αναλυτικη γεωμετρια
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Transcript of αναλυτικη γεωμετρια
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2012
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. 1996 2004 2005 . . . . . . , . . . 32 . . .
, , . , .
2012
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.....................................................................................................................31.1 ..................................................................................................................31.2 ...........................................................................................................51.3 ................................................................................................71.4 ,...............................................................121.5 ...............................................................................................12.....................................................................................................................................................20
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.....................................................................232.1.................................................................232.2 ........................................................................................................242.3 ..............................................................................252.4 .....................................................................................26..............................................................................................................................................28
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.....................................................................................................................313.1 ........................................................................................................313.2 ............................................................................323.3 ........................................................................343.4 ............................................................................................................353.5 ............363.6. ......................................................................................................................393.7 ....................................................................................39Ax+By+0....................................................................................................393.8 ...................................................................................................40.....................................................................................................................................................42
IV....................................................................................................................43
...............................................................................................434.1 .....................................................................................................434.2 ......................................................454.3 Ax+By+z+=0.........................................................................................464.4 ................................................494.5 ...........................................................................................................504.6 .............................................................................................534.7 ......................................................................................................534.8 ....................................................................................................................54
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.....................................................................................................................................54......................................................................................................................544.9 ........................................................................................................564.10 ...........57.....................................................................................................................................................59
V....................................................................................................................61
...........................................615.1 ................................................................................................................................................615.2 .......................................................................625.3 .............................................................................................................................................645.4 .......................................................................655.5 ............................................................................................665.6 ...................................................675.7 ..........................................................................................................................................695.8 ..............................................................................................715.9 .........................................................................................725.10 ,.................................................................725.11 ................................................................................765.12 ....................................................................................................................................785.13 .........................................................................................805.14 ...................................................................................805.15 .............................................................................................81.....................................................................................................................................................82
V...................................................................................................................85
.........................................................................................856.1 ................................................................................................................................................856.2 ...........................................................................................866.3 ............................................................................................876.4 ..............................................................................................................89.....................................................................................................................................................92
VII..................................................................................................................95
.......................................................................................................957.1...................................................................................................................................................957.2,..................................................967.3 ...............................987.4 ............................................................99...................................................................................................................................................102
VIII................................................................................................................103
...........................................................................................................................1038.1 .........................................................................................................................................1038.2 ........................................................................................................................1038.3 ..................................................................................................................1058.4 ...........................................................................................................107...................................................................................................................................................119
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...............................................................1219.1 ..............................................................................................................................................1219.2 ...................................................................................................................1219.3 ...................................................................................................................1239.4 ...........................................................................................1249.5 ..................................................................................................................1259.6 .....................................................................1279.7 ,..........................................1289.8 ......................................................................................1309.9 SerretFrenet.............................................................................................1319.10 T......................................................................132...................................................................................................................................................135
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IV................................................................................................140
V...............................................................................................141
B...............................................................................................................143
..............................................................................143...............................................................................................................143B.1 ....................................................................................................................143B.2 ........................................................................................................145B.3 ...............................................................................................147
1.......................................................................149
2.......................................................................155
3........................................................................158
4........................................................................161
5.......................................................................171
6.......................................................................179
7........................................................................187
8.......................................................................193
9........................................................................200
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, ' ' . (1, (Descartes), 1637 (x,y,z), . , , . , .. , , ... , . x, y z , . , (2. (x,y,z) : F(x,y,z)=0 .. F(x,y,z)=x2+y2+z2-R2=0, , R. . F(x,y,z)=0 x, y, z : z=f(x,y) x=g(y,z) y=h(x,z) . .. , z :
2 2 2z R x y= 2 2 2z R x y=
(1 RENE CARTESIUS (1596-1650). . , , , (La Geometrie 1637). O Cartesius, Fermat, . (2 , . , ...
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2
. : F(x,y,z)=0 G(x,y,z)=0 . . OXY, z=0 f(x,y)=0. .. x2+y2=R2, z=0 , x2+y2+z2=R2 OXY, ( z=0). , . . , . . : , . . 1788 Lagrange(3 Mecanique Analytique, ( ), . . , , .
(3 JOSEPH LOUIS LAGRANCE (1736-1813), . 19 1766 . 1787 . , , , , , .
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1.1 .
() , . () , . ( ), , (,). () (4. , , , ||. B : || B = (1) -|| (,). x . . (,) Ox. , .. OY, , 1 .
(4 .. , v,u,w, ..
x
()
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4
: ) (x,y) . ) (x,y) . , , 1 ,
OY 2. 1 x, 2 OY y, . (x,y). , (x,y), 1 2 OX OY
1 2 OY . . x, y Y, . ,
Y.
, ( ), . OX, OY, OZ . . , ( 2). 1.
OXZ OXY, OZ 2 3 . 1, 2 3 x, y, z,
P2(y)
P1(x) X
Y
P((x,y)
O
1
O
X
P
P2(y)
P1(x)
P3(z)
Z
2
-
.
5
1, 2, 3 . (x,y,z).
, (x,y,z), 1, 2, 3 1, 2, 3 OYZ, OXZ, OXY . .
x, y, z OXYZ.. x , y z . (x,y,z) (x,y,z) (x,y,z). , .
1.2
OXYZ , r. . , r , r, r . R3 R2. R3 . R2 . R3 . r, r (r) r, ( 3). OXYZ , OY, OZ, i, j, k . 1, 2, 3 x, y, z . :
OP = r = OP1 + OP2 + OP3 = xi + yj +zk
xi, yj, zk r {i, j, k}, R3. r ,
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6
. o r (5 :
2 2 2| | x y z= + +r (3) 1(x1, y1, z1) 2(x2, y2, z2) 12 : 12 = 1 + 2 = 2 - 1 = [x2i + y2j +z2k] - [x1i + y1j +z1k] = = (x2-x1)i+(y2-y1)j+(z2-z1)k (4) 12 : x2-x1, y2-y1, z2-z1 , . H 1 2, d(P1, P2), 12,
2 2 21 2 1 2 2 1 2 1 2 1( , ) | | ( ) ( ) ( )d P P x x y y z z= = + + P P (5) 1: ,
(5 (585-565 . .) 47 . , (), . .
Z z
3
P1 x
i
j
k
O Y y
P2
P(r)
r
P3
-
.
7
. , ( ), . . , .
1.3
1.3.1 v w : O, , v w , , - v w, 4. r, , v w. v w (vx, vy, vz) (wx, wy, wz ) , v w, :
r=v+w=(vx+wx, vy+wy, vz+wz) (1)
.
1.3.2 v+v+v 3v . 3v 3 v v 3v v. , v , v v v. , v , v. =0, v=0 0 , ( , 0=(0,0,0) ). v v, .
x
z
y O
A
w
v
r
4
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8
. :
v-w=v+(-1)w
v -w, w .
1.3.3 v w : v-w=0. v w ,
v=w (vx,vy,vz )=(wx,wy,wz ) vx = wx, vy = wy, vz = wz (1) , (), . , (.. ) , , : , . . , , . , .
1.3.4 v w, (6 vw , :
cos =v w v w (1) v w. (1) :
vw=wv (2) (1) 5:
(6 .. , (v,w)
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9
|w|cos== w v, :
vw= |v|( w v) vw= |w|( v w) : |v|=1 v.w= w v vv=|v|2 (3) , , , : (v+w)u=vu+wu (4) OXYZ i, j, k OX, OY, OZ, v w :
v=vxi+vyj+vzk w=wxi+wyj+wzk (5)
vw (4) ij=ik=jk=0 ii=jj=kk=1, ( i, j, k, ), :
x x y y z zv w v w v w = + +v w (6) (2), (6). : (1) v w , (), v w , . . (6) , vx, vy, vz wx, wy, wz. . v w vw=0, v w :
vw=|v||w|cos=0 cos=0 =/2. : , ( ), , .
A
B
w
v
5
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10
1.3.5 v w, vw, , |v||w|sin, , v w v, w, vw , 6. :
vw=-wv (1) . vw=0, =0, v w . .
,
v(w+u)=vw+vu (2) v w :
vw=(vxi+vyj+vzk)(wxi+wyj+wz k) vw=(vywz -vzwy)i+(vzwx-vxwz)j+(vxwy-vywx)k
y z x yz xx y z
y z x yz xx y z
i j kv v v vv v
i j k v v vw w w ww w
w w w = + + =v w (3)
1.3.6 , . , . , ( ), , ( ). . : v w vw= ()
vw
wv v
w
6
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11
, , . , (), w v, vw=. , v*=v+u u w, uw=0 (). : v* w=(v+u )w=vw+uw=+0= . .
1.3.7 ) v, w, u :
( ) v w u (1) : v,w,u, , 7. |wu| , |wu|= =|w||u|sin, w u . |v|cos. - V :
V=|w||u|sin|v|cos=|wu||v|cos= v(wxu) >/2, v(wu) . : V = |v.(wu)|
: ( )x y z
x y z
x y z
v v vw w wu u u
=v w u
) v, w, u :
w
7
uv
wu
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12
( ) v w u
: ( ) ( ) ( ) = v w u v u w v w u
1.4 ,.
, . , . . ' . , , . rF F , r rF F . ' r . . rF F . F. ' F . .
1.5
. . . .. , . , ( ), . , , ( ), , , ( ), .
