Методическое пособие по курсу ''Интерактивные графические системы''

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  • -

    ., ... ..

    -, 2002

  • . .., .., .. , .. , .. , .. : ..., . ..

    , 2002

  • 1.

    1.1. 1.2 1.3. - 1.4

    1.4.1. . 1.5 1.6

    1.6.1. 1.6.2. 1.6.3 . .

    1.7 2.

    2.1. 2.2. , 2.3. 2.4.

    3. 3.1 3.2 .

    3.2.1 . 3.2.2 3.2.3

    3.3 . 3.3.1 3.3.2 z- 3.3.3 3.3.4 3.3.5 3.3.6 3.3.7

    4. 4.1 4.2 4.3

  • 1.

    1.1. . : , . , .

    - ( ), , , , , .

    , , , , , (, ).

    : (xi, f(xi)), i = 0,1,...,N f(x) [ ]a b, . .

    1.2 ( 1.1)

    =

    =N

    iiiN (x)) Lf (x (x) L

    0 (1.1)

    )x)...(xx)(xx)...(xx(x)x)...(xx)(xx)...(xx(x

    (x) LNiiiiii

    Niii

    =

    +

    +

    110

    110 . (1.2)

    a=x0 xi xN=bx

    y

    .1.1 .

    Li (x) ,

  • Li (xj) =ij , i, j [ ] 0, N , ij - - :

    =

    =.ji 0,

    j;i 1,ij

    - N, . [ ]x xN0 , : N ( 1), . . - [ ]x xN0 , . , . , . . , , .

    1.3. -

    : = x0 < x1

  • ai0 = fi-1; ai0 + ai1 hi + ai2 hi2 + ai3 hi3 = fi; ai1= f i 1 ; ai1 + 2ai2 hi + 3ai3 hi2 = f i . : ai0 = fi-1; ai1= f i 1 ;

    ai2 = 3(fi -f i-1)

    hi2

    2fi-1hi

    f ihi

    ; (1.4)

    ai3 = 2(fi-1-fi )

    hi3

    fi+1hi

    fihi

    2 2+

    +

    .

    u = (x - xi-1) / hi. (1.4) (1.3) fi-1, fi, f i 1 f i ,

    () = [ ]f (u) f (u) f (u) f (u) hi-1 i i 1 i o i+ + + 1 0 1 . (1.5) , o (u) = 1- 3u

    2 + 2u3; 1 (u) = 3u

    2 - 2u3; 0 (u) = u - 2u

    2 + u3; (1.6) 1 (u) = -u

    2 + u3 . .1.2. 1.3.

    0( )

    u 1 0

    11( )

    u10

    1

    u10

    0( )1

    u10

    1( )1

    . 1.2 .

    a=x0 xi xN=bx

    y

    .1.3 -

  • . (1.5), fi-1 fi, - f i 1 f i ,

    () [xi-1, xi]. : , f i

    ? :

    = + +fif f

    hf f

    hi i 1

    i

    i 1 i

    i

    , i = 1,2,...,N-1

    = +

    =

    + +

    f0f f

    hf f

    h

    f f fh

    f fh

    1 0

    11

    2 1

    2

    NN-1 N-2

    N-1

    N N-1

    N

    ( )

    ( )

    1

    1

    1

    1 1

    N N

    (1.7)

    i = + +h

    h hi

    i i 1

    , i i1= . (1.8)

    . , . .

    1.4 - -. , ( ). - . . - ( ) - 2, .. . 1. , [, b] , 0. spline. - , . , , (.1.4).

  • a=x0 xi xN=bx

    y

    .1.4 -

    -,

    : EI s () = - (),

    s - ; () - , ; EI - . , , s (x), , . , 1 . , , s (0) = s (N) = 0. , . -, , . -, , . , , , , s (0) = f (0) s (N) = f (N) s (0) = s (N) = 0, ,

    .

