

^ ^ ^ ^ ^

) cos ;

cos( ) cos( ) cos( ) ( ) sin( )

= +

sin = cos( ) ( ) cos .^

pi sin Y Z1

(3.77)

(3.77) cos, :

cos( )( ) ( )

^

YY ZL e

YL

LX Y

Z YL e X Ya a

a

a

1 2 2 2 2 2 21 1=

+=

+A A

A A

A A

A A

.

cos( )^

YX1 . , XA 0 ( )^

YX1 - [0, /2] cos( )

^

YX1 0 , sin( )^

YX1 0 . XA < 0 ( )^

YX1 [/2, ] cos( )

^

YX1 0< , sin( )^

YX1 0 . cos( )^

YX1 c - / :

cos( ) ( ) cos ( ) cos ( )

( )

^ ^ ^

YX X YY YZ

X Z Y

1

2

1

2

1

2

1

1

= =

=

sign

sign

A

A

A A

22

2 2 2 2 2

2

2

2 2 2 2 2

1

1

L e X YYL

X X e X Y Z

a a( ) ( )