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. 13
) OXYZ, . OXYZ , , OXYZ. , OY OY OZ OZ. OXYZ (,,) OXYZ, ( 8) (x,y,z) OXYZ (x,y,z) OXYZ. : = + = xi + yj +zk, = i +j +k, = xi + yj +zk : , , , , x x y y z z x x y y z z = + = + = + = = = (6)
OXYZ OXYZ .
) OXY, . OY (7, ( 9).
(7 , .
8 X
XI
Z
Z
Y
Y
P(x,y,z) P(x,y,z)
O(,,)
i
jk
j
k
8
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14
OXY i, j. (x,y) , (x,y) , i, j i, j. : i = 1i + 1j j = 2i + 2j (7)
1, 2, 1, 2. , i j =(i,j). i j,
ii=1ji sin((i, i))=1sin((j, i)) ji=2ji sin((j, i))=2sin((j, i)) ij=1ij sin((i, j))=1sin((i, j))
jj=2ij sin((j, j))=2sin((i, j)) (8) (i, i)=, (j, j)=, (i, j)= : (i, j) = (i, i) + (i, j) = - (8 (j, i) = -(i, j) = -(i, j) - (j, j) = --
(i, j) = (i, i) + (i, j) + (j, j) = -++=
(i, j) = (i, j) + (j, j) = + ... ( ). (8) : sin(-)=1sin(-) 1= sin()/sin() sin(--)=2sin(-) 2= sin(+)/sin() (8 (v,u), .
x
Y
Y
P(x,y) P(x,y)
x X
y
y
J
i
i X
j
. 9 9
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. 15
sin(-)=1sin() 1=sin(-)/sin() sin(-)=1sin() 2=-sin()/sin() (9)
(x,y) OXY (x,y) OXY. : OP = xi + yj = xi + yj = x(1i + 1j) +y(2i + 2j) (7) xi + yj = (1x+2y)i + (1x+2y)j x = 1x+2y , y = 1x+2y (10) 1, 2, 1, 2 (9) (10), :
x= sin( )sin x-
sinsin
y
y= sinsin
x+
sin( )sin + y (11)
:
x= sin( )sin + x+
sinsin
y
y=- sinsin
x+
sin( )sin y (12)
=+-. O (11) (12) :
sin( ) sinx xsin sin
sin sin( )y y
sin sin
= +
(11)
sin( ) sinx xsin sin
sin sin( )y y
sin sin
+ =
(12)
(11) (12) .
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16
OXY , =/2, (11) (12) : x = xcos - ysin y = xsin + ycos (13)
x cos sin xy sin cos y
= (13)
cos sinx x y
cos( ) cos( ) = + (14)
sin cosy x y
cos( ) cos( ) = +
cos( ) sinx xcos( ) cos( )
sin cos( )y y
cos( ) cos( )
=
(14)
OXY , , = : x = xcos - ysin , y = xsin + ycos (15) x = xcos + ysin , y = -xsin + ycos
x cos sin x x cos sin xy sin cos y y sin cos y
= = (15) ) . Euler OXYZ OXYZ . OXYZ OXYZ OXYZ OXYZ. ,
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. 17
, Euler(9. OXYZ (,,), 10. OXY, OXY . . OZ . OXY, OXY. , , Euler . . , ( ), . OXYZ OXYZ, , , Euler, : ) 11Z1 1=, ( 11). ) 1 OX222 2=1 2=, ( 11). ) OXYZ, ( 11). (9 LEONHARD EULER, (1707-1783), Bernoulli. 1727 . 1771. , , , .
. 10
Z Y
Z
Y
10
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18
:
1
1
1
x cos sin 0 xy sin cos 0 yz 0 0 1 z
= r1 = Rz()r (17)
2 1
2 1
2 1
x 1 0 0 xy 0 cos sin yz 0 sin cos z
= r2 = Rx()r1 (18)
2
2
2
x cos sin 0 xy sin cos 0 yz 0 0 1 z
= r = Rz()r2 (19)
11
2
1
11
X1
1
ZZ2
O
X X2
Y2
Y1
Y
11
1
1
1
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. 19
OXYZ OXYZ (17), (18), (19) :
x cos cos cos sin sin cos sin 1 cos cos sin sin sin xy sin cos cos sin cos sin sin cos cos cos cos sin yz sin sin sin cos cos z
+ = + (20)
( ) ( ) ( ) z x z =r R R R r (21)
) . . : ( ) ( ) ( ) z x z = +r R R R r c (22) c c = (,,) .
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20
1) (3, 8), (-11, 3), (-8, -2)
. 2) (7, 5), (2, 3), (6, -7)
. 3) : (3,2), (5, 8/3),
(9, 4). 4) (1, 7), (8, 6), (7, -1)
(A. (4, 3))
5) P(x, y) 1(1, 7), 2(6, -3) r=2/3. (. (3,3))
6)
OXY.
7) x2+y2=R2.
, , . (. x2+y2=R2)
8) : (v+w )u =vu+wu 9) : |v+u| |v|+|u| 10) Cauchy-Schwarz: |vu||v||u| 11)
) : |v-u|2=|v|2+|u|2-2|v||u|cos v u .
) |v+u|2 -|v-u|2=4 v.u ) |v+u|2+||v-u|2=2|v|2+2|u|2 12) : ) v(uw) = u(vw)-w(vu)
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. 21
) (vu)w = u(vw)-v(uw) ) (vu)(wr) = (vw)(ur)-(vr)(uw) 13) Jacobi: v(uw) + u(wv) + w(vu) = 0 14) v0 :
vu=vw vu=vw u=w. , uw.
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2.1
f(x,y)=0 (,). (,) f(x,y)=0 f(,)=0. , f(x,y)=0 . , f(x,y)=0, . , : 1) f(x,y)=0 . 2) , f(x,y)=0 . . , , , . 1: x2+y2=R2 . x2+y2 P(x,y) . R . . x2+y2=R2 R. 2: 1(,) 2(,). . P(x,y) . :
d(P1, P) = d(P2, P) (x-)2 + (y-)2 = (x-)2 + (y-)2 2(-)x+2(-)y = (2+2)-(2+2)
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24
Ax+By+=0 , .
2.2
f(x,y,z)=0. P(,,) f(x,y,z)=0 P(,,) , f(,,)=0 . 1) f(x,y,z)=0 . 2) , . , f(x,y,z)=0, . , f1(x,y,z)=0 f2(x,y,z)=0. , f1(x,y,z)=0 f2(x,y,z)=0. 1: (x-)2+(y-)2+(z-)2=R2 P(x,y,z) (,,) R. , (,,) R. 2: : x2+y2+z2=R2 Ax+By+z+=0 , Ax+By+z+=0 x2+y2+z2=R2 . 1: f(x,y)=0. f(x,y)=g(x,y)h(x,y), g(x,y)=0 h(x,y)=0. f(x,y)=0 g(x,y)=0 h(x,y)=0. f(x,y,z)=g(x,y,z)h(x,y,z) f(x,y,z)=0 g(x,y,z)=0 h(x,y,z)=0 3: (x2+y2-1)(x2+y2-4)=0 1 2 . , f(x,y)=0 fx,y,z)=0, (x,y,z) .
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25
2.3
. , (, , ..), . f(r)=0 , , . . r t,
( ) ( ) ( ) ( ) [ ]1 2 , ,t x t y t z t t t t= = + + r r i j k r(t) t , . r (u,v),
( ) ( ) ( ) ( ) ( ) [ ] [ ]1 2 1 2, , , , , , , ,u v x u v y u v z u v u v u u v v= = + + r r i j k r(u,v) u, v , . 1 : H r=r(t)=Rcosti+Rsintj t[0,2) R. 2: r=r(u,v)=Rcosusinvi+Rsinusinvj+Rcosvk (u,v) [0,2)[0, ] R. , ( ), , , ( ), , ( ). r=r(t) :
r=r(t)=x(t)i+y(t)j+z(t)k ( ) ( ) ( ), , x x t y y t z z t= = = (1) x=x(t), y=y(t), z=z(t) . r=r(u,v) :
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26
r=r(u,v)=x(u,v)i+y(u,v)j+z(u,v)k ( ) ( ) ( ), , , , ,x x u v y y u v z z u v= = = (2) .
2.4
r(t) t0 :
0 0 00 h 0( ) ( ) ( )( ) limd t t h ttdt h
+ = = =r r rr
= 0 00 0 0 0h 0 h 0 h 0( ) ( )( ) ( ) ( ) ( )lim lim limy yx x z z
r t h r tr t h r t r t h r th h h
+ + + + +i j k = =rx(t0)i+ry(t0)j+rz(t0)k :
( ) ( ) ( ) ( )0 x 0 y 0 z 0( ) r t r t r td tt dt = = + +rr i j k
.
2, , r(t). : (x,y,z) , t, x,y,z t : x=x(t), y=y(t), z=z(t).
