-













































:

:

.

- . , - . , .

, , L N . U( N, L ) :

l2

U ( N, L ) = ( 1/ p)×e (y)×w ×cosy(N,L)×dS(N,L)×ò Sl×W(l,T,y,z)×t0(l)×ta(l)×dl ( 1 );

l1

w - ;

y - dS( N,L ) ;

W(l,T,y,z) - dS(N,L) , T;

e(y) - ;

Sl - ;

l1 ,l2 - ;

t0(l), ta(l) - ;

y,z - dS(N,L) [ 2 ] .

, , () E(y, z), :

00 j×2×p×(n×y+m×z)

E(y, z)= t0×w×òò L(n, m)×h0(n,m)×h(n,m)×h(n,m)×h(n,m)×e dn×dm. (2)

-00

堠 w - t;

h0(n,m),h(n,m),h(n,m),h(n,m) - , , ;

y, z - dS ;

L(n,m) - - ;

(n,m) - , .

. , , ( 1 ), . ? .1. , :

dS1 × cos y1 = dS 2 × cos y2 = dS3 × cos y3 ( 3 )

.1 , , . dS dS, , , , (yÞ0, cosyÞ1), dS, , (yÞ900, cosyÞ0).

, dS cosy. . .

2.

.

 

2.1.

- 

.

, :

f(x,y,z) = 0.

, ( .2 ) , . dS, . dS 󠠠

y r. n r . : eïï, ( n*r ), eûë , . dS - R , q j.

- - [ 3 ], , dS(N, L) :

é U0 ( N, L) + U90 ( N, L) ù

Ui( N, L ) = ê U0 ( N, L) - U90 ( N, L) ê , ( 4 )

ê U45 ( N, L) - U135 ( N, L) ç

ë 0 û

i = 1, 2, 3, 4;

U0, U45, U90, U135 - , , , 00, 450, 900, 1350 ( ).

00, 450, 900, 1350. U0, U90 , e(y) e÷÷ eûë , :

U0 (N, L) = A (N, L) ×[e÷÷ (y) × (n * j)2 + eûë(y) × (eûë × j)2 ], ( 5 )

U90 (N, L) = A (N, L) ×[e÷÷ (y) × (n * k)2 + eûë(y) × (eûë × k)2 ]. ( 6 )

堠 l2

A ( N, L ) = ( 1/ p)×e (y)×w ×cosy(N,L)×dS(N,L)×ò Sl×W(l,T,y,z)×t0(l)×ta(l)×dl.

l1

, , , tn=0, , 00, 900, :

P (N, L) = [ U0 (N, L) - U90(N, L)] / [U0 (N, L)+U90(N, L)], ( 7 )

P (N, L) - tn=0.

- , (4) :

é 1 ù

U1(N, L) = U(N, L) ô P(N, L) ×cos2×t(N, L) ê , ( 8 )

ô P(N, L) ×sin2×t(N, L) ê

ë 0 û

P(N, L) - dS(N, L) ;

t(N, L) - dS(N, L).

(7) (8) :

P(N, L) = P(N, L) × cos2 ×t(N, L). ( 9 )

(5) (6) (7), P(N, L):

e÷÷ (y)×[(n*j)2 - (n*k)2] + eûë(y)×[(eûë*j)2 - (eûë*k)2]

P(N, L) = ------------------------------------------------------------------ , ( 10 )

e÷÷ (y)×[(n*j)2 + (n*k)2] + eûë(y)×[(eûë*j)2 + (eûë*k)2]

j , k - OY OZ;

eûë, e÷÷ - , , dS.

(10) :

[e÷÷ (y)/eûë ]×[(n*j)2 - (n*k)2] + [(eûë*j)2 - (eûë*k)2]

P(N, L) = ------------------------------------------------------------------ , ( 11 )

[e÷÷ (y)/eûë ]×[(n*j)2 + (n*k)2] + [(eûë*j)2 + (eûë*k)2]

:

P(y) =[ e÷÷ (y) - eûë (y)] / [ e÷÷ (y) + eûë (y)] ,

e÷÷ (y) eûë (y) P(y):

e÷÷ (y)/eûë (y)= [1+ P(y)] / [1- P(y)]. ( 12 )

, P(y) :

P(y) = a × (1- cosy),

- , .

