. , , ,

,,,

,

1.      . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.      . . . . . . . . . . . . . . . . . . . .4

3.      . . . . . . . . . . . . . . . . . . . . . . . . 9

4.      . . . . . . . . . . . . . . . . . 15

5.      . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

6.      . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.    

, , . , , , , , . , , . .

, , , , , , .

3- , .. . , , , 3- , . 3- , .

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

2.    

, (2D) (2-dimention).

, . (, ) (. 1). , (x*, y*).

0

Y

X

M (x, y)

. 1

:

x* = ax + by +l, (2.1)

y* = gx + by + m, (2.2)

a, b, g, l, m -- , :

a b

= 0. (2.3)

g d

(2.1) (2.2) : (. 2) , (, ) | (*, y*), (. 3) (2.1) (2.2) , (, ) * (*, *), .

X*

0

Y

X

M 0*

Y*

. 2

0

Y

X

M

M*


. 3

, (2.1) (2.2) , .

, . (2.1) (2.2) , .

1.      j (. 4) :

* = x cosj - y sinj, (2.3)

y* = x sinj - y cosj. (2.4)

2. () :

x* = ax, (2.5)

y* = dy, (2.6)

a > 0, d > 0. (2.7)

() , a > 1 (a < 1). . 5 a = d > 1.

3.   

x* = x, (2.8)

y* = -y. (2.9)

4.      l, m. :

x* = x + l, (2.10)

y* = y + m. (2.11)

M*


M

0

Y

X


. 4

M*

0

Y

X

M


. 5

M

0

Y

X


. 6

M*

M*


M

0

Y

X


. 7

.

1.        ( , ).

2.        , (2.1) () 1 4 ( ).

, : (2.1) , (2.3) (2.11).

. , 1 3, :

cos j sin j a 0 1 0

-sin j cos j 0 d 0 -1

3.    

, . 1, 2, 3, :

x1 / x3 = x, x2 / x3 = y (3.1)

: (, ) (, , 1) .

, , , (0, 0, 0), (, , 1), (hx, hy, h).

, h = 0. hx, hy, h , (0, 0, 0) (, , 1). z = 1 (, , 1), (, ) .

(, ) (hx, hy, h), h = 0, , hx, hy, h .

( , , ).

:

: : 1 (3.2)

, ,

1 : 2 : 3 (3.3)

( , 1, 2, 3 ).

.

, , , . ( ), h (, h = 1) (0.5, 0.1, 2.5) . h , . , h = 10 (5, 1, 25).

. (80000, 40000, 1000) , , h = 0.001. (80, 40, 1).

. .

.

, h = 1, :


a g 0

(x * y * 1) = (x y 1) b d 0 (3.4)

l m 1

, , , (2.1) (2.2) 1 = 1. .

. , , . .

, 1 4, .

.

. (rotation)


cos j sin j 0

[ R ] = -sin j cos j 0 (3.5)

0 0 1

. - (dilatation)


a 0 0

[ D ] = 0 d 0 (3.6)

0 0 1

. (reflection)


1 0 0

[ M ] = 0 -1 0 (3.7)

0 0 1

. (translation)


1 0 0

[ T ] = 0 1 0 (3.8)

l m 1

.

1. (a, b) j (. 9).

0

Y

X

j


. 8

1- . (-a, -b) ;


1 0 0

[ T-A ] = 0 1 0 (3.9)

-a -b 1

.

2- . j;

cos j sin j 0

[ Rj ] = -sin j cos j 0 (3.10)

0 0 1

.

3- . (a, b) ;


1 0 0

[ TA ] = 0 1 0 (3.11)

a b 1

.

, :

[ T-A ] [ Rj ] [ TA ].

, ( ) :


cos j sin j 0

(x* y* 1) = (x y 1) -sin j cos j 0 (3.12)

-a cos j + b sin j + a -a sin j - b cos j + b 1

( ) . .

2. a b (a, b).

1- . (-a, -b) ;


1 0 0

[ T-A ] = 0 1 0 (3.13)

-a -b 1

.