: r(t) = x(t)i + y(t)j + z(t)k
: v(t)= ( ) ( ) ( ) ( )d t dx t dy t dz tdt dt dt dt
= + +r i j k
: 2 2 2 2
2 2 2 2
( ) ( ) ( ) ( ) ( )( ) d t d t d x t d y t d z ttdt dt dt dt dt
= = = + +v ra i j k : (t)r(t), r(t)v(t) r(t)w(t)
z
O
r(t0)
r(t0+h)
2
y
x
r(t0) r(t0+h)-r(t0)
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27
: . :
[ ]d d (t) d (t)(t) (t) (t) (t)dt dt dt
= + rr r (2.7)
[ ] ( ) ( )( ) ( ) ( ) ( )d d t d tt t t tdt dt dt
= + r vr v v r (2.7)
[ ] ( ) ( )( ) ( ) ( ) ( )d d t d tt t t tdt dt dt
= + r vr v v r (2.7) , . 1: r(t) , :
( )( ) 0d tt
dt =rr tI
: . |r(t)|=r(t)=.
r2(t)=r.r=. [ ]( ) ( ) 0d t tdt
=r r ( ) ( )( ) ( ) 0d t d tt tdt dt
+ =r rr r
( )2 ( ) 0d ttdt
=rr ( )( ) 0d ttdt
=rr
: ( )( ) 0d ttdt
=rr . : r2(t) =rr 2 ( )( ) 2 ( ) 0d d tr t tdt dt = =
rr
r2(t)=. r(t)=|r(t)|=. 2: r(t) , :
( )( ) d tt
dt =rr 0 tI
: : :
r0(t)=( ) ( )
| ( ) | ( )t tt r t
=r rr
|r0(t)|=1 , r0(t) r0(t)=0 r(t)=r(t)r0(t) r(t)=r(t)r0(t)+r(t)r0(t)=r(t)r0(t) r(t)r(t)=r(t)r(t)r0(t)= =r(t)r(t) ( ) ( )
( ) ( )t r t
r t r t=r r(t)r(t)=0 r(t)r(t)=0
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28
: r(t) ( )d tdtr =0. :
0( ) ( )( )
d t d tdt dt r t
=r r = 2( ) 1( ) ( )( ) ( )r t t tr t r t +r r 2
3
( ) ( ) ( ) ( ) ( )( )
r t r t t r t tr t
+= r r (A) r2(t)=r(t)r(t) 2r(t)r(t)=2r(t)r(t) r(t)r(t)=r(t)r(t) (B) (A) (B) :
[ ] [ ]03
( ) ( ) ( ) ( ) ( ) ( )( )( )
t t t t t td tdt r t
+ = r r r r r rr = { } { }31 ( ) ( ) ( ) ( ) ( ) ( )( ) t t t t t tr t = r r r r r r{ }31 ( ) ( ) ( )( ) t t tr t = r r r
r(t)r(t)=0 0 ( )d tdt
=r 0 r0(t)=. r(t) .
1) :
[ ]d d(t) d (t)(t) (t) = (t)+(t)dt dt dt
rr r
[ ] ( ) ( )( ) ( ) ( ) ( )d d t d tt t t tdt dt dt
= + r vr v v r
[ ] ( ) ( )( ) ( ) ( ) ( )d d t d tt t t tdt dt dt
= + r vr v v r 2) r(t)
t r(t), ( , r(t) t ), :
) (x-5)2+(y-3)2=9 ) 4x2+9y2=36 ) y=x2 ' ) y=x3 ' 3) : r1(t)=(et-1)i+2sintj+ln(t+1)k r2(t)=(t+1)i+(t2-1)j+(t3+1)k
. .
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29
4) r(t) [0,2], r(0)=i
r(t) : 2 2
2 2 1x y+ =
: ) ) ) )
-
3.1
, . , (), , ( ). . . : ) ) , ( ). ) , (. ). ) , .. . 1 r1 v R3 ( 1). 1 v. r. 1 v t 1=tv . 1=r-r1 :
1 t= +r r v (1)
(1) .
P
P1
O
r1 R
V
Y
X . 1 1
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32
1 2 r1 r2 , , v 12 1 2 . (1) :
( )1 2 1t= + r r r r (2) (1) t, r-r1 v : ( )1 =r r v 0 (3) 0 . (2) : ( ) ( )1 2 1 =r r r r 0 3.2
(1) :
r=r1+tv xi+yj+zk = x1i+y1j+z1k +t(v1i+v2j+v3k) 1 1 1 2 1 3, ,x x tv y y tv z z tv= + = + = + (4) (4) , 1 v . (2) :
( ) ( ) ( )1 2 1 1 2 1 1 2 1, ,x x t x x y y t y y z z t z z= + = + = + (5) , 1, 2 . t (4) :
1 1 11 2 3
x x y y z zv v v = = (6)
(6) , 1(x1,y1,z1) v(v1,v2,v3) . (5) :
1 1 12 1 2 1 2 1
x x y y z zx x y y z z = = (7)
1(x1,y1,z1), 2(x2,y2,z2) . 1: (6) (7) .
-
3.7 33
2: OXY, (6) (7)
1 1 1 11 2 2 1 2 1
x x y y x x y y
v v x x y y = = (8)
: 1 1 1 11 2
2 2
10 1 0
1
x yx x y y
x yv v
x y
= = (8)
(8) :
v2x-v1y+v1y1-v2x1=0 (y2-y1)x-(x2-x1)y+(x2-x1)y1-(y2-y1)x1=0
:
A B 0 0 x y+ + = + (9) : =0 0 (9) y=-/B (10)
, -/ OY (0,-/). y=0 . =0 0 : x=-/ (11)
OY Q(-/,0). x=0 OY. (,)(0,0) . (9) :
0x
y = (12)
(12) (-/,0) v(B,-A) (,,)(0,0,0) (9) . (9) x=-/ Q y=-/. =-/ =-/ . (9) :
-
34
1
A
x y+ =
1x y + = (13)
, . (13) () . =0, , (0,0). v(B,-A) (9).
3.3
OXY, , , . , 2, v(v1,v2)= :
: =tan= 1 21 1 1 2
P Psin v sinPAOA OP P A v v cos
= =+ + (14)
x+By+=0 v=(,-), v1=B, v2=-A :
Asin
B Acos = (15)
V
P1 A X
2
2
-
3.7 35
, =/2 :
0 = (16)
:
y x = + = (17) 1 2
12 =(y2-y1)/(x2-x1) :
y-y1=(x-x1) y-y1= 2 1 12 1
( )y y x xx x 1 1
2 2
11 01
x yx yx y
= (18)
3.4 .
(1) (2) :
(1): A1x+B1y+1=0 (2): A1x+B1y+1=0 (19)
(1), (2) , v1(B1,-A1) v2(B2,-A2) :
1 12 2
= 1 1
2 2
= (20)
. 1: (19) . (1) (2) :
1 21 2
= 1 2
1 2
= 1 1 1
2 2 2
= = (21)
: 2: (19) .
-
36
(20), , 12-210 (19) x y:
2 1 1 2 1 2 2 11 2 2 1 1 2 2 1
A A,x y = = (22)
3.5 . .
(1), (2), (3) :
(1): A1x+B1y+1=0
(2): A2x+B2y+2=0 (23)
(3): A3x+B3y+3=0
(23) x y. (10 r() r() :
1 1
2 2
3 3
=
1 1 1
2 2 2
3 3 3
AAA
=
: 1) r()=1, r()=1 (21)
.
2) r()=1, r()=2, (20) . ( ).
3) r()=2, r()=2, (23) . , ( ).
(10 , ( ' ), r : 1) r . 2) r+1 .
(1)
(2)
-
3.7 37
4) r()=2, r()=3, :
) ) .
:
||=1 1 1
2 2 2
3 3 3
AAA
=0 (24)
. : 1: (24). P1(x1,y1), P2(x2,y2) :
(4)
(3)
(4)
-
38
1 12 1 2 1
x x y yx x y y = :
1 12 1 2 1
0x x y yx x y y = 1 1
2 2
11 01
x yx yx y
= (25)
P3(x3,y3,z3) , (25)
3 3
1 1
2 2
11 01
x yx yx y
= 1 1
2 2
3 3
11 01
x yx yx y
= (26)
(26) . 1 : . . 2: (1): 1x+B1y+1=0 (2): 2x+2y+2=0 (27) , (), , :
1 1 1 1 2 2 2 2( B ) ( ) 0 x y x y + + + + + = (28) 1, 2 . : (28) . (27) , (28), (28) . (27) , :
1 2 1 1 2 21 2 1 1 2 2
+ = = + (29) (28) (27), . 1=0 2=0 (2) (1). : (27) (28). 1, 2 1/2, ( 2/1). P1(x1, y1) , ,
1 2 1 2 1 22 1 1 1 1 1
BB
x y x y
+ += + + 1 (28) (28) , .
-
3.7 39
3.6.
:
(1): 1x+B1y+1=0 (1): 2x+2y+2=0 (30)
: v1=(B1,-A1) v2=(B2-A2) . . :
1 2 1 2 1 2 1 2 1 22 2 2 2 2 2 2 2
1 2 1 1 2 2 1 1 2 2
B B ( A )( A ) A A B Bcos| || | A B A B A B A B
+ + = = =+ + + +
v vv v
(31)
cos=0 12+12=0 (32)
1, 2 (1) (2), (32) :
12=-1 (33)
1=-1/1 2=-2/2. 1, 2 (1) (2) (1) (2) =1-2
tan=tan(1-2)= 1 2 1 21 2 1 2
tan tan1 tan tan 1
=+ + (34) (34) . 1, 2 (34)
2 1 1 21 2 1 2
tan = +
3.7 .