, y dS r , :

P(y) = [ 1-(n*r) ] × a . ( 13 )

(12) :

e÷÷ (y) 1+ [ 1 - (n*r)] × a

--------- = ------------------------- . ( 14 )

eûë (y) 1 - [ 1 - (n*r)] × a

, (12), (11) , P(N, L) n :

1+ [ 1 - (n*r)] × a

------------------------ [(n*j)2 - (n*k)2] + [(eûë*j)2 - (eûë*k)2]

1- [ 1 - (n*r)] × a

P(N, L) = ---------------------------------------------------------------------- . ( 15 )

1+ [ 1 - (n*r)] × a

------------------------- [(n*j)2 - (n*k)2] + [(eûë*j)2 + (eûë*k)2]

1- [1 - (n*r)] × a

. n . ( - ) f(x,y,z) = 0, :

[( df/dx ) × i + ( df/dy ) × j + ( df/dz ) × k ]

n = ---------------------------------------------------- . ( 16 )

[( df/dx )2 + ( df/dy )2 + ( df/dz )2 ] 1/2

r l R :

r = ( l - R ) / | ( l - R ) |, ( 17 )

l - , H;

R - - dS , x, y, z i, j, k.

- R :

R = x × i + y × j + z × k . ( 18 )

, l , l :

l = l × i , ( 19 )

l - ;

i - .

(17) :

r = [( l-x)i + y × j +z × k ] / [( l-x)2+ y2 + z2]1/2 . ( 20 )

eûë , n r ( ), :

eûë = [ n* r ] / | [ n* r ] | . ( 21 )

, P , - .


2.1.

.

. 3.

, dS t .

P U0 U90 dS t=00 t=900. U0 U90 dS t , ( ) ( OY ). , t dS , U0 U90 :

U0(N, L) = Umax × cos2 t + Umin × sin2 t = A(N, L) × ( e÷÷ × cos2 t + eûë × sin2 t) ; ( 22 )

U90(N, L) = Umax × sin2 t + Umin × cos2 t = A(N, L) × ( e÷÷ × sin2 t + eûë × cos2t) ; ( 23 )

堠 Umax= A(N, L) × e÷÷ , Umin= A(N, L) × eûë.

(6) P(N, L) dS :

P(N, L) = [ e÷÷ - eûë ] / [ e÷÷ + eûë] × cos(2 × t) = P × cos(2 × t) , ( 24 )

P = [ e÷÷ - eûë ] / [ e÷÷ + eûë ] - dS .

cosy = ( n* r ), (12) :

P(N, L) = [ 1- ( n* r ) ] × × cos(2 × t); ( 25 )

, nyz , n xyz, :

cos t = ( nyz*j ) , ( 26 )

,

cos(2 × t) = 2 × cos2t - 1,

(25) :

P(N, L) = ×[ 1- ( n* r ) ] × [ 2 × ( nyz*j )2 -1 ]. ( 27 )

, (15) (27) (16) - (21) - [5,6]. , , :

P(N, L) = ×[ 1- ( n* r ) ] . ( 28 )


2.3.

, .

, . , , , . , ( . 4).

:

f(x,y,z) =x2+ y2+ z2- R2= 0. ( 29 )

n = (x × i + y × j + z × k ) /R - ,

堠 R = (x2+ y2+ z2)1/2 - .

r (17):

r = [( l-x) × i - y × j - z × k ] / [R2+ l2 + 2 × l × x]1/2 . ( 30 )

:

e = [ n* r ] = ( ny × rz - nz × ry) × i +( nz × rx - nx × rz) × j +( nx × ry - ny × rx) × k ;

:

_____________

eûë = ( lz × i - ly × j ) / (R × Ö R2+ l2 - 2 × l × x ), ( 32 )

(15):

_____________

( n* r ) = (x × l -R2) / (R × Ö R2+ l2 - 2 × l × x ), ( 33 )

( n* j )2 = y2 / R2 ; ( 34 )

( n* k )2 = z2 / R2 ; ( 35 )

( eûë * j )2 = l2 × z2/ (R2 × ( R2+ l2 - 2 × l × x ); ( 36 )