    [ ] [ ] s (x) dx f (x) dx2 2

    a

    b

    a

    b

    , (1.9)

    f (x) =s (x). s (i) = fi. s (xi) = mi, i = 0,1,...,N. [xi-1, xi] (1.5) s x f u f u m u m u hi i i i i( ) ( ) ( ) [ ( ) ( )]= + + + 1 0 1 1 0 1 . (1.10)

    .

  • =

    + +

    + + s (x)

    f fh

    (6 12u)m ( 4 6u)

    hm ( 2 6u)

    hi i 1

    i2

    i 1

    i

    i

    i

    , (1.11)

    :

    + =

    +

    =

    ++

    +

    +

    +

    +

    s x

    s x

    i

    i

    ( )

    ( )

    0 64 2

    0 62 4

    1

    12

    1

    1

    12

    1

    f fh

    m mh

    f fh

    m mh

    i i

    i

    i i

    i

    i i

    i

    i i

    i

    (1.12)

    i (i = 1,2,...,N-1) s (i + 0) = s (i - 0) iiiiii cmmm =++ + 11 2 . (1.13)

    c f fh

    f fhi i

    i i

    ii

    i i

    i

    =

    ++

    +

    3 11

    1( ) .

    m0, mi,..., mN, ( ). ( ):

    1. s () = f (), s (b) = f (b) 2. s () = f (), s (b) = f (b). mi:

    22

    0 0 1 0

    1 1

    m m cm m m ci i i i i i

    + =+ + = +

    * * ;; (1.14)

    N N N Nm m c* * . + =1 2

    (1.14) 1: 0 0

    * *= =N , c f0 02

    * = , c fN N

    * = 2 , 2 (1.12):

    0 1* *= =N ,

    cf f

    hh

    f01 0

    1

    103 2

    * =

    ,

    cf f

    hh

    fNN N

    N

    NN

    * =

    32

    1 .

    (1.14) . , , .

  • 1.4.1. . i :

    a b

    c a b

    c a b

    c a

    y

    y

    y

    y

    N N N

    N N

    N

    N

    N

    N

    1 1

    2 2 2

    1 1 1

    1

    2

    1

    1

    2

    1

    0 0 0 0

    0 0 0

    0 0 0

    0 0 0 0

    ...

    ...

    ...

    ...

    =

    M M

    . (1.15)

    . yi = vi yi+1 + ui , i = 1,2,...,N-1. (1.16) yi-1 (1.16) yi-1 = vi-1 yi + ui-1 , i- (1.15): ci yi-1 + ai yi + bi yi+1 = i . (ai + ci vi-1) yi + bi yi+1 = i - ci ui-1. (1.17) (1.17) (1.16), vi , ui ( ): v0 = u0 = 0;

    v ba c vi

    i

    i i i

    = + 1

    ; (1.18)

    u c ua c vi

    i i i

    i i i

    =+

    11

    , i = 1,2,...,N-1.

    , yN = uN. (1.16) ( ).

    1.5 y = f (x) c . , . , , . , , F (x, y) = 0, .

  • F (x, y) = 0 .

    . , y = f (x) F (x, y) = 0, x = x (s) y = y (s) s ( 1.5 a,). 1.5.

    a)

    1

    x(s)

    2

    3 4 5

    6 7

    8

    s )

    1

    y(s) 2 3

    4 5 6

    7 8

    s

    c)

    1

    y 2 3

    4

    56

    7

    8

    . 1.5 -.

    -

    r = r (s) = x (s) i + y (s) j + z (s) k, (1.19) i, j k - . r = r (s) - , : x = x (s), y = y (s) z = z (s) ( 1.6 ,,).

    a)

    1

    x(s)

    2

    3 4

    6

    5

    7

    8

    s )

    1

    (s)

    2 34 5

    6 7

    8

    s )

    1

    z(s)

    2 3

    4

    5 6

    7

    8

    s

    1

    y

    2 3

    4

    56

    7

    8

    z

    . 1.6 (3D) -.