Ax+By+0 () Ax+By+=0 OXY.
(x,y) () Ax+By+ . : Ax+ By+ . 1(x1,y1) P2(x2,y2) () .
1 2
. 3
(2) (1)
3
-
40
0(x0,y0) () (). 10 02 , :
10= 02 (x0-x1)=(x0-x2) (y0-y1)=(y0-y2)
1 20 1x xx
+= +
1 20 1
y yy
+= + (35)
0 Ax0+By0+=0 (35) : Ax1+By1++(Ax2+By2+)=0 0 1 2, >0 Ax1+By1+ Ax2+By2+ , 0 12 0
-
3.7 41
d(P0,). v(,) () () 0 () :
x=x0+tA, y=y0+tB (36)
t () () :
A(x0+tA)+B(y0+tB)+=0 0 0
2 2
Ax Byt + += + (37)
(36) t (37), (x1, y1) 1, :
2
0 01 2 2
A A y B x )xA B
+ = +(
20 0
1 2 2
(B x A y )yA B
+ = +
d(P0, ) d(P0, P1).
( ) ( )2 2 0 00 0 1 1 0 1 0 2 2Ax By
d(P , ) d(P , P ) x x y y+ + = = + = +
(38)
(38) x1 y1.
: v 10,
:
) v( 10)=A(x0-x1)+B(y0-y1)=Ax0+By0+ ( AX1+By1=-) ) v( 10)=|v||P1P0|= 2 2A +B d(P0,1)
0 00 1 2 2| Ax y |d(P ,P )
A B+ += +
1
5
()()
d(P0,)
P0
v
O
r0
X
-
42
1) (-2, 3)
2x-3y+6=0. (. 3x+2y=0)
2) 1(7, 4) 2(-1, -2). (. 4x+3y-15=0)
3) (4, -2) d=2. (. 4x+3y=10 y=-2) 4) () 12x-5y-15=0.
() () d=4. (. 12x-5y+37=0 12x-5y-67=0)
5) ) -4 ) (4, 1) ) 7 ) 8
) .
(. . 4x+y-c=0 . cx-y+1-4c=0 . cx-y+7=0 . (8-c)x+cy-8c+c2=0 . 2x+y-c=0) 6) (x1,y1), B(x2, y2), (x3,y3) , ,
E , :
1 1
2 2
3 3
11 12
1
x yE x y
x y=
-
IV
4.1
, . . , ..: ) ) ) ) . .
) 1(r1) v1, v2, 1, 1. (r) . 1, v1, v2 , , 1=v1+v2 .
r=r1+v1+v2 (1)
(r), (1) , (1) 1, v1, v2 . v1v2 . , ( ), (1) v1v2 :
1 1 2( ) [ ] 0 =r r v v (2)
Y
v1
v2
1
P1
P
Z
X
-
44 V
.
) . P1(r1), P2(r2) v 12. 1 12, v . :
( ) = + +1 2 1r r r r v (3) :
( ).[( ) ] 0 =1 2 1r r r r v (4)
) P1(r1), P2(r2), P3(r3). 1 12, 13, : ( ) ( ) = + + 1 2 1 3 1r r r r r r (5) : ( ) [( ) ( )] 0 =1 2 1 3 1r r r r r r (6)
)
1(r1) N , 2. (r) , 1 1=0. :
( ) 0 =1r r N (7) .
P
Y
2
N
X
Z
P1
-
45
4.2
(1), :
1 1 2
1 1 2
1 1 2
x xy yz z
= + += + += + +
(8)
r1=(x1,y1,z1), v1=(1,1,1) v2=(2,2,2). (8) . (8) (2) , :
1 1 1
1 1 1
2 2 2
0x-x y-y z-z
= (9)
(3) :
1 2 1
1 2 1
1 2 1
( ) ( ) ( )
x x x xy y y yz z z z
= + += + += + +
(10)
:
1 1 1
2 1 2 1 2 1 0x-x y-y z-zx -x y y z z
= (11)
() :
( ) ( )( ) ( )
1 2 1 3 1
1 2 1 3 1
1 2 1 3 1
( ) ( ) x x x x x xy y y y y y
z z z z z z
= + + = + + = + +
(12)
:
1 1 1
2 1 2 1 2 1
3 1 3 1 3 1
0x x y y z zx x y y z zx x y y z z
=
(13)
H (13) :
-
46 V
1 1 12 2 2
3 3 3
11
011
x y zx y zx y zx y z
= (14)
(14) .
() (7) : (x-x1)N1+(y-y1)N2+(z-z1)N3=0 (15) 1: (9), (11), (14), (15) :
Ax+By+z+=0 (16)
, (16) . , P1(x1, y1, z1), P2(x2, y2, z2), (16) (,,) . :
Ax1+By1+z1+=0 Ax2+By2+z2+=0
:
A(x2-x1)+B(y2-y1)+(z2-z1)=0 (17)
(17) (,,) 12. (16). (16) . 2: =i+Bj+k Ax+By+z+=0. f(x,y,z) f=f(x,y,z) f(x,y,z) f(x,y,z)=c=. f=f(x,y,z)= =Ax+By+z+=0 =Ai+Bj+k (17).
4.3 Ax+By+z+=0
Ax+By+z+=0 ||+||+||0, , , . :
-
47
) ==0 0, (16) x=-/ Ax+By+z+=0 (-/, y,z) (-/,0,0), OYZ.
) ==0 0, (16) y=-/ Ax+By+z+=0 (x, -/,z) Y (0,-/,0), O.
) ==0 0, (16) z=-/ Ax+By+z+=0 (x, y,-/) (0,0,-/), OZ.
x
z
y -/
()
x
z
y
-/
()
()
-/
z
y
x
-
48 V
) =0, (16) :
Ax+By+=0 (18)
OXY (1) (18). (18) (1) (1) OXY . ) =0, (16) :
Ax+z+=0 (19)
OX (2) (19). (19) (2) (2) OX .
) =0, (16) :
y+z+=0 (20)
OY (3) (20). (20) (3) (3) OY
x+By+=0
() x
z
y
y+z+=0
() x
z
y
x+z+=0
() x
z
y
-
49
X. ) =0, (16) :
x+By+z=0 (21)
) ,,,0, (16) :
1x y z+ + = (22)
=-/, =-/, =-/ . (22) .
4.4
1(x1,y1,z1) v :
1 1 11 2 3
x x y y z zv v v = = (23)
1(x1,x2,z1) P2(x2,y2,z2) :
1 1 12 1 2 1 2 1
x x y y z zx x y y z z = = (24)
(23), :
v2x-v1y-v2x1+v1y1=0, v3y-v2z-v3y1+v2z1=0 (25)
x/+y/+z/=1
() x
z
y
y
x+y+z=0
() x
z
-
50 V
(24) :
(y2-y1)x-(x2-x1)y-(y2-y1)x1+(x2-x1)y1=0
(z2-z1)y-(y2-y1)z-(z2-z1)y1+(y2-y1)z1=0 (26)
:
A1x+B1y+1=0 2y+2z+2=0 (27)
(27) . , OXY OYZ :
(1): A1x+B1y+1z+1=0
(2): A2x+B2y+2z+2=0 (28)
. , (28). 1=1i+1j+1k 2=2i+2j+2k (1) (2) . : ) 1 2 , :
1=2 1=2, 1=2, 1=2
1 1 12 2 2
= = (29)
(1) (2) 12. :
1 1 1 12 2 2 1
= = = (30)
. ) 12, (1) (2) .
4.5 .
:
(1): A1x+B1y+1z+1=0
(2): A2x+B2y+2z+2=0 (31)
(3): A3x+B3y+3z+3=0
, :
-
51
) (1), (2), (3)
) (1), (2), (3)
) (1), (2), (3)
) (1), (2), (3)
1 2 3
()
3
2
1
()
1 2
3
()
()
()
1
2
3
-
52 V
) (1), (2), (3) .
, .
1 1 1
2 2 2
3 3 3
=
1 1 1 1
2 2 2 2
3 3 3 3
BBB
=
r(), r(B) , : ) r(A)=r(B)=1
(30) (1), (2), (3) . ) r(A)=1, r(B)=2 (29)
(30) . ) r(A)=2, r(B)=3 (31) ) r(A)=3, r(B)=3 (31) ,
. ) r(A)=2, r(B)=2 (31)
.
:
1 1 1
2 2 2
3 3 3
0 =
(32)
:
(4): A4x+B4y+4z+4=0 (33)
1 2
3
()
-
53
(31) (33)
1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
0
=
(34)
(34) .
4.6
. .
(1): A1x+B1y+1z+1=0 (2): A2x+B2y+2z+2=0 (35)
, :
1 1 1 1 1 2 2 2 2 2(A B ) (A B ) 0x y z x y z + + + + + + + = (36)
(36) . . 0 (1) (2) 1=A1i+B1j+1k, 2=A2i+B2j+2k (1) (2) . : (r-r0)N1=0, (r-r0)N2=0. : (r-r0)N=0, 1 2 =11+22 :
(r-r0)N=0 (r-r0)(11+22)=0 1(r-r0)N1+2(r-r0)N2=0 1(A1x+B1y+1z+1) + 2(A2x+B2y+2z+2)=0
, . . , :
1 1 1 1 1 2 2 2 2 2 3 3 3 3 3(A B ) (A B ) (A B ) 0x y z x y z x y z + + + + + + + + + + + = (37) (37) .