( e÷÷* k )2 = l2 × z2/ (R2 × ( R2+ l2 - 2 × l × x ); ( 37 )

(30) - (37) (15), :

l × x - R2

2 - ---------------------------------

R2 × ( R2+ l2 - 2 × l × x )1/2 æ y2- z2 ö é l2 × z2 - l2 × y2 ù

----------------------------------------- × ï --------- ê + ï --------------------------- ç

l × x - R2 è R2 ø ëR2 ×( R2+ l2 - 2 × l × x ) û

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

R2 × ( R2+ l2 - 2 × l × x )1/2

P (N, L) = ---------------------------------------------------------------------------------------------- .

l × x - R2

2 - ---------------------------------

R2 × ( R2+ l2 - 2 × l × x )1/2 æ y2+ z2 ö é l2 × z2 + l2 × y2 ù

----------------------------------------- × ï --------- ê - ï --------------------------- ç

l × x - R2 è R2 ø ëR2 ×( R2+ l2 - 2 × l × x)û

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

R2 × ( R2+ l2 - 2 × l × x )1/2

:

P(N, L) = [( y2 - z2 ) / ( y2 + z2 )] ×( 1 - x/R ). ( 38 )

.

:

X = R × sinq × cosj ;

Y = R × sinq × cosj ;

Z = R × cosq .

(38) :

sin2q × sin2j - cos2q

P(N, L) = --------------------------- ( 1 - sinq × cosj) . ( 39 )

sin2q × sin2j + cos2q

.

. :

f(x,y,z) =x2 / b2+ y2 / a2+ z2 / c2- 1= 0. ( 40 )

K = b/a - ( b - , a - ), :

________________

P(N, L) = [( y2 - z2) / ( y2 + z2)] ×[ 1 - ( x / Ö x2 + k2 × y2 + k2 × z2)] . ( 41 )

C :

X = b × sinq × cosj ;

Y = a × sinq × cosj ;

Z = a × cosq .

:

sin2q × sin2j - cos2q é sinq × cosj ù

P(N, L) = -------------------------- × ê 1- ------------------------------------------------------ ç(42)

sin2q × sin2j + cos2q ë Ö sin2q × cos 2j + k2 ×( sin2q × sin2j + cos 2q) û

, ( 42 ), , k := 0.1, .. , 10- ; ( 42 ) k = 1. , , ( 42 ) k. q j L N . :

q = L × p / L0 ; ( 43 )

j = ( N × p / N0 ) - p/2 ; ( 44 )

L0 - ;

N0 - .

2.4. .

, - OZ ( . 5).

:

f(x,y,z) = x2 / a2+ y2 / a2 - z2 / c2 = 0. ( 45 )

- ;

- .

n (16), :

[(-2×z/c2)×k+ (2×x/a2)×i+ (2×y/a2)×j ]

n = ------------------------------------------------- . ( 46 )

Ö (2 × x / a2 )2+ (2 × y / a2 )2+ (2 × z / c2 )2

:

.

r = - x × i - y × j - ( l - z )× k / Ö x2 + y2 + ( 1 - z2) , ( 47 )

, , , . , l , r = - k.

( n* r) :

.

( n* r ) = (2×z/c2) / Ö(2 × x / a2 )2+ (2 × y / a2 )2+ (2 × z / c2 )2 ( 48 )

, k = c / a,

.

( n* r ) = z / Ö( x2 + y2 ) × k4+ z2 . ( 49 )

, ( 27 ), OZ :

P(N, L) = [ 1 - ( n × r )] × [ 2 ×( nxy × i )2 - 1 ]. ( 50 )

.

nxy = (x × i + y × j) / Ö x2 + y2 ; ( 51 )

.

( nxy × i ) = x / Ö x2 + y2 ; ( 52 )

( 49 ) - ( 51 ), :

.

P(N, L) = [ 1 - z / Ö( x2 + y2 ) × k4+ z2 ] × [ 2 × x2 / (x2 + y2) - 1 ] . ( 53 )

P(N, L) , :

X = sinq × cosj ;

Y = sinq × cosj ; ( 54 )

Z = cosq .

.