  • , , , , (, ). , s , , , , . - () , .. s0 = 0, si = si-1 + | ri - ri-1 |, i = 1,2,...,N, (1.20) | ri - ri-1 | = ( ) ( ) ( )x x y y z zi i i i i i + + 1 2 1 2 1 2 . , , , , , . , . , s (xi, si), (yi, si), (zi, si) . . . , (1.18) vi ai + ci vi-1 . , , . - , , | mi | = | ri | 1. , rg, :

    =

    r

    r r

    r rig i

    g i

    . (1.21)

    2 r = 0 . , ( , ), . . , , (N + 1) ( + 1) , K N . , . - 1, - N. (1.10) [si-1, si]:

  • r s r u r u m u m ui i i i i i( ) ( ) ( ) [ ( ) ( )]h= + + + 1 0 1 0 1 , (1.22) hi = si - si-1, u = (s - si-1) / hi , mi - , .

    mi-1 = mi = r rr r

    i i

    i i

    1

    1

    , hi = | ri - ri-1 |. (1.23)

    (1.22) , r s r u r u r r u u

    r u u u r u u u r u r ui i i i i

    i i i i

    ( ) ( ) ( ) ( )[ ( ) ( )][ ( ) ( ) ( )] [ ( ) ( ) ( )] ( )= + + +

    = + + + = +

    1 0 1 0 1

    1 0 0 1 1 0 1 1

    - . , . ( ), , ( ).

    1.6

    1.6.1.

    ,

    , - . . , . , ,

    L2 =4 2 8 ( ) ,

    L = 2 2 . (1.24) - (.1.7);

    = +

    ( ) /x yx y y x

    2 2 3 2

    . (1.25)

    1.6.2. - . , ( ).

    /

    .1.7

  • , , , , . .

    P1 P2

    P3

    P0

    P6

    P4 P5P7

    P8

    PC

    BA

    D

    . 1.8

    , . . (.1.8) P0, P1, P2, P3, P4, P5, P6, P7, P8, Pk. (1.26) : P0, P6, P1, P2, P4, P5, P3, P8, P7, Pk. (1.27) ( ): P0, P6, P1, P2, P4, P5, P3, P7, Pk. (1.28) . , ; , - . , (1.28). : [P6, P1], [P2, P4], [P5, P3], [P7, Pk]. (1.29) - ( ). Ax + By + Cz + D = 0, ( ) s: F (s) = Ax (s) + By (s) + Cz (s) + D = 0. (1.30) , . .

  • (1.30) , . , .

    1.6.3 . .

    . , - , , . .

    : 1. , - ,

    . 2. , . .

    .

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

    :

    1. . 2.

    . 3. , ,

    . 4.

    . ,

    , .

    . , .

    . (.. )

    ,, - [,,1], 1 -

  • , , [x y z ] , - .

    [x y z ] , , ,, , , . .

    = x / , = y / , = z / . , , ,

    - .

    1. Ax + By = C

    [ x y ]

    BA = C [x y 1 ]

    CBA

    = 0 . :

    [ x y ]

    CBA

    = 0 . [ x y 1 ] -

    CBA

    -

    . 2. x + y = 1 3x - 5y = 0

    :

    [ x y ]

    5131 = [ 10 ] [x y 1 ]

    0151

    31 = [ 0 0 ] .

    32 , . , ,

    [ x y ]

    101051031

    = [ x y ].

    .

    .

    3. x = y2 x = u2 . y = u . :

    [ x y 1 ] = [ u2 u 1]

    100010001

    = [ u2 u 1].

    33 . , , . .

  • : x` = a1x + b1y + c1z + d1 , x` = a1x + b1y + d1 ,

    y` = a2x + b2y + c2z + d2 , y` = a2x + b2y + d2 , z` = a3x + b3y + c3z + d3 .

    . ( ) .

    , .