4.7
. P1(x1,y1,z1)
-
54 V
Ax+By+z+ P2(x2,y2,z2) . =i+Bj+k , (;). (r-r0)N=0, (r0) . P1(x1,y1,z1) . 1 , ( 3) 01 . :
( )1 00 1d = = r r NP P NN N
(38)
1 0 1 0 1 02 2 2
A( ) B( ) ( )d
A B
x x y y z z + + = + +
Ax0+By0+z0=-
1 1 1
2 2 2
A Bd
A B
x y z+ + + = + + (39)
0 v1, v2 (38) :
0 1 1 2 1 2 1 21 2 1 2
( ) ( ) ( )d
= = P P v v r r v v
v v v v (40)
v1v2 .
4.8 .
.
.
. ,
(1): A1x+B1y+1z+1=0
(2): A2x+B2y+2z+2=0
,
1=1i+B1j+1k 2=2i+B2j+2k
3
P1
P0
N
1 2
1 2
4
-
55
:
1 2 1 2 1 2 1 22 2 2 2 2 2
1 2 1 1 1 2 2 2
cos| || |
+ + = = + + + + N NN N
:
12+12+12=0 (41)
(1) (2). 12=0
1 1 12 2 2
= =
(29) :
(1): 1 1 11 2 3
x x y y z zv v v = =
(2): 2 2 21 2 3
x x y y z zw w w = =
v=v1i+v2j+v3k w=w1i+w2j+w3k (1) (2) . :
1 1 2 2 3 32 2 2 2 2 21 1 1 2 2 2
v w v w v wcos| || | v v v w w w
+ += = + + + +v wv w
(42)
()
() :
1 1 11 2 3
x x y y z zv v v = =
Ax+By+z+=0 . - v=v1i+v2j+v3k, ( ), . :
(2)
(1)
v1
v2
5
()
6
-
56 V
1 2 32 2 2 2 2 21 2 3
v v vcos| || | v v v
+ + = = + + + + v Nv N
(43)
,, v OX, OY, OZ ,
cos=cos(i,v)= 1| |vv
, cos=cos(j,v)= 2| |vv
, cos=cos(k,v)= 3| |vv
(), .
, , , OX, OY, O , :
2 2 2Acos cos( , )
A B = = + + i N
2 2 2
Bcos cos( , )A B
= = + + j N
2 2 2
cos cos( , )A B
= = + + k N
Ax+By+z+=0
xcos+ycos+zcos+d0=0 (44)
d0 ,
0 2 2 2d = + +
:
(44) Hesse.
4.9 .
() r=r0+tv, P0(r0) v, P1(r1) , 7. 1 (), 1, 01 :
P1
K
P0
v ()
7
-
57
d=KP1=P0P1sin (A). |vP0P1|=|v||P0P1|sin (B). (A) (B) :
0 1 0 1| | | ( ) |d d
| | | | = =v P P v r r
v v (45)
4.10 .
(1) (2) :
(1): r=r1+tv1 (2): r=r2+tv2
12, v1, v2 ( 8). :
12(v1v2)=0 ( ) ( ) 0 =2 1 1 2r r v v (46)
(1) (2) , (46) . (1) (2) v1, v2 v1v2. H (1) (2) 1P2 v1v2,
( )2 1 1 21 2
| ( ) |d
| | =
r r v vv v
(47)
() (1) (2). v1v2 . (2) v1v2, . (1) . :
(r-r2).[v2(v1v2)]=0 (48) H (1) (48) , t=t0 :
()
(1)
(2)
d
P1
v1
P2v2
8
v1v2
-
58 V
(r1+t0v1-r2).[v2(v1v2)]=0 t0v1.[v2(v1v2)]+ (r1-r2).[v2(v1v2)]=0
221
212120
)]([)(vv
vvvrr
=t
:
1221
212121
)]([)( vvv
vvvrrr +
:
( )211221
212121
)]([)( vvvvv
vvvrrrr ++= t (49)
P1(x1, y1, z1), P2(x2, y2,z2), v1(1, 1, 1), v2(2, 2, 2) (1), (2) :
(1): 1 1 11 1 1
x x y y z z = = (2): 2 2 2
2 2 2
x x y y z z = =
(46) :
2 1 2 1 2 1
1 1 1
2 2 2
0x x y y z z
= (50)
(47) :
2 1 2 1 2 1
1 1 1
2 2 2
2 2 21 2 2 1 1 2 2 1 1 2 2 1
d( ) ( ) ( )
x x y y z z
= + + (51)
-
59
1) , ,
1(-3,2,1), 2(9,4,3) . (. 6x+y+z=23)
2) , 1(1,-2,3)
x-3y+2z=0. (. x-3y+2z=13)
3) , 1(1,0,-2)
2x+y-z=2 x-y-z=3. (. 2x-y+3z+4=0)
4) , 1(1,1,-1),
2(-2,-2,2), 3(1,-1,2) (. x-3y-2z=0)
5) d (-2,2,3) 8x-4y-z-8=0. (A. d=35/9) 6) 3x+2y-5z=4 2x-3y+5z=8. (. cos=25/38) 7) ,
3x+y-5z+7=0 x-2y+4z-3=0 (-3,2,-4). (. 49x-7y-25z+61=0)
8) : 7x+4y-4z=-30 36x-51y+12z=-17 14x+8y-8z=12 12x-17y+4z=3
. 9) t : OP=r(t)=(1-t)i+(2-3t)j+(2t-1)k
) . ) t
2x+3y+2z+1=0 ; ) , t=3.
(. () , () t=1, () 2x+3y+2z+20=0)
-
60 V
10) , 0(1,2,-3) 3x-y+2z=4. ;
(A. 3x-y+2z+5=0, d=9/14) 11) ,
2x-y+2z+4=0, 0(3,2,-1) .
(A. 2x-y+2z-8=0) 12) (1,2,3) (3,3,1) v=i-2j+2k . ()
, v. ) (). ) (), , .
(. (5/9,26/9,19/9) 13) (,,0), (0,1,0), (0,0,1), (,,).
) , , , . )
. (. () 2x+2y+4z-2=0, () d= 1 34 8
)
14) () :
(1): 4x-3y+8z-5=0, (2): -3x+2y-3z-1=0 d=A () () .
(A. 1059d194
= , 0 0 00 0 0A A A
x x y y z zx x y y z z = = (xA,yA,zA)=
319 833 1025, ,192 192 192
(x0,y0,z0)=(0,0,0) ) 15) () : x+y-z=1, 2x+3y+2z=2
Oxz. (A. x=1+5z)
16) (): 2x+3y+z+4=0 .
, v1=i+j+k v2=i+j . (. =)
-
V
5.1
OXY, , Ax+By+=0 x y. x y, : Ax2+2Bxy+y2+2x+2Ey+Z=0 (11 (1)
(1) . (1) :
) , ( ) ) )
, ( 1), o . ,
1
, . .
, , , , , , (1) x y. x y , (1) , ( (12).
(11 2 2Bxy, 2x, 2Ey . (12 , B
-
62 V
(1) , :
AB 0
=
(2)
0 (1) : Ax2+2(By+)x+y2+2Ey+Z=0 (3)
x. (3) , (3) x , y. (3) x :
2 2(By ) (By ) A( y 2Ey Z)x
A + + + += (4)
(4) y, ( ), . :
(B2-)y2+2(-)y+2- (5)
(-)2-(2-)(2-)=0 (6)
(6) (2).
, (2) (1) , 0 y x.
5.2
(1):
Ax2+2Bxy+y2+2x+2Ey+Z=0
P1(x1, y1), P2(x2, y2). P(x, y) , : x=x1+t(x2-x1), y=y1+t(y2-y1) (7) (7) (7) :
A[x1+t(x2-x1)]2+2B[x1+t(x2-x1)][y1+t(y2-y1)]+[y1+t(y2-y1)]2+2[x1+t(x2-x1)]+
-
63
+2E[y1+t(y2-y1)]+Z=0 (8)
(8) :
2L 2M N 0t t+ + = (9) : L= A(x2-x1)2+2B(x2-x1)(y2-y1) +(y2 -y1)2 (10)
M= Ax1(x2-x1)+B(x2-x1)y1+B(y2-y1)x1+(y2-y1)y1 +(x2-x1)+E(y2-y1) (11)
N= Ax12+2Bx1y1+y12+2x1+2Ey1+Z (12)
(9) t, (7) , 1 2, . . , (9) , 1 2 . (9) :
2M LN 0 = (13) (13) , 1, 2, . , 1(x1, y1) , P(x, y) . 1, (13). 1 (12) : N=0 (13) : 2=0 =0
1 1 1 1 1 1A B( ) ( ) E( ) Z 0x x x y y x y y x x y y+ + + + + + + + = (14) (14)
: f(x,y)=Ax2+2Bxy+y2+2x+2Ey+Z, : f(x,y)=(2Ax+2By+2)i+(2y+2Bx+2E)j f(x,y). P1(x1,y1), r1(x1,y1), f(x,y)=0 r(x,y) P(x,y) P1(x1,y1). :
(r-r1)f(r1)=0 (x-x1)[2Ax1+2By1+2]+(y-y1)[2y1+2Bx1+2E]=0 (14) P1(x1,y1) , , P1(x1,y1) ,
-
64 V
: P(x,y) . P(x,y) P1(x1,y1) (13), :
[Ax1(x-x1)+B(x-x1)y1+B(y-y1)x1+(y-y1)y1 +(x-x1)+E(y2-y1)]2-
-[A(x-x1)2+2B(x-x1)(y-y1) +(y -y1)2][Ax12+2Bx1y1+y12+2x1+2Ey1+Z]=0 (A) () (2). , P1(x1,y1) .