( n* r ) = 1 / Ö 1 + k4 tg2q .

q , :

tgq = a / c. ( 55 )

( 55 ) ( 54 ), :

.

( n* r ) =1 / Ö 1 + k2, ( 56 )

( nxy × i ) = cosj. ( 57 )

.

P(N, L) = [ 1 - 1 / Ö 1 + k2 ] × cosj. ( 58 )

, ( 58 ) - . j L N :

j = arctg[( L - L0 ) / ( N - N0 )]. ( 59 )

.

2.5.

.

. k, В 0 1 0 -1 .

( >1 ) | P | < 0.09 , 1< | P | < 0.09 . . P , , | P |, 1.

. , , . . . , , | P| 1 .

, , P .

2.6.

.

. , . , 00 3600 . - - . :

l2

( N, L ) = ( 1/ p)×w ×cosy(N,L)×dS(N,L)×ò Sl×W(l,T,y,z)×t0(l)×ta(l)×dl ( 60 );

l1

- , 2.1 ( 4 ), :

é 1 ù

Uj (N, L) = U0 × | P × cos 2 × t | ( 61 );

| P × sin 2 × t |

ë 0 û

堠 U0 - t=00 t=900. U0 = U0 + U90;

P - ;

t - .

- :

é 1 ù

Uj (N, L) = [ W(l,T,y,z) / p ] × | P × cos 2 × t | , ( 62 )

| P × sin 2 × t |

ë 0 û

, :

é 1 cos 2 × d sin2 × d 0 ù

tij = t × | cos 2 × d cos2 2 × d sin2 × d × cos2 × d 0 | , ( 63 )

| sin 2 × d cos 2 × d × sin 2 × d sin22 × d 0 |

ë 0 0 0 0 û

t - ;

d - , .

ࠠ 蠠 蠠  ࠠ 񠠠

d = 00 d = 450, tij :

é 1 1 0 0 ù

tij(0) = t × | 1 1 0 0 | ; ( 64 )

| 0 0 0 0 |

ë 0 0 0 0 û

é 1 0 1 0 ù

tij(45) = t × | 0 0 0 0 | ; ( 65 )

| 1 0 1 0 |

ë 0 0 0 0 û

- , , :

4

Li(l,T,P) = S tij × Lj(l,T,P). ( 66 )

j =1

:

l2

U1 = ò c(l) × Li(l,T,P) × dl, ( 67 )

l1

堠 c(l) = w × cosy × dS × Sl × t0(l) × t a(l) .

, - , d=00 d=450, :

é 1 + P × cos 2 × t ù

Li(0) = t × W(l,T,y,z) / p ] × | 1 + P × cos 2 × t | , ( 68 )

| 0 |

ë 0 û

é 1 + P × sin 2 × t ù

Li(45) = t × W(l,T,y,z) / p ] × | 0 | , ( 69 )

| 1 + P × sin 2 × t |

ë 0 û

, - , :

l2

U1 = t×(1+P×cos2×t)×[(1/ p)×w ×cosy×dS ]×ò Sl×t0(l)×ta(l)×W(l,T,y,z) ×dl

l1

( 70 ).

l2

U2 = t×(1+P×sin2×t)×[(1/ p)×w ×cosy×dS ]×ò Sl×t0(l)×ta(l)×W(l,T,y,z) ×dl

l1

U1 U2 :

 

l2

B( T ) = t×[(1/ p)×w ×cosy×dS ]×ò Sl×t0(l)×ta(l)×W(l,T,y,z) ×dl

l1

( 70 ) :

 

U1 = B( T ) × ( 1 + P × cos2×t )

( 71 )

U2 = B( T ) × ( 1 + P × sin2×t ).

( 71 ), B( T ):

U1 = 1 + P × cos2×t ;

( 72 )

U2 = 1 + P × sin2×t .

, 2.1 2.2 , :

U1(N, L) = 1 + P(N, L) × cos2×t ;

( 73 )

U2(N, L) = 1 + P(N, L) × sin2×t .

, . 2.3 2.4 q j , L N .

.

, - ( zk -z), - ( yk - y ), zk, z, yk, y -

( .6 ). , OZ OX, 6. . .

2.7. -

.

