# Компьютерная графика: методические указания к лабораторному практикуму

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• ,

.. , ..

-

2005

• ,

, . , , , , .

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

.. .. . , .

• 3

, ,

, , . . , .

, , . , .

- . . , , , , , , . .

, , .

-. , . , .

• 4

1 1

1.1

. , ( ) . , . , .

1.2 1.

.

, , - , .

: Y = kX + b, (1) k=0, k=1, k=10, k=0.2, k=1,5;

. dy/dx = y/x = const, (2) const k, .

(, Bezier).

, , .

X = x1(1-t) + x2 t, Y = y1(1-t) + y2 t, (3)

, 1(1,1) , 2(2,2) ,

t=0,...,1 ( t

• 5

1.3 2.

. , .

: (X - a)2 + (Y - b)2 = R2, (4) =150, b=100 R=50 (=0, b=0 R=80);

X = a + RCos(), Y = b + RSin(), (5) , ( =3.14159);

dy/dx = - x/. , , , (- X) Y:

X n+l = Xn + y n, Y n+l = Y n - n, (6)

,

. , 2-n (2n-1

• 6

, (8). q , Sin( ) Cos( ) .

:

Xn+l = nCos()+YnSin(); Yn+l = YnCos() - nSin(), (8) = 2 /n (n -

). .

, . , .

1.4 3. .

. 3+ .

. , , .

.

: ( ) (). () .

.

: = R(), (9) :

X= Cos(), Y= Sin(); (10)

• 7

(), :

X=a+RCos()+ACos(), Y=b+RSin()+BSin(), (11)

(),

: X=a+RxSin(/Dx+Dy),

Y=b+RySin(/Dx+Dy) (12) 1.5 4.

, , 1.

1

2, .

2

• 8

2 2

2.1

(1-)S1 + S2 = 0 , (13) S1 S2 ( ) , -

1 0 . S1 S2 4 ,

P(x,y) : = S1(xp,yp)/[ S1(xp,yp) - S2(xp,yp)] (14) , , S,

3.

3 4

: (k1X + b1Y + C1)(k2X + b2Y + C2) = 0 (15) , , (1-)t1t2 + t3t4 = 0 , (16) t1, t2, t3 t4 - . .

[ 1(x1,y1), 2(x2,y2) ] [ t1 ,t2 ],

• 9

l . , ( 4),

(1-)t1t2 + l2 = 0 , (17)

F(xF,yF) :

= t1t2(xF,yF)/[ t1t2(xF,yF) - l2 (xF,yF)]. (18)

2.2 1.

, 5, . (-40,0), D(-20,80) (0,100) , . (-40,0), G(-20,-30) F(0,-40) , .

5 6

2. 6 . 0.2.

3. , (0,0), (1,0), (1,1), D(0,1) (3/2,1/2). .

2.3 2.

= R (), (19)

• 10

X = Cos() Y = Sin() (20) 2.4 3.

,

: = a *(5+2 *Cos() + 3*Cos(m)sign(Cos()))--b*(Sin(18*))2 *Cos(/2))8, = a * Cos(4*t) + b = t c*Sin(4 + t ), = a *(1 + Cos(3*) + b* Sin(3*))2

• 11

3 3

3.1 1.

. R1 R2. , R3 , ( ).

R2 R1, . R3 R2.

7 3.2 2

(

) Bezier 1, 2 3- ,

Px(t)=Cimti(1-t)m-iXi,

Py(t)=Cimti(1-t)m-iYi, (21) m - , i=0,...,m. 3.3 3

(

) Bezier . , .

• 12

4 4

4.1 ()

,

. , .

4.2 1

, , 1:1

, . 4.3 2

,

Bzier . , . .

4.4 3

-

. , . .

P(t) = P1(t) + P2(t) + P3(t), (22)

tPPP iit 211 *][21)( += ,

PttP it 122 *)(*2)( = ,

)1 2213 (*][21)( += tPPP iit .

• 13

t 0,...,1. (x,y) .

• 14

5 5

5.1

, . .

8

U () V ().

. V U ( 1).

.

F(x,y,z) = 0, z=f(x,y), :

x = X(u,v), y= Y(u,v), (23) z = Z(u,v).

u v .

.

U () V ().

. -

• 15

V U .

.

: F(x,y,z) = 0, z=f(x,y), x = X(u,v), y= Y(u,v), z = Z(u,v).

u v . 5.2 1.

. oZ. . . , , oY.

.

5.3 2.

. , .

- Bzier ( ), 1(0,100), 4(100,0) 2(30,20), 3(80,140).

. 120. 1.

: = 30)( CosxyX , += 30)( SinxyzY a (24) : Sin(30)= 0.5, Cos(30)= 0.866

.

• 16

6 6 6.1

, ,

( ) ( ).

( ).

, . , , .

6.2

u.

9

. , , r0, u1 u2. r0. ( ) u1 u2 ( 9). r u r', r=r-zu.

x' ' , r'= r0+x'u1+y'u2 r'=r-z'u= r0+x'u1+y'u2.

