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Δημήτρης Τμήμα Φυσικής Πάτρα 2012

αναλυτικη γεωμετρια

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  • 2012

  • . 1996 2004 2005 . . . . . . , . . . 32 . . .

    , , . , .

    2012

  • 5

    ...................................................................................................................5

    ...........................................................................................................................1

    ...............................................................................................................................3

    .....................................................................................................................31.1 ..................................................................................................................31.2 ...........................................................................................................51.3 ................................................................................................71.4 ,...............................................................121.5 ...............................................................................................12.....................................................................................................................................................20

    .....................................................................................................................23

    .....................................................................232.1.................................................................232.2 ........................................................................................................242.3 ..............................................................................252.4 .....................................................................................26..............................................................................................................................................28

    ....................................................................................................................31

    .....................................................................................................................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

  • 6

    .....................................................................................................................................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

  • 7

    ..................................................................................................................121

    ...............................................................1219.1 ..............................................................................................................................................1219.2 ...................................................................................................................1219.3 ...................................................................................................................1239.4 ...........................................................................................1249.5 ..................................................................................................................1259.6 .....................................................................1279.7 ,..........................................1289.8 ......................................................................................1309.9 SerretFrenet.............................................................................................1319.10 T......................................................................132...................................................................................................................................................135

    ...............................................................................................................137

    ................................................................................137

    ......................................................................................................137

    ................................................................................................................138

    ..........................................................................................................139

    ...........................................................................................139

    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

    ...........................................................................................................209

  • 8

    ....................................................................................213

    ......................................................................................................................233

    ................................................................................................................236

  • , ' ' . (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 , . , ...

  • 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 . , , , , , .

  • 1.1 .

    () , . () , . ( ), , (,). () (4. , , , ||. B : || B = (1) -|| (,). x . . (,) Ox. , .. OY, , 1 .

    (4 .. , v,u,w, ..

    x

    ()

  • 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 ,

  • 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

  • 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)

  • 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

  • 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

  • 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

  • 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

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

  • . 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

  • 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

  • . 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) .

  • 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. ,

  • . 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

  • 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

  • . 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 = (,,) .

  • 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)

  • . 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.

  • 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)

  • 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) .

  • 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) :

  • 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)

  • 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

  • 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

    . .

  • 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

  • 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