, , - . - .

g, - g=0, - g , . - - :

é 1 ù

U(N, L) = U0 | P(N, L) × cos2×t × cos2×g | . ( 74 )

| P(N, L) × sin2×t × cos2×g |

ë P(N, L) × sin2×g û

, - U1 U2 - :

U1 = 1 + P × cos2×g × cos2×t ;

U2 = 1 + P × cos2×g × sin2×t . ( 75 )

, ( 75 ) ( 16 ) - ( 19 ) - .

2.8.

, .

, 2.3 , , , .6. , ( z0, y0 ) - , R - , rt - - , Z Y . rt :

.

rt = Ö( y-y0)2 + ( z-z0)2 . ( 76 )

dS ( y, z ) . , , . 7, :

x2 ( y-y0)2 ( z-z0)2

f( x, y, z ) = ---- + --------- + --------- = 1, ( 77 )

R2 R2 R2

dS :

.

x = Ö R2-( y-y0)2 + ( z-z0)2 . ( 78 )

( 76 ) ( 78 ) :

.

x = Ö R2 - rt2 . ( 79 )


( 16 ) - ( 19 ) :

df df df 2 × x 2 ×(y-y0) 2 ×(z-z0)

n = ----- × i + ----- × j + ------ × k = ------- × i + ----------- × j + ----------- × k ; ( 80 )

dx dy dz R2 R2 R2

 

2 ×(y-y0) 2 ×(z-z0)

nyz = ----------- × j + ----------- × k ; ( 81 )

R2 R2

 

é (y-y0) ù y-y0

t = arccos | ------------------- | = arccos ------------ ; ( 82 )

ë Ö (y-y0) + (z-z0) û rt

( n* r ) .

cosy= -------------- = x / Ö x2 + rt2 . ( 83 )

| ( n* r ) |

( 12 ) :

ì x ü

= a ( 1- cosy) = a × | 1 - ---------- | . ( 84 )

î Ö x2 + rt2 þ

- ( 80 ) - ( 84 ) ( 73 ) ( ( 74 ) - ).

( 82 ) . , t y z, . , , Y Z ( .. ) , , t ( , , ) .

. ( 82 ) . , z=z0 y>y0 , t=0 z=z0 y< y0 , t = p; y=y0 z> z0, t= - p /2; y=y0 , z<z0 , t= p /2.

, , , , 7.

- , 8 :

x2 ( y-y0)2 ( z-z0)2

f( x, y, z ) = ---- + --------- + --------- = 1, ( 85 )

a2 b2 c2

 

 

:

b = c = R ; ( 86 )

a = k × R, ( 87 )

k - .

( 85 ) :

 

x2 ( y-y0)2 ( z-z0)2

f( x, y, z ) = -------- + --------- + --------- = 1, ( 88 )

k2 × R2 R2 R2

, ( 88 ), :

.

x = k × Ö R2 + rt2 . ( 89 )

, , ( 88 ), :

t = arccos [(y - y0) / rt ]. ( 90 )

( , z ) ( 16 ) - ( 19 ) ( 25 ) - ( 27 ):

.

cosy= x / Ö x2 + k4 ×( y-y0 )2+ k4 ×( z-z0 )2= x / Ö x2 + k4 × rt2 . ( 91 )

, ,

.

P = a ×( 1 - x / Ö x2 + k4 × rt2 ) . ( 92 )

, ( 73 ) ( ( 75 ), ) d = 00 d = 450 . , = 1 . ( 73 ) ( 75 ) = 0.1, . .

2.9.

.

, . 9, :

f(x, y, z) = - ( h- x )2 / h2 + ( y - y0 )2 / R2 + ( z - z0)2 / R2 = 1, ( 93 )

R - ;

h - .

:

x = h × ( 1 - rt / R) . ( 94 )

. n rt :

n = - 2 ×( h - x ) × i / h2 + 2 ×( y - y0) × j / R2 + 2 ×( z - z0) × k / R2, r = i. ( 95 )

.

cos y = ( h - x) / [ h2 × Ö ( h-x)2 / h4+ r2 / R4 ] . ( 96 )

( h - x) / h = rt / R,

.

cos y = 1 / Ö 1+ ( h /R)2 . ( 97 )

k = h / 2 / R, ( 97 ) :

.

cos y = 1 / Ö 1+ 4 × k2 . ( 98 )

, ( 73 ) ( ( 75 ) - ) ( 16 ) - ( 19 ) ( 25 ) - ( 27 ), 00 450, .