. u1 u2

', ', z'. ,

• 17

(25) u1 u u2 u, x' '.

, ,)uu(uu))(ur(rx

12

10

=

,)uu(uu))(ur(ry

12

20

= (26)

, . . u=u1 u2, : x' = (r-r0)u1, y' = (r-r0)u2, z' = (r-r0)u3.

, .

. xA-yA-zA 10.

10

oZ

Ya :

= - 1+x2, Ya =-z1-z2+za (27)

kx,ky kz :

=-kxCos +ykyCos, Ya =-kxSin -ykySin +zaky (28)

• 18

, kx=ky=kz ==30 , , :

=(y-)Cos30 ,

Ya =z-(y+)Sin30 (29)

6.3 ()

r', r rn (

) 11.

11

, , r0 -

, u1 u2, - r'=zr+x'u1,+y'u2. , , r

rn , z r'=zr+(1-z)un . : r'= zr+x'u1,+y'u2 =zr+(1-z)un .

u2 (r-rn ), u1 (r-rn ) u1 u2, x, y z.

, , , .

d, rn =r0+du, u=u1 u2,.

,

, (30)

• 19

6.4 1

. ( ) . S(500,300,100).

12

.

6.5 2

. . ( )

. , .

.

• 20

13

• 21

7 7

7.2 1.

. . .

7.3 2

( ). .

• 22

8 8

8.1

, . , , , , .

, , . , , .

. . , , .

8.2 1. :

.

. =

2b2a + : kx = 0,5, ky =0.5 = 49.

14

• 23

8.3 2 .

15

: (X-a)2 + (Y-b)2 = R2. , , n=4. f = 0,3. , , , = 2. 8.

8.4 3 .

: :

yCosxSinYySinxCosX

+=+=

cyxxdYxdyxxcX

+=

+=

)()(

0

00

• 24

n=4. 16

8.5 4

.

17

( 1/3), . . n = 4.

• 25

9 9

9.1

. , , : , .

9.2

. -, , . , , .

, .

, , , , . , , , .

, , , . .

(*.DXF) , . . (*.DWG), .

• 26

, DXF - , . (SHX-).

9.3

7 DXF

. DXF .

• 27

10 10

10.1

, . . , , , . , . , , , . , , , . 0,3 - 0,4 . , (). : Auto CAD , . .

• 1. , . [] /. - .: , 1986.- 398 .

2. , .. AutoCAD 14 []/ .. , .. , .. - .: 2000

3. , . []/ . - .: 1992.

4. , . []/ . , . - .: , 1976. 253 .

5. , . . []/. , . - .: , 1982.

6. , .. []: / .. , .. , .. - : , 2003.

7. , . AutoCAD. []/ .. - .: , 1991.- 284 .

• ()

4

.1 1,13,25

.2 2,14,26

t

d1

R

hRx

d1

b

Y

Y1

R2

R1

R3

d

YR1

b

a

d1

R

X

t

h

Rx

R2

R 3

15

• 30

.3 3,15,23

.4 4,16,20

R2

b

R

Rx

t

R3

A

Bd

h

d1

R1

xS

t

d

45

hR2

R1

b1

a 1

R

R3

Y

b

RX

• 31

.5 5,17,29

.6 6,18,30

x

d1

h

b

Y

Y1

R2

R 3

R1

t

Rx

d 1

R

R2

h

Rx

R3

15

d

Y

R 1b

a

d1

R

X

t

• 32

.7 7,19

.8 8,24

45

xS

t

dh

R3

Yb

RX

R2

R1

a1

b 1

R3

tR

R2

b

h

d

R1 A

Rx

d1

B

• 33

.9 9,21

.10 10,22

Rx

h

d1

t

d1

R

R2Y1

R1 R3

Y

x

b

d

Y

R 1b

a

d1

R

X

t

h

R x

R2

R3

15

• 34

.11 11,27

.12 12,28

Rx

B

A

d1

h

d

R1

R

R2

t

R3

b

R

x S

t

d

hY

b

RX

R3

45

a1

b1

R2

R1

• ()

.1 .

R R1 R2 R3 S a a1 b b1 d d1 h t x Y Y1

1,13,25 90 26 45 26 - 65 - 37 - 34 37 10 38 82 53 -

5,17,29 85 37 41 26 - 53 - 34 - 30 41 9 34 86 51 -

9,21 82 30 37 30 - 50 - 30 - 26 37 8 30 90 45 -

2,14,26 71 26 26 19 - 52 - 37 - 30 45 9 33 75 64 30

6,18,30 75 41 41 22 - 64 - 41 - 34 48 10 38 90 71 34

10,22 67 30 30 19 - 56 - 34 - 30 45 9 33 82 67 26

3,15,23 75 26 22 15 86 45 33 30 37 33 - 10 40 101 40 -

7, 19, 71 37 30 13 90 41 30 33 39 37 - 12 42 97 34 -

11,27 67 48 38 11 94 37 27 36 41 41 - 10 40 93 28 -

4, 16,20 90 75 38 22 - 60 - 38 - 30 34 10 38 30 26 -

8, 24 86 82 56 30 - 67 - 41 - 34 38 12 42 34 30 -

12,28 84 80 47 27 - 63 - 44 - 32 42 9 40 36 32 -

• 36

()