.
5.3
F1 F2. , F1, F2 ( 2). 2c 2 >c. OXY, F1(-c,0), F2(c,0) P(x,y) . :
|PF1|+|PF2|=2 2 2 2 2(x c) y (x c) y 2+ + + + = (15) (15)
2 2
2 2 2 1x y c
+ = (16)
>c 2-c2 2-c2=2. (16) :
2 2
2 2 1x y
+ = (17)
X
2
F1(-c,0) F2(c,0)A1 A2
B1
B2 P(x,y)Y
-
65
(17):
) x -x y -y. OX OY . ) (17) y2/2=1-x2/2 1-x2/20 -x. -y. (,), (-,), (-,-), (,-) .
, F1, F2 . 1(-,0) A2(,0). A1A2 . OY 1(0,-) 2(0,) 12 . |2F1|+|2F2|=2 |2F1|=|2F2|=. 1, 2, 1, 2 . c=0, = . x2+y2=2. c=, =0, 1, 2 , () 12 . . c , 12 .
5.4 .
F1, F2 2,
: 2. ( 3) .. |F1|+|F2P|=2, . , , . .
3
F2 F1
P P
-
66 V
: 2 2
2 2 1x y
+ = >
, 4.
, . , Q . . . (x, y), (x, y), (x, y) , , , , : x2+y2=2, x2+y2=2, x/x=y/y x, y . t x, y
cos , sinx t y t = = (18) (18) . t , (x=rcos, y=rsin).
5.5
5.2 P1(x1,y1). , (14) :
B1(0,-)
Y
X
B2(0,)
1(-,0) 2(,0)
N(x,y) P(x,y) M(x,y)
O Q(x,0)
t
4
-
67
12 11
2
x x y y
+ = (19)
P1(x1,y1) , () . 5.2 , P1(x1,y1) :
1 1 12 2 2 2 2
1 1 1 12 2 2 2 2 2
x x y y x y x y+ - - + - + - =0
(19)
(19) =-2x1/2y1. 1 =2y1/2x1 : 2 21 1 1 1( ) ( ) 0y x x x y y = (20) f=f(x,y). f , :
f(x,y)=fx(x,y)i+fy(x,y)j = 22x i+ 22y j
1(x1,y1) f(x1,y1) : r=r1+tf(x1,y1) x=x1+t 122x y=y1+t
12
2y
t (20).
5.6 .
. , ( ), e :
2ce 1 = = (21)
0
-
68 V
, . e . 5.
r1 r2 P(x,y) F1 F2 , r1=|PF1|, r2=|PF2|, 6, : r12=(c+x)2+y2, r22=(c-x)2+y2 r12-r22=4cx
r1+r2=2 r1-r2=2cx/
r1=+cx/, r2=-cx/
e=c/
2
1r e cx
= +
2
2r e cx
= (22)
(22) 1 22 2
r r e
c cx x= = +
(23)
(d1): x=-2/c, (d2) : x=2/c
2/c+x 2/c-x P(x,y) (d1) (d2) . (23) :
5
(d1)
O
B1
6
(d2) B2 P(x,y)
A1 A2F1 F2 X
r1 r2
-
69
F1, ( F2) (d1), ( (d2)), . .
(d1), (d2) . , >c 2/c>. : (x0,y0) , :
2 2
0 02 2
( ) ( ) 1x-x y-y
+ = : ) . ) , ( ), , .
5.7
, , . F1 F2 . , F1, F2 , . F1, F2 . , |F1F2|=2c | |F1P|-|F2P| |=2, . c> . , OXY F1F2 ( 7). F1, F2 (-c,0) (c,0) . P(x,y) ., :
-
70 V
2 2 2 2( ) ( ) 2x+c y x-c y + + = 2 2 2 2( ) 2 ( )x+c y x-c y+ = + +
(x+c)2+y2=42+(x-c)2+y24 2 2(x-c) +y 2 2(x-c) +y = cx
x2-2cx+c2+y2=2 2
2
c x -2cx+
2 2 2
2 2 2 22
c x y c + =
12 2
2 2 2
x y+ = -c
c> c2-2=2 :
2 2
2 2 1x y
= (24) (24) . , . . x2/21 |x| x=- x=. OY 1(-,0) 2(,0), . 12
1(0,-), 2(0,) 12, (24), 7 F1, F2 OY, ,
2(0,)
7
F1(-c,0) F2(c,0) A1(-,0) A2(,0)
1(0,-)
x=- x=
X
Y
-
71
12 2
2 2
y x-
= (25)
(24), ( 8) :
12 2
2 2
x y-
=
12 2
2 2
y x-
= (26) (24) (26) (24) (26). , 9.
(0,0), c= 2 2 + .
5.8
F1, F2 2.
: -2. .. F1 . F2, ( 10). ,
2(0,)
9
F1(-c,0) F2(c,0) A1(-,0) A2(,0)
1(0,-)
X
Y
-
72 V
F1. , 10 :
|F1P|-|F2P|=(|F1P|+PB|)-(|F2P|+PB|)=-(-2)=2
5.9
cosh2t-sinh2t=1 :
2 2
2 2
x y 1 =
:
cosh sinhx t y t = = -
-
-
73
Ax2+2Bxy+y2+2x+2Ey+Z=0
, :
=1/2, =0, =-1/2, =0, =0, =-1
(8) 5.2 :
Ax1x+B(x1y+y1x)+y1y+(x1+x)+E(y1+y)+Z=0
:
1 12 2x x y y- =1
(28)
(28) P1(x1,y1). (28) =(2x1)/(2y1) P1(x1,y1) =-(2y1)/(2x1) :
2 21 1 1 1( ) ( ) 0y x x x y y + = (29)
() . 5.2 P1(x1,y1) , , :
2
1 1 1 11 1 1 02 2 2 2
2 2 2 2 2 2
x x y y x y x y- - -
= (30)
y (24) :
2 2y x= 2
21xy
x = (31)
x , 2
21 x
(31) :
y x= (32)
(32) , ( 11). :
-
74 V
y x= (33) , (x>0, y>0),
2 2y x= , d(P1, P2) 1 2 (32), :
d(P1, P2)= ( )2 2 2 2 2 2x x x x x x = = + x. (x
-
-
75
, . 1 y=-x/ OY1 y=x/. 1 tan=-/ OY1 Y tan=-/, ( 12)
(13) 1.5. :
x=Xcos-Ysin y=Xsin+Ycos (24) :
( ) ( )2 2
2 2
Xcos Ysin Xsin Ycos1
+ = 2 2 2 2
2 22 2 2 2 2 2
cos sin sin cos sin cos sin cosX 2 XY Y 1 + + =
tan=-/, tan=-/ 2 2
2 2
cos sin 0 = , 2 2
2 2
sin cos 0 = , 2 2sin cos sin cos + = 2 2
2 + (36)
2 2 2cXY
4 4 += =
2cXY=4
(37)
. . , (37) .
12
y=x/ y=-x/
1
1
F1
F2
-
76 V
5.11
. e=c/ .
e>1, ( e
-
-
77
r1=+cx/, r2=-(-cx/) r1=-(+cx/), r2=-cx/ (38)
c/=e=, :
r1=e(x+2/c), r2=-e(x-2/c) r1=-e(x+2/c), r2=e(x-2/c)
: (d1): x=-2/c (d2): x=2/c. (x+2/c), (x-2/c) (13 (x,y) (d1) (d2) . (38) :
1 22 2r r e
x xc c
= = + (39)
F1, ( F2), (d1), ( (d2)), . . (d1), (d2) . 2/c
-
78 V
) . GPS . .
) . .
) . .
) . , . ( 8 8.4).
) . . .
5.12
, F (d). F (d) . F (d) OY F ( 14). p ,
O
. 14
F(p/2,0) X
Y
P(x,y)
(d)
M
-
-
79
F(p/2,0) x+p/2=0 P(x,y) , :
|PF|=|PM| 2
2
2 2p px y x + = +
2 22
2 2p px y x + = +
2 22 2 2
4 4p px px y x px + + = + +
2 2py x= (44) (44) p . , , , . (0,0), . x>0 OY.
F , y2=-2px OY, 15. : x2=2py ( 16) x2=-2py ( 17).
15
Y
O XF
(d)
Y
16
O
(d)
X
-
80 V
5.13
y2=2px. : , , ( 18). F. . , . . : +=F+ =F
y2=2px x=2pt2, ( 2px>0), y=2pt. : x=2pt2 y=2pt
.