2.10. -

.

. , l, , , , . , - , l, .10. , y y - a - :

y= y + a . ( 99 )

y ( 82 ), a .11:

a = arctg [ rt / ( l - x)]. ( 100 )

:

.

y= arccos [ x / (Ö x2+ rt2 )] + arctg [ rt / ( l - x)]. ( 101 )

, l, ( 84 ) y ( 101 ). . 11. l, y ( 99 ) :

y= y + a .

, y ( 91 ), a .11:

a = arctg [ rt / ( l - x)]. ( 102 )

.

.12, y , , a. , , , l, y, y ( 101 ) a ( 100 ).

2.11. .

1 2, . . , . n , . y dS, ( 72 ) , d = 00 d = 450 .

( 72 ) :

U1 = 1 + P × cos2×t ;

U2 = 1 + P × sin2×t .

, U1 U2 :

2 × t = arctg [(U2 - 1) / ( U1 - 1 )] ; ( 103 )

P = ( U1 - 1 ) / cos [ arctg (U2 - 1) /(U1 - 1)] . ( 104 )

y :

y = arccos [ 1 - ( U1 - 1 ) / cos [ arctg (U2 - 1) /(U1 - 1)] ; ( 105 )

, . .13.

= drt - ;

y - , U1 U2 ( 105 );

n - ( i+1) ( i- ). .13 , drt << 1

dx = drt × tg b, ( 106 )

b = Ð. , Ð = y - , :

dx = drt × tg y. ( 107 )

( i + 1) :

xi+1 = xi + dx = xi + drt × tg y. ( 108 )

( 108 ) , , , .

2.12. ,

.

.13 : , l. 2.9, l - , .

.13, l, ( 99 ):

y= y + a .

, :

é U1 - 1 ù

y = arccos | 1 - ---------------------------------------- | + arctg [( rt / ( l -x)] . ( 109 )

ë cos [ arctg (U2 - 1) /(U1 - 1)] û

.

2.13.

.

( 75 ) U1 U2 :

U1 = 1 + P × cos2×g × cos2×t ;

U2 = 1 + P × cos2×g × sin2×t .

, :

g = arctg[( 1 - P ) / ( 1 + P )]; ( 110 )

2 × t = arctg (U2 - 1) /(U1 - 1) ; ( 111 )

U1 - 1

P × cos 2 × g = ----------------------------------------- . ( 112 )

cos [ arctg (U2 - 1) /(U1 - 1)]

U1 - 1

-------------------------------------- = ,

cos [ arctg (U2 - 1) /(U1 - 1)]

 P×cos 2×g = ;

1 - tg2g 1 - [( 1 - P ) / ( 1 + P )]2 4 ×P

cos 2×g = ---------------- = ------------------------------------ = -------------- ;

1 + tg2g 1 + [( 1 - P ) / ( 1 + P )]2 2 ×(1+ P2)

P × [4 ×P /2×(1+ P2)] = A ;

.

/ A / U1 - 1

P = / -------- = / ----------------------------------------------------- . ( 113 )

Ö 2 - A Ö 2 × cos [ arctg (U2 - 1) /(U1 - 1)] - (U1 - 1)

( 113 ) - .

- .

2.14.

.

, .. .

. :

P = a ×( 1 - cos y) ,

y - . , , ( ) , .

:

òò P × dz × dy

Pc = ( z, y) . = a òò ( 1 - cos y) × dz × dy/ p × R2 ( 114 )

p × R2 (z, y)

, .

2.15.

.

U1 , d = 00, U2 d = 450.

d = 00 , . , , U1 . , d = 00, U1 . , .

, U2 , U1 450 . , U1 d = 00, U2 d = 450.

2.16. -

.

- [ 2, 9 ]. -, . , - ( ). - :

00 - 2 × j × p × (n × x + m × y)

P(n, m) = òòP( x, y) × e × dx × dy. ( 115 )

-00

P( x, y) - ;

n, m - , , y.

- , . - , .


2012 , .