    , ( x`, y`, z` ) ( x , y , z ) :

    [ x` y` z` 1 ] = [ x y z 1 ]

    1ddd0cc0bbb0aaa

    321

    321

    321

    321

    c; [ x` y` 1 ] = [ x y 1 ]

    100

    21

    21

    21

    ddbbaa

    .

    , , , () .

    : 1. . ( )

    ( x , y , z ) ( x`, y`, z` ) :

    [ x` y` z` 1 ] = [ x y z 1 ]

    1010000100001

    zyx

    ; [ x` y` 1 ] = [ x y 1 ]

    1010001

    yx

    ,

    x , y , z - x , y z .

    2. . ( x , y ) :

    [ x` y` 1 ] = [ x y 1 ]

    1000cossin0sincos

    .

    . , ( x ,

  • y ) - ( x , y) .

    , ( x , y ) , ( x , y)

    [ x` y` 1 ] = [ x y 1 ]

    1010001

    MM yx,

    (x`,y`) :

    [ x`` y`` 1 ] = [ x` y` 1 ]

    1000cossin0sincos

    .

    , .. :

    [ x``` y``` 1 ] = [ x`` y`` 1 ]

    1010001

    MM yx .

    x , y (). :

    [ x`` y`` 1 ] = [ x y 1 ]

    1010001

    MM yx

    1000cossin0sincos

    1010001

    MM yx .

    . .

    . . .

    X, (0,0,0), :

    [ x` y` z` 1 ] = [ x y z 1 ]

    10000cossin00sincos00001

    .

    , , X .

    Y :

  • [ x` y` z` 1 ] = [ x y z 1 ]

    10000cos0sin00100sin0cos

    .

    Z, , :

    [ x` y` z` 1 ] = [ x y z 1 ]

    1000010000cossin00sincos

    .

    :

    1) (x,y,z) ( );

    2) X (0,0,1) (a,b,c) X Y ( R Q);

    3) X ( F);

    4) , ( R-1 Q-1);

    5) , ( -1) .

    6) 4- 5-

    . ,

    R Q F R-1 Q-1 -1 . 3. .

    .

    [ x` y` z` 1 ] = [ x y z 1 ]

    1000000000000

    z

    y

    x

    SS

    S

    .

    Sx , Sy , Sz , .

    Sx , Sy , Sz .

  • 1.7

    ( , ) () , . .

    x

    . 1.9 .

    , , . , , , (.1.9). , - . , , . , , , , . :

    (u) = ( ) [ ( ) ] + =

    u dxR

    u x ya

    b

    ii i

    i

    N2 2

    0

    1 . (1.31)

    Ri > 0 - , , ; u () - .

    ( ) u dxa

    b2 (. 1.9),

    . , . . . , s () = s (b) = 0 (1.32)

  • , , , (1.31) s (). , mi Mi = Si . s () [xi-1, xi]:

    s ()= M x xh

    Mx x

    hii

    ii

    i

    i

    +

    11 . (1.33)

    hi = xi - xi-1, Mi - . M0 = MN = 0. . (u - s ), u (x) - , s () - , ( (1.32)) : (u - s ) = (u ) - ( s ) - 2 , = +

    = =

    S u SR

    S y u Sii

    N

    X

    X

    ii i i i

    i

    N

    i

    i

    ( ) ( )( )1 0

    1 1 . (1.34)

    , = 0, (u - s ) = (u ) - ( s ) 0 (1.35) ( (u - s ) - ). , ( s ) (u), .. . (1.33) Si . = 0 : s (xi) = yi - Ri Li , i = 0,1,...,N, (1.36) Li =

