3D动画的设计技术

1、第十一章3D动画的设计技术图11-0103点的进程动画with(plots):with(plottools):N:=24:F:=i->point(cos(i*2*Pi/12),sin(i*2*Pi/12),3,color=blue,symbol=DIAM0ND,symbolsize=55)：display(seq(F(i),i=0.N),insequence=true,axes=FRAME,orientation=-12,35);dl:=%:PLOT3D(ANIMATE(seq(F(i),i=0.N),AXESSTYLE(FRAME);d2;spacecurve(cos(i*2*Pi/1

2、2),sin(i*2*Pi/12),3,i=0.N);d3:=%:display(dl,d3);display(d2,d3);图i利用对象函数制作的孤点进程动画(连环画)with(plots):N:=24:h:=4:F:=i->pointplot3d(cos(4*Pi*i/N),sin(4*Pi*i/N),h/N*irgrid=5r3,symbol=circle,symbolsize=23,style=POINT,color=blue):display(seq(F(i),i=0.N)zinsequence=true,axes=box);dl:=%：；PLOT3D(ANIMATE(seq(

3、op(F(i),i=0.N);d2:=%:;spacecurve(cos(i*4*Pi/24),sin(i*4*Pi/24)f4*i/24zi=0.24rcolor=red,style=LINE);d3:=%:display(d2,d3zaxes=frame);图2利用画图函数制作的孤点进程动画(连环画)利用变换函数和对象函数制作孤点进程动画with(plots):with(plottools):a:=3:b:=2:k:=6:n:=2:m:=0:N:=72:t:=2*Pi*i/N:L:=point(-n,mAm,color=blue,symbol=circle,symbolsize=23):L

4、I:=i->translate(L,a*cos(k*Pi*i/N),b*sin(k*Pi*i/N),0)*display(seq(LI(i),i=0-N),axes=normalzlabels=xzy,z,insequence=truezorientation=-80,65);PLOT3D(ANIMATE(seq(LI(i)zi=0.N)fORIENTATION(-80r65)XES(FRAME);内层动画with(plots):R:=24:a:=0:b:=4*Pi:f:=n->cos(a+(b-a)*n/R):g:=n->sin(a+(b-a)*n/R):h:=n->

5、(a+(b-a)*n/R)/2:F:=n->POINTS(evalf(f(n),g(n),h(n)zSYMBOL(CIRCLEf36),COLOR(RGB,1,0,0):PLOT3D(ANIMATE(seq(F(n),n=0.R),AXES(FRAME),VIEW.,.z0.7),ORIENTATION(9,71);display(seq(F(n)zn=0.R)zinsequence=truezaxes=box,view=7,orientation=9z71);图3用内层方式设讨的点进程动画图11-0411点的踪迹动画利用对象函数制作点踪迹动画(匀步长)A,利用对象函数plottools

6、pointwith(plots):with(plottools):N:=54:h:=4:r:=l-k/N:Fl:=i->seq(point(r*cos(k*6*Pi/N)zr*sin(k*6*Pi/N),h/N*k,symbolsize=54fsymbol=CIRCLE,color=black),k=0.i):spacecurve(r*cos(6*Pi*k/N)zr*sin(6*Pi*k/N)Zh/N*k,k=0.Nzcolor=green):d01:=%:display(ANIMATE(seq(Fl(i),i=0.N)zinsequence=true,axes=FRAME,orient

7、ations65,65,view=-l.1f-1.1z0.4):;dl1;display(dOlzdll);B,利用对象函数pIottooIscurveF2:=i->curve(seq(r*cos(k*6*Pi/N),r*sin(k*6*Pi/N),h/N*k,k=0.i),thickness=3,color=red,style=POINT,symbolsize=36,symbol=DIAMOND):spacecurve(r*cos(6*Pi*k/N)zr*sin(6*Pi*k/N)rh/N*k,k=0.N,color=magenta):d02:=%:display(ANIMATE(se

8、q(F2(i)zi=0.N),insequence=true,axes=boxzorientation=65,65zview=-1.1,-1.1,0.5):;dl2:=%:;display(d02,dl2);C,利用对象函数plottoolspolygonF3:=i->polygon(seq(r*cos(k*6*Pi/N)zr*sin(k*6*Pi/N),h/N*k,k=0.i)zthickness=3zcolor=blue,style=POINTrsymbolsize=36fsymbol=DIAMOND):spacecurve(r*cos(6*Pi*k/N)zr*sin(6*Pi*k/

9、N)h/N*k,k=0.Nzcolor=cyan):d03:=%:display(ANIMATE(seq(F3(i)zi=0.N),insequence=true,axes=box,orientation=65,65zview=-1.1,-1.1,0.5):;dl3:=%:;display(d03fdl3);利用三个不同对象函数的方室，目左至右：A,B,C图4成效相同的连环画利用画图函数制作点踪迹动画(匀步长)A,利用pointpIot3d函数with(plots):with(plottools):N:=54:h:=4:r:=(l-k/N)A2:Fl:=i->pointplot3d(se

10、q(r*cos(k*6*Pi/N),r*sin(k*6*Pi/N),h*r/N*k,k=0.i)zsymbol=DIAMOND,color=blue,symbolsize=12):spacecurve(r*cos(6*Pi*k/N),r*sin(6*Pi*k/N)rh*r/N*kzk=0.N):d0:=%:display(seq(Fl(i),i=0.N),insequence=true,axes=box,view=-1.1z-1.1r0.zorientation=67,73):;dll:=%:fdisplay(dO,dll);B,使用polygonplot3d函数with(plots):wit