.1- 5

1

+= 2

21 xarcSinY 16

22 )34( += xxY

2

2252 2 += xxY 17 )14(2 += xxCosY

3 )1)(1( 2 += xxY 18 )21( 2xarcCosY +=

4 tgxxxY )1)(1( += 19

xxxY 2)1( 3 +=

5 451 2 ++= xxY 20

2)2)(1( xxxY +=

6 )1(

22 +

=xctgxY 21 )1ln(

2 = xxY

7 x

XY ln= 22 12 +

=XtgXY

8 )1ln( 2 ++= xxY 23 x

xxY

=2

9 1

5

=x

xtgY 24 xxxY )1)(1( +=

10 xxY

+

=11ln

21 25 )1)(1(

2 ++= xxY

11 )4(

422

3

+=

xY 26 xxY += 4

12 23axY = 27

3353 2 = xxY

13 CosXSinXY 3+= 28 )3

(2 += xSinY

14 xY 423 = 29

)32( 2 = xctgY x

15 )5.123( 2 += xxSinY 30 )1ln( 2 = xxY

• 37

()

6

.1 - 1,15,30

.2 - 2,16

.3 .3 - 3,17

• 38

.4 - 4,18

.5 5,19

.6 6,20

• 39

.7 7,21

.8 8,22

.9 9,23

• 40

.10 10,24

.11 11,25

.12 12,26

• 41

.13 13,27

.14 14, 28

.15 15,29

• 42

()

: ) ; ) ; ) . ,

: : (Command:).

( , ..) , (), . , , < >.

AutoCAD : 1) ; 2) ,

(). "".

.1

: )-,

OX OY ; ) -

, : , , .. , , ..., .

-.

. , ... (UNITS), . :

• 43

) (UNITS) - , ;

) (PRECISION) - , ;

) (ANGLES)- , .

.3

(LIMITS). . .

.4 AutoCAD , - , . .

.4.1 : (Command:)_(Line:) :(From point:)30,10 :(To point:) 30,30

.4.2

( ) : , , .. , 50,50 30 .

: (Command:)_(CIRCLE) 2T(2P)/3T(3P)/KKP(TTR): 50,50 (Diameter):: 30 .4.3

, , .

- : (Command:)_ (PLINE) - : (From point:) 0, 0

• 44

- (urrent line-width is):0.0 - (Specify next point or)

[(Arc)/(CLose)/(Halfwidth)/(Length)/(Undo)/(Width)]: (W)

- (Starting width):1 - (Ending width): 1 - (Specify nd point of line): 20, 40 - (Specify nd point of line): 90, 60 - ,

, 1. - (Arc).

(Specify nd point of arc or) [ (Angle)/(CEnter)/(CLose)/(Direction)/ (Halfwidth)/(Line)/(Radius)/ (Second pt)/(Undo)/(Width)]: (S) , . . (Specify Second point of arc) . (Specify nd point of arc)

.5 , , AutoCAD, , . , , 3 :

) ; ) ; ) . (EDIT)

. : (Select object):

- . . .

• 45

(Select object): (W), .

, , .

.5.2 ()

(ERASE). : (ERASE) : (Select object):

. (ERASE)

. (TRIM)

. , . , , . 6 .

: (TRIM) :

(Select cutting edge(s) (Select object): . , (Select cutting to trim): , .

, , , .

(TRIM) (EXTEND).

.5.3 (MOVE), (COPY),

(ROTATE), (SCALE) . : (Select object): . (Base point or displasement): . (Second point or displasement): . (MIRROR) . .

• 46

: (MIRROR) (Select object): . (First point or mirror line): . (Second point): . ?(Delete old objects? ): .

.5.4

(PEDIT) , (PLINE).

: (PEDIT) (Select polyline): . (Close) - . (Join) - , . (Width) - . (Edit vertex) - . (Fit) - . (Spline) - -. .(Decurve) - . (Ltype gen) - . (Undo) - . (eXit) - . .6 .6.1

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

(HATCH) (Boundary Hatch).

(Pattern...). (Hatch pattern palette). , . ...

• 47

(Boundary Hatch) , , , (Apply).

.6.2

...(Dimensoin Styles) . (Dimension Style Manager) (Modifi) : (Text)- , , ..;

(Dimension lines) , , ..;

(Linear dimensions) , .

.6.3

(DIMENSIONS) . (DIMLINEAR) -

. : ENTER (First

extension line origin or ENTER for select): .

(Second extension line origin): .

( / / / //)

Dimension line location (Mtext/Text/Angle/Horizontal/Vertical/ Rotated): , .

(DIMALIGNED) - , , . .

:

(DIMBASELINE) -

• 48

(DIMCONTINUE) - .

(DIMANGULAR) - , , , .

: , , : Sel...