    M Mh

    M Mh

    i i

    i

    i i

    i

    +

    +

    1

    1

    1 - (1.37)

    i - . : M-1 = MN+1 = 0, h0 hN+1 0. (1.36) s (xi) = yi . Mi (i = 1,2,...,N-1) , s () () , .. s (xi - 0) = s (xi + 0), s (xi - 0) = s (xi + 0). (1.38) (1.33) (1.36) (1.38), N - 1 : ci-2 Mi-2 + bi-1 Mi-1 + ai Mi + bi Mi+1 + ci Mi+2 = Fi ; (1.39) a1 = c0 = b0 = 0; bN-1 = cN-1 = cN-2 = 0;

    ai = 1 1 1 1 1

    32 12

    12 1 1h

    Rh h

    Rh

    R h hi

    ii i

    ii

    i i i+ + + + ++ +

    + +( ) ( ) ;

    i = 1,2,...,N-1

    bi = - ( ) ( )1 1 1 1 1 1

    61 1 21

    11h h

    Rh h

    Rh

    hi i

    ii i

    ii

    i+ + +

    +

    + + ++

    ++ ; (1.40)

  • i = 1,2,...,N-2

    ci = 1

    1 21h h

    Ri i

    i+ +

    + ; i = 1,2,...,N-3; Fi = ( )y y

    hy y

    hi i

    i

    i i

    i

    +

    +

    1

    1

    1 .

    (R0, R1,..., RN) , (1.39) ( Ri ) ( Ri) Mi (1.36). Ri = 0 (1.39) . , yi , Ri. k, , Rk = 0. yi, .. | s (xi) - yi | i (i = 0,1,...,N). (1.41) Ri. Ei =s (xi) - yi , (1.42) (1.36) (1.41)

    Ri = EL L

    i

    i

    i

    i

    . (1.43)

    (1.43) Ri , :

    Ri(j+1) = ii

    jL ( ), (1.44)

    j - . Ri(0) = 0 , . , Li(j) = 0, Ri(j+1) = 0. , (1.44) . , (1.44) :

    Ri(j+1) = Ri(j) i

    ij

    ijR L( ) ( )

    = Ri(j) ii

    jE ( ). (1.45)

    , j- i- (1.41) ( |Ei(j) | > i ), Ri

    (j+1) < Ri(j) , .. (j + 1)- Ri . i. , j- |Ei(j) | < i Li

    (j) 0, Ri , (1.41) .

  • , s (xi) . , yi , . xi , i- . , i : i = Ai | Li |. (1.46)

    Ai = 1 32

    12

    1

    /h hh h

    i i

    i i

    +

    ++. (1.47)

    ,

    Ai = 1 3 11

    3

    /h h

    h hi i

    i i

    +

    ++

    . (1.48)

    r (s) = x (s) i + y (s) j + z (s) k . . , ~si . .1.10

    x

    y

    . 1.10

  • 2.

    2.1. , , . . . . . - . , , . r = r (u, v), (2.1) r (u, v) - - . (2.1) x = x (u, v); = y (u, v); (2.2) z = z (u, v). u v - , .

    2.2. , , u v 0 1 (.2.1).

    ru(0,0) rv(0,0)

    ruv(0,0)r(0,0)

    rv(u,0)

    ru(0,v) r(0,v)

    u=0 r(u,v)

    r(1,0)ru(1,0)

    rv(1,0) ruv(1,0) r(1,v)

    ru(1,v)

    r(1,1) ru(1,1)

    ruv(1,1) rv(1,1)

    r(u,0)

    r(0,1)

    rv(0,1)ru(0,1)

    ruv(0,0) r(u,1)

    u=1

    v=0

    rv(u,1)

    v=1

    .2.1

  • r (u, v), 0

  • - ;

    Q =

    r r r r

    r r r r

    r r r r

    r r r r

    v v

    v v

    u u uv uv

    u u uv uv

    ( , ) ( , ) ( , ) ( , )

    ( , ) ( , ) ( , ) ( , )

    ( , ) ( , ) ( , ) ( , )

    ( , ) ( , ) ( , ) ( , )

    0 0 0 1 0 0 0 1

    1 0 1 1 1 0 1 1

    0 0 0 1 0 0 0 1

    1 0 1 1 1 0 1 1

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