# 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