,; ,

:,

,:,,: (1) PjPk,PjPkPjPk,0,0>0,0,

(2)Pi,Pj,Pk,PiPjPjPk,0,0,PjPiPjPjPk

n+1,0,1,2,,n;0;0;k0k0

1. i0,j0,kk0,0,s1{i,0,j,ij,jk}2.jk,PjPkm,km{,k=j+1}3.jL2jk,i=jL1L2,6;{L2j,L1}

4.Pi,Pj,PkL1L2,,ji2,{j,ik}5.ij,L1jk,j,ss+16.jk,kk+k0,k>nkn,jn-12, 7.n,

P0P1P2P3P4P5P6i0,j0,kk0 (3) PjPkP0P3,j k,k k+ k0 , PjPkP3P6

,,,,, (xb,yb),(xe,ye),,,wlwr

P0,P1,,Pn,w0,:1. P0,Pn,P0Pn,2. P0PnP0PnPk,P0PkPkPn1,,3. ,w=wl+wr w0

5(3,7)(9,12),(15,4),(18,5),(20,7) ,w0=1,

w*,w0,w0w*,: ,W=wl+wr w*,;,W> w*,,,

S1S2:(1) (2) ,S1S2,S2S1(3) ,,

,,, T1T2,: T1T2,; T1T2,; T1T2,T1T2,T2T1

T1T2,,;,;,,,, ,

,1,0,2n2n :2n2n ,,.,0,1,2,3

,,1,,0

RAabcdBCDefghR

,,,2,

0

{t} {x,y}{x(t),y(t)} {x,y,z}{x(t),y(t),z(t)}

: :

: : R3

: {(x,y,z)|f(x,y,z) 0}f

V(E)(F),

:,;, ,:;

V20, >0,2--1=0,=1/2(1+ )1.618034 XYZ

1 (CSG:Constructive Solid Geometry)() CSGCSGTx

BNFCSG ::=|| CSG CSG

CSGCSGBrep CSGCSG ()CSGCSG

2

BNF =||||;=||||||||||=|||=||

3

Brep Brep BrepBrep BrepBrep

VC,C2n,V C ,: CCV=C,VVC,C,V,,,,,.

V,,V, V,, ,

, , ,,,,,,,

R1,R2,R=R1R2, "",,,,,,450,

,T1, R,T2,1. T1,,22. ,C,;33. T2,2T2,:

(1) ,,,(2) ,"",(3) ,,,3,, ,,

,, nn,, ,: {0x,10,12,13,14,2x,4x,6x,7x}

C1UC2 :C1={122,123,301,302,303,305,307}C2={12x,300,302,304,306} C2C1,C2C1,,"",:C1UC2={12x,300,301,302,303,304,305,306,307} ={12x,30x}

x1>0,y10,2,26034715

zl>0,y10 26034715{0x,10,12,13,14,2x,4x,6x,7x} :{2x,6x,Ox,4x,7x,12,10,13,14}

(Cantor) E0[0,1],E00x1x;E1E01/3,E1[01/3][1/32/3];E2E11/3,E2,Ek2k1/3kFEk, FEkkkEkk,EkF

[0,1]3:a13-1+a23-2+a33-3 ai02,1,E0E1,ai=1,E1E2,a2=1, ,(1) FE1[01/3][1/32/3],FF,1/3

(2) F (3) F (4) F,F,(5) F, (6) ,F,F,,F

von Koch

F,:(1)F (2) F(3) F,(4) ,F"" (5) ,F,

,1,2,4,22=4,23,1,2,8,23=8 ,Df,L,K,,Df,L,K:LDf=K Df,:Df=lnK/lnL DfHausdorff,Hausdorff

,,,9(1/3)2=1DfN,r,NNrDf=1 ,: (1)(2),Hausdorff

von Koch,4,1/3,Hausdorff Df

kE0,:

1.i0,j1,Q,A0Eo;{iEim 2,EimEi+1,imiA0,jii+1,Q;}2.,A0,mA1,A2, Am;3.Al,A2, ,Am;4.Q (A1,A2,,Am){;}

5.A0Q;{;}6.ijj+1,j>mi,j1;ii+1; 7.i>k,; 2. von Koch E0, P0,P11,A0=E0=(Po,Pl ),2,Po,Pl ,P2,P3,P4 m=4,A1=(P0,P2),A2=(P2,P3),A3=(P3,P4),A4=(P4,P1)3,P0P2,P2 P3, P3 P4, P4P1,P2P4

Von Koch DRn,S:D DDDx,y,c,0

,S1,S2RR,

P0P1(0,0)(1,0),P2,P3,P4 S1S2,:

S1P0,P1,P3,Po, P3,P2,:

S2P0,P1,P3,P3,P1,P4

von Koch1.x10,y10,s1,u1;{(x1,y1)}2.

3.(x2,y2)(x3,y3);4.Ps(x2,y2),Ps+1(x3,y3),ss+2;5.(x1,y1)Pu,uu+1;6.u>k,2

JuliaMandelbrot :f(z)=z2+cc,:zn+1= zn2+c,n=0,1,2, :1.z0,c|zn|2.c,z0,|zn|?

,c=0,:1. ,zz212. ,,""1;3. 1,, ,c 0,,,,Julia

zz2+ccMandelbrot :zk+1=zk2+c,zk =xk+yki,c=p+qi,

ab,k+1,0k,0pq,(x,y),Julia1.xmin -1.5,ymin -1.5,xmax 1.5,ymax 1.5,M 100,x (xmax - xmin )/(a-1), y (ymax- ymin)/(b-1);{(xmin , ymin)(xmax, ymax),M,;K}2.(nx,ny),nx=0,1,,a-1, ny=0,1,,b-1,

2.1x0 xmin + nxx ,y0 ymin + ny y ,k 0;2.2 k k+1; 2.3r r>M,k,2.4;k=K,0,,2.4;r Mk

(x,y),pq,(p,q),Mandelbrot(p,q)-2.25

2.1p0 pmin + nxp ,q0 qmin + nq q ,k 0; x0 0,y0 0 2.2 k k+1; 2.3r r>M,k,2.4;k=K,0,,2.4;r Mk

P145: 510