2- . a b ;


a 0 0

[ D ] = 0 d 0 (3.14)

0 0 1

3- . (a, b) ; :


1 0 0

[ TA ] = 0 1 0 (3.15)

a b 1

[ T-A ] [ D ] [ TA ],

a 0 0

( x* y* 1) = (x y 1) 0 d 0 (3.16)

(1 - a)a (1 - d)b 1

, , [ R ], [ D ], [ M ], [ T ], .

4.    

(3D) (3-dimension) .

, , (x, y, z), ,

(x y z 1)

, ,

(hx hy hz), h = 0.

( ) ; .

.

, , . (, ).

. .

j:


[ Rx ] =

1 0 0 0

0       

(4.1)

cos j sin j 0

0 -sin j cos j 0

0 0 0 1

y:


[ Ry ] =

cos y 0 -sin y 0

(4.2)

0 1 0 1

sin y 0 cos y 0

0 0 0 1

c:


[ Rz ] =

cos c sin c 0 0

-sin

(4.3)

c cos c 0 0

0 0 1 0

0 0 0 1

- .

. -:


a 0 0 0

[ D ] =

(4.4)

0 b 0 0

0 0 g 0

0 0 0 1

a > 0 () ;

b > 0 () ;

g > 0 () .

.

:

1 0 0 0

[ Mz ] =

(4.5)

0 1 0 0

0 0 -1 0

0 0 0 1

yz:


-1 0 0 0

[ Mx ] =

(4.6)

0 1 0 0

0 0 1 0

0 0 0 1

zx:


1 0 0 0

[ My ] =

(4.7)

0 -1 0 0

0 0 1 0

0 0 0 1

. ( (l, m, n) - ):


1 0 0 0

[ T ] =

(4.8)

0 1 0 0

0 0 1 0

l m n 1

, .

.

3. j L, (a, b, c) (l, m, n). , :

l2 + m2 + n2 = 1

. 10 , .

Z

Y

L


X

. 10

. .

1- . (-a, -b, -c)


1 0 0 0

[ T ] =

(4.9)

0 1 0 0

0 0 1 0

-a -b -c 1

, L .

2- . L .

1- y ( ). , L L X = 0 (. 11).

Z

X

L L q

Y

Y


0


. 11

L

(0, m, n).

,

cos y = n / d, sin y = m / d, (4.10)

d = m2 + n2 (4.11)

:


1 0 0 0

(4.12)

[ Rx ] =

n/d m/d 0

0 -m/d n/d 0

0 0 0 1

, , (l, m, n) . ,

(l, m, n, 1)[ Rx ] = (l, 0, d, 1). (4.13)

2- q,

os q = l, sin q = -d (4.14)

C :

l 0 d 0

[ Ry ] =

(4.15)

0 1 0 0

-d 0 l 0

0 0 0 1

3- . L j.

L , :


[ Rz ] =

cos j sin j 0 0

-sin

(4.16)

j cos j 0 0

0 0 1 0

0 0 0 1

4- . -q.

5- . -y.

. , , .

6- . (a, b, c).

, :

[ T ][ Rx ][ Ry ][ Rz ][ Ry ]-1[ Rx ]-1 [ T ]-1.

, , .


l2 + cos j(1 l2) l(1 cos j)m + n sin j l(1 cos j)n m sin j 0

l(1 cos j)m n sin j m2 + cos j(1 m2) m(1 cos j)n + lsin j 0

l(1 cos j)n + m sin j m(1 cos j)n lsin j n2 + cos j(1 - n2) 0

0 0 0 1

,

a1 a2 a3 0

[ ] =

(4.17)

b1 b2 b3 0

g1 g2 g3 0

l m n 1

.

4. .

[ A ]. ,

Vi ( xi, yi, zi), i = 1,,n,


x1 y1 z1 1

V = . . . . . . . . . . (4.18)

xn yn zn 1

, , [ V ][ A ], (. 12).

Z


0

Y

X

. 11

5.    

, , , . , , .

- (). , . , . 3- . , , , , , , . , , .

, . , , .

6.     .

1.     

2. . ., . . . ( 97). - . , 1997., . 52 53.

1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. . . . . . . . . . . . . . . . . . . . .4 3. . . . . .

 

 

 

! , , , .
. , :