5.14
:
Ax2+2Bxy+y2+2x+2Ey+Z=0
, :
(d)
F
17
X
(d)
A
B
18
P
F
-
-
81
=0, =0, =1, =-2p, =0, =0
(8) 5.2 : Ax1x+B(x1y+y1x)+y1y+(x1+x)+E(y1+y)+Z=0 :
1 1p( )y y x x= + (45) (45) P1(x1, y1). P1(x1, y1) , () . 5.2 P1(x1, y1) :
( ) ( )( )2 2 21 1 1 12 2 0y y p x x y px y px + = . (45) p/y1 P1(x1, y1) -y1/p :
11 1 1 1 1 1( ) p p 0yy y x x y x y y x yp
= + =
5.15
, :
: P1(x1, y1) P1F , 1 ( 19). : y2=2px P1(x1, y1). n(p,-y1) P1(x1, y1) (45). P1F (p/2-x1, -y1), 1 i.
21 1
1 22 2 2
1 1 1
p - p+2cos( , )
p - + +p2
x y
x y y
=
P F n
2 21
pcos( , )+py
=n i
(d) P1 n E
F
Y
X
19
-
82 V
21 1 1 1 1p p p- p+ - p+2p p +2 2 2
x y x x x = =
2 2 22
1 1 1 1 1 1p p p p- + = - +2p = + = +2 2 2 2
x y x x x x
1 2 21
pcos(P F,n)= =cos(n,i)+py
. . , , . , , , .
: , , e . e 0
-
-
83
. (. 3x2+4y2=192)
5) H . 93 1/62. .
(. 91.500.000 - 94.500.000)
6) ,
: x2+y2=1 x2+y2-4x-21=0. (. (x-1)2/9+y2/8=1 (x-1)2/4+y2/3=1) 7) 12 .
=8 . , , . . (. x2+4y2=64)
8)
(1) 4x-3y+11=0 (2) 4x+3y+5=0 144/25. . (. (x+2)2/9-(y-1)2/16=1)
9)
1(0,4) 4/3 () 4y-9=0. . (. y2/9-x2/7=1)
10) , , (6,0) 4x-3y=0. (. x2/36-y2/64=1)
11) y2+8y-6x+4=0.
, . (. (-2,4), (-,-4), x=-7/2)
12) 0
-
84 V
14) x2-y2=1. ,
) . ,
) . ) .
(. =0/2, 0 . )
15) C ,
, =. C (4,5) . (. x2-y2=-9)
16)
x+y+1=0. (. x2-2xy+y2-2x-2y=1) 17) 3x2+y2=1
r(t)=x(t)i+y(t)j. ) ) ) .
(. , dy(t)/dt=3x(t), =2/3) 18) x2+c(y-x)=0 c>0,
t . (c,0) (0,0), T1 (c,0) (c/2,c/4) . (. 1=3/4)
c/2
c/4
c
-
V
6.1 .
, . , . :
) (1)
) (1) .
(1) , (x,y) (-x,-y). . , xy x y , , :
Ax2+2Bxy+y2+2x+2Ey+Z=0 (1)
, , , , , (1) :
) , ( )
)
)
) , ( ).
, (1), . (1) (1). : (x,y) (x,-y) (x,y) (-x,y).
-
86 V
(1) . (1), ( ), .
6.2 . OXY OXY x0i+y0j . , , (x,y) (X,Y) :
x=x0+Xcos-Ysin y=y0+sin+Ycos (2)
(2) (1) :
AX2+2BXY+Y2+2X+2EY+Z=0 (3) =Acos2+2sincos+sin2 =-sincos+B(cos2-sin2)+sincos (4) =Asin2-2Bsincos+cos2 =(Ax0+By0+)cos+(Bx0+y0+E)sin E=-(Ax0+By0+)cos+(Bx0+y0+E)sin (5) Z=(Ax0+By0+)x0+(Bx0+yo+E)y0+x0+Ey0+Z (6) , , . (4) :
+=+ (7) (4) :
=- 12
(-)sin2+cos2 (8)
(4), :
12
(-)= 12
(-)cos2+sin2 (9)
(8) (9) :
-
87
2 2 2 21 1B (A ) B (A )
4 4 + = + (10)
(7):
2+ 14
(-)2- 14
(+)2=2+ 14
(-)2-14
(+)2 (11)
(11) :
2-=2- (12)
A A BB B = (13)
:
=
(14)
:
1 2 3J A , J , J = + = =
(15)
.
6.3
(x0,y0) , OXY ==0 (5) (x0,y0) : Ax0+By0+=0 Bx0+y0+E=0 (16)
2J 02 = = (17)
J2=0 , . :
Ax2+2Bxy+y2+2x+2Ey+Z=0
-
88 V
(17): -20. -20, (3) : AX2+2BXY+Y2 +Z=0 (18) , =0, ( xy ). (4) :
2Btan2
A = (19)
. = 0, ( ), (19) tan2 cos2=0 =/4. , (19), xy. :
AX2+Y2 +Z=0 (20) -2==-2 0, 0, 0. 0, (20) :
2 2X Y 1Z Z
A
+ =
(21)
|/|=2, |/|=2 (21) :
2 2
2 2
x y 1+ = (22)
2 2
2 2
x y 1+ = (23)
2 2
2 2
x y 1 = (24)
2 2
2 2
x y 1 + = (25) =0, (20) ) ,
) Y= A X , .
-
89
2 2 2
22
Z Z Z Z ZA A A B J = = =
J2>0 0, (22) (23) J20, (21) , -/>0, -/>0 =J2>0
Z Z 0A
AZ 0 +
3 1
2 2
0J JJ J J1J30 (28) J2
-
90 V
2 22E ZX
A A 2E 2A E + = +
(32)
2Z,
A 2E 2A E + ,
XA=
EpA= XY.
, : x2=2py (33) =0 (30) :
2 AZXA
= (34) , 2->0, 2-0, 2-=0.
, :
Ax2+2Bxy+y2+2x+2Ey+Z=0
J1, J2, J3 :
J1J3 0 J1J3 >0 J3 = 0
-
91
J3 0 J20 2->0 ) 2-=0 2-=0 ) 2-
-
92 V
J3=A 1 2 5B 2 1 4 18
5 4 7
= =
J20, J1J2
-
93
5) Ax2+Bxy+y2=0 y=m1x y=m2x :
) 0 ) m1+m2=-/ m1m2=/ ) 2=4 ) +=0
-
VII
7.1
, , . , 1. ||, , r , , , . . , ( ). (r,) . , (r,), , , r . r , + -. r0, (r,) , r=0 .
. . OY
r
O
P
X 1
|r|
P
r
-
96 V
, 3. (x,y) (r,) :
rcos , sin x y = = (1) . (1) r :
2 2 , arctan 0yr x y xx
= + = (2) (2) .
F(r,)=0 r=f(),
C={(r,) / F(r,)=0 r=f()} . : ) , .
) F(r,)=0 . F(r,)=0 . : ) , , , : r=r0=. r0 =0=. 0 . ) -r r, .
) - , - =/2.
.
7.2,
) , 0 ||=p, (p,0), 4. (r,) , :
Y
y P(r,)
x
r
3
-
97
p==rcos(-0) 0
prcos( )
= (3) (3) , x+By+=0 x,y (1):
Arcos+Brsin+=0 (4)
:
0cos p = 0
BtanA
=
: 0
0 0
cos pArAcos Bsin cos( ) cos( )
= = = +
(3). p=0 (3) cos(-0)=0 =k, .
) (r0,0) R. 5 :
R2=r02+r2-2rr0cos(-0) (5)
(5) . (1) x=rcos, y=rsin (x-x0)2+(y-y0)2=R2. (5). ) (1) :
Ax2+2Bxy+y2+2x+2Ey+Z=0 . , p, 6. (r,) . || ,
0
P(r,)
r
-0
5
r0
P(r,)
(p,0)
-0
0
p
4
r
6
R
P
-
98 V
e , :
|OP|/||=e ||=r ()=()=()+)=p+rcos
:
r ep r cos
=+ epr
1 ecos= (6)
(6) . e1 . , :
epr
1 ecos= +
7.3 .
, . (r,) :
(r,+2k) (-r,++2k) kZ r=f() f(r,)=0 :
() r=f(+2k) f(r,+2k)=0 kZ () -r=f(++2k) f(-r,++2k)=0 kZ
k=0 () , k , (). 1 Arcos+Brsin+=0, ( (4) . 7.2) . cos(+2k)=cos sin(+2k)=sin, () . cos(++2k)=-cos sin(++2k)=-sin () . 2 r=cos(/2). ) () r=cos(/2+k), kZ :
r=cos(/2) k=0 r=-cos(/2) k=1
) () r=cos(/2+/2+k), kZ :
-
99
r=sin(/2) k=0 r=-sin(/2) k=1
7.4
() (), . : 1)
=k, kZ r. 2) ,
. 3) . f(r,)=0
, : ) f(r,)=f(r,-) ) =/2 f(r,)=f(r,-) ) f(r,)=f(r,+) ) =0 f(r,0-)=f(r,0+) 4) ,
. 0 r, , 0 . :
limr a =
r= . 5) . r0 0
=0 . r=0 .
1 : r=f()=2(1-cos) ) (0,0) (4,),
=k , r=2(1-cosk). : k= r=0 k= r=4. =/2 =k+/2. k (2,/2), k (2,3/2). r 4.
) ) f()=f(-)
. , , .
-
100 V
) . r . 0r4 . r . .
.
cos R
02 10 02
2 0-1 24
32 -10 42
32 2 01 20
:
r=(1cos) >0. r=(1sin) =/2. 2 :
r=f()=sin2 >0 ) =k =k+/2 r=0
.
) . ) r=f()=sin2=sin2(+)=f(+)
. =/4 =3/4 f(/4-)=f(/4+) f(3/4-)=f(3/4+) . ()
r=-sin2
r=2(1-cos)
-
101
: f1(r,-)=r-sin2=0 f2(r,-)=r+sin2=0 : i) f2(r,-)=r-sin2=f1(r,)
( ).
(ii) f2(r,-)=r+sin2(-)= r-sin2=f1(r,) ( =/2)
) =0 =/2 . .
, , .
sin2 r
02 01 0
4 2
4 10 0
24 3
4 0-1 0-
34 4
4 -10 -0
44 5
4 01 0
54 6
4 10 0
64 7
4 0-1 0-
74 8
4 -10 -0
r=sin2 =2
-
102 V
1) ,
(1,-1). (. (2,7/4+2k) ) 2) (1,/3) (2,).
r=1 r=2 .
(. d=7, 1rcos( / 3)
= , 2r
cos( )= )
3) (4,/3), (
), R=4. . (A. r2-8rcos(-/3)=0, (x-23)2+(y-2)2=16 )
4) epr
1 ecos= e=1 .
5) :
) r=1- 11+ ) r
2=2 ) r=-cos
-
VIII
8.1
. .. , , ... . , , . .. , . (x,y,z) , :
F(x,y,z)=0 (1)
H (1) , .. z
z=f(x,y) (2)
, :
x=x(u,v), y=y(u,v), z=z(u,v) (3)
. , , . , . .. , ... , , .
,
:
) , F(x,y,z)=0 F(x,y,z) x,y,z ) .. z-sin(xy)=0
8.2
(), 0 (c).
-
104
. , . P0(x0,y0,z0), (c), , :
1 2( , , ) 0, ( , , ) 0f x y z f x y z= = (1) :
0 0 0x x y y z z = = (2)
(1), (2) , , : (2) t :
x=x0+t, y=y0+t, z=z0+t
(1):
f1(x0+t, y0+t, z0+t)=0, f2(x0+t, y0+t, z0+t)=0
' t :
F(,,)=0 (3)
, (2), ,,. ,, (3) (2) . : : , r=rc(u) (c) . :
OM=OP0+P0M
0 0 0=v0 :
OM=O0+v0 = O0+v(ON-O0)=(1-v)O0+vrc(u)
:
( ) ( ) ( )0 c, 1v u v v u= +r O r (A)
(c)()
0rc(u)
1
-
105
1: :
y2=2px, z= p, . (4)
P0(x0,y0,z0)=(0,0,0), f1(x,y,z)=y2-2px=0, f2(x,y,z)=z-=0 :
x y z= =
(5)
,, . (5) t, :
x=t, y=t, z=t (5)
(4) :
2t2=2pt, t=
t :
2=2p (6)
,, (6) (5), (5) ,,: =x/t, =y/t, =z/t, (6) :
222p 2py z x y xz
t t t = =
: () 0=0, ( ), rc(u)=(u2/2p)i+uj+k, ( y=u), :
( ) 2,2puu v v u = + + r i j k
p .. p=3 =10, .
8.3
(), ( ), () (c), ( ), . :
2
-
106
( ) ( )1 2, , 0, , , 0f x y z f x y z= = (7) w(,,) (). w, :
1 1 1x x y y z z = = (8)
x1, y1, z1 . x1, y1, z1 (7), :
f1(x1,y1,z1)=0, f2(x1,y1,z1)=0 (9)
x1, y1, z1 (8) (9) . (8) -t :
x1=x+t, y1=y+y, z1=z+t
(9) : 1 2( , , ) 0, ( , , ) 0 f x t y t z t f x t y t z t + + + = + + + = (10) H t (10) F(x,y,z)=0 . : : , r= rc(u) (c), w () M . :
OM=O+M
NM w NM=vw
OM= rc(u)+vw
:
c( , ) ( ) v u u v= +r r w (B)
M
(c)
()rc(u)
w
()
3
-
107
1: , w(1,2,-1) x2/2+y2/2=1, z=0, , .
: =1, =2, =-1, f1(x,y,z)=x2/2+y2/2-1=2, f2(x,y,z)=z=0
x1=x+t=x+t, y1=y+y=y+2t, z1=z+t=z-t
x, y, z x1, y1, z1 :
2 2
2 2
(x t) (y 2t) 1+ ++ = , z-t=0 ' t
: 2 2
2 2
(x z) (y 2z) 1+ ++ =
.. =4 =8, .
: (B)
rc(u)=cosui+sinuj, w=i+2j-k,
r(u,v)=rc(u)+vw=(cosui+sinuj)+v(i+2j-k)=(cosu+v)i+(sinu+2v)j-vk
8.4 .
. (C) . 5 .
x
y
z
C
5
R
P0
4
-
108
, , . :
0 0 0x x y y z z = = (11)
1 2( , , ) 0, ( , , ) 0f x y z f x y z= = (12) . P0(x0, y0, z0) R , . :
(x-x0)2+(y-y0)2+(z-z0)2=R2 x+y+z= (13)
R, . (12) (12) (13) x, y, z. R, , x, y, z (12) (13), :
F(R2, )=0 () (14)
(13) (14) :
( )2 2 20 0 0( ) ( ) ( ) , x+y+ z 0F x x y y z z + + = (15) . , (15) . : : ' , , w .
=1i+2j+3k, |w|=1 w=0. l=w | |l = I .
0 l= II 0 =
.
-
109
, w.
:
=cosu0+sinul0
(c): rc(v)=x(v)i+y(v)j+z(v)k
(): r=r0+tw, |w|=1
:
=(rc(v)-r0)-[ (rc(v)-r0)w]w l=w l=w[(rc(v)-r0)-[ (rc(v)-r0)w]w]= w(rc(v)-r0)
( )( )
c 00
c 0
(v)(v)
= w r r
lw r r
( ) ( )( ) ( )
c 0 c 00
c 0 c 0
(v) (v)| | (v) (v)
= = r r r r w w
r r r r w w
=(v)cosu0(v)+(v)sinul0(v)
, (c) () :
l0
x
y
z
rcr0
rc-r0
w
()
0
6
-
110
( )( )0 c 0v = + + r r r r w w ( )( )0 c 0 0 0v (v)cosu (v)+(v)sinu (v) = + + r r r r w w l : () , w=k, r0=0.
(c) ,
rc(v)=y(v)j+z(v)k.
:
=rc(v)-[rc(c)k]k=y(v)j, 0=j, l0=kj=-i, =y(v)cosuj+y(v)sinu(-i) r(v,u)=z(v)k+y(v)[cosuj-sinui]=-y(v)sinui+y(v)cosuj+z(v)k
: rc(v)=Rcosvk+Rsinvi, v[0,] w=k. : :
( )( )0 c 0 0 0v (v)cosu (v)+(v)sinu (v) = + + r r r r w w l r0=0, w=k, =(rc(v)-r0)-[ (rc(v)-r0)w]w=Rsinvi+Rcovk-Rcosvk=Rsinvi =Rsinv, 0=i, l=w=k(Rsinvi)=Rsinvj, l0=j :
r(v,u)=Rsinvcosui+Rsinvsinuj+Rcosvk x=Rsinvcosu, y=Rsinvsinu, z=Rcosv v[0,)
x2+y2+z2=R2 0z , .
1: , x=y/2=z/4. : (x0,y0,z0)=(0,0,0), =1, =2, =4 :
x2+y2+z2=R2, x+2y+4z=
:
f1(x,y,z)=x=0, f2(x,y,z)=y=0
x, y, z . :
-
111
z2=R2, 4z= 2=16R2 (15) :
(x+2y+4z)2=16(x2+y2+z2)
:
( ) ( )
( ) ( )
4 2 21u,v v 1 cosu vsin u21 21
8 21 vv 1 cosu vsin u 16 5cosu21 21 21
= + + + + +
r i
j k
7 .
2: , 14: y=, x-z=0 (16)
, . : :
(x0,y0,z0)=(0,0,0), =0, =0, =1, f1(x,y,z)=y-=0, f2(x,y,z)=x-z=0
:
x2+y2+z2=R2, z= (17)
x, y, z (16) (17) :
2
2 2 2 22 R
+ + = 2
2 2 22 1 R
+ = (18)
R, (17) (18), :
2 2 2 2 212
2
z x y z
+ = + + 2 2 2
2 2 1x y z + = (19)
.
, ( =1 =2:
14
x=y/2=z/4
7
-
112
( ) ( ) ( )( )
u, v 1 v cos u sin u 1 v sin u cos u
3 2v
= + + + +
r i j
k
8. , , , . , . . 2 OXZ :
2 2
2 2 1x z- =
, y=0 (20)
(19) . , ( ), . . .. , :
f(x,z)=0, y=0 (21)
. :
x2+y2+z2=R2, x+y+z= =0, =0, =1 x2+y2+2=R2, z= x2+y2=R2, z= R2=R2-2 (22) x, y, z (21), (22) :
f(R, )=0 (23) R, (22) (23) :
( )2 2 , 0f x y z + = (24) :
: (. f(x,z)=0), (.. OXZ), , (.. ), ,
-
113
( x), ,
2 2x y+ . , f(x,y)=0, z=0 :
( )2 2, 0f x y z + = :
( )2