11、h(plottools):N:=54:h:=4:r:=(l-k/N)A2:F2:=i->polygonplot3d(seq(r*cos(k*6*Pi/N),r*sin(k*6*Pi/N),h*r/N*k,k=0.i)zsymbol=DIAMOND,color=bluerstyle=POINTZsymbolsize=12):spacecurve(r*cos(6*Pi*k/N)zr*sin(6*Pi*k/N)Zh*r/N*kzk=0.N):d0:=%:display(seq(F2(i)zi=0.N),insequence=truezaxes=box,view=-1.1-1.1r0.zorie

12、ntation=67,73):dl2:=%:display(dO,dl2);C,利用spacecurve函数with(plots):with(plottools):N:=54:h:=4:r:=(l-k/N)A2:F3:=i->spacecurve(seq(r*cos(k*6*Pi/N)rr*sin(k*6*Pi/N),h*r/N*kzk=0.i),style=POINT,symbolsize=12zsymbol=DIAMOND,color=blue):spacecurve(r*cos(6*Pi*k/N)zr*sin(6*Pi*k/N)Zh*r/N*kzk=0.N):d0:=%:displ

13、ay(seq(F3(i)zi=0.N),insequence=truezaxes=box,view=-11,-11,0.,orientation=67,73):dl3:=%:display(dOzdl3);利用三个不同画图函数的方型,自左至右：A,B,C图5成效相同的连环画利用变换函数制作点踪迹动画（匀步长）with(plots):with(plottools):N:=54:h:=4:r:=k/N:a:=l:b:=2:c:=0:point(azb,czsymbolsize=12zsymbol=CIRCLE,color=black):G:=%:F:=i->seq(translate(G,r

14、*cos(k*6*Pi/N)zr*sin(k*6*Pi/N),h/N*k),k=0.i):spacecurve(a+r*cos(6*Pi*k/N),b+r*sin(6*Pi*k/N),c+h/N*k,k=0.Nfcolor=green):d0:=%:display(ANIMATE(seq(F(i),i=0.N)finsequence=true,axes=FRAME,orientation=65f65,view=a-1.a+1,b-l.b+1zc.c+4):dl:=%:display(dO,dl);Q:=%:图6用变换函数制作的点踪迹动画(匀步长)点的步进式踪迹动画(匀步长)with(plots

15、):R:=8:a:=0:b:=2*Pi:w:=3:f:=i->cos(a+(b-a)*i/R/w):g:=i->sin(a+(b-a)*i/R/w):h:=i->(a+(b-a)*i/R/w)/2:F:=n->POINTS(seq(evalf(f(i)zg(i)zh(i)zi=0.w*n),SYMBOL(CIRCLE,18)ZCOLOR(RGB,1,0,0):display(seq(F(n)zn=0.R)rinsequence=true,axes=box,orientation=70,50zview=-1.1,-1.1,0.);Q:=%:PL0T3D(ANIMATE(s

16、eq(F(n)fn=0.R)zAXES(FRAME):;图7点元步进式踪迹(匀步长)动画点的步进式踪迹动画(变步长)with(plots):R:=8:K：=21:a:=0:b:=2*Pi:f:=i->cos(a+(b-a)*i/K):g:=i->sin(a+(b-a)*i/K):h:=i->(a+(b-a)*i/K)/2:F:=n->POINTS(seq(evalf(f(i)zg(i)zh(i),i=0.K/R*n)ZSYMBOL(CIRCLE,18)zCOLOR(RGBzOr0,1):display(seq(F(n)rn=Q.R)zinsequence=true,ax

17、es=frameforientation=30,65);Q:=%:PLOT3D(ANIMATE(seq(F(n)fn=0.R),AXES(FRAME):;图8点元步进式踪迹（变步长）动画点的链伸式踪迹动画(全元族)with(plots):K:=18:R:=8:a:=0:b:=2*Pi:f:=(i,n)->cos(a+(b-a)*i/K*n/R):g:=(i,n)->sin(a+(b-a)*i/K*n/R):h:=(izn)->(a+(b-a)*i/K*n/R)/2:F:=n->POINTS(seq(evalf(f(irn),g(i,n)fh(i,n),i=0.K)zSY

18、MBOL(CIRCLE,18),COLOR(RGB,1,0,0):display(seq(F(n)fn=0.R)rinsequence=true,axes=box);PLOT3D(ANIMATE(seq(F(n),n=0.R),AXES(FRAME);图9全元族链伸踪迹动画点的链伸式踪迹动画(变元族，匀步长)with(plots):K:=15:R:=8:w:=2:u:=w*R/K:a:=0:b:=2*Pi:f:=(i,n)->cos(a+(b-a)*i/K*n/R/u):g:=(i,n)->sin(a+(b-a)*i/K*n/R/u):h:=(izn)->-(a+(b-a)*

19、i/K*n/R/u)/2:F:=n->POINTS(seq(evalf(f(ifn),g(i,n),h(i,n)zi=0.w*n),SYMBOL(CIRCLE,18),COLOR(RGB,1,0z0):display(seq(F(n)fn=0.R)zinsequence=true,axes=box,orientation=73,58):;PLOT3D(ANIMATE(seq(F(n),n=0.R),AXES(FRAME);图10点元链伸式踪迹（变元族，等步长）动画点的链伸式踪迹动画(变元族，变步长)with(plots):K:=19:R:=8:a:=0:b:=2*Pi:f:=(i,n)-

20、>cos(a+(b-a)*i/K*n/R):g:=(i,n)->sin(a+(b-a)*i/K*n/R):h:=(izn)->-(a+(b-a)*i/K*n/R)/2:F:=n->POINTS(seq(evalf(f(i,n),g(i,n),h(i,n),i=0.K/R*n),SYMBOL(DIAMOND,18)rCOLOR(RGBz0,0z1):display(seq(F(n)fn=0.R)zinsequence=true,axes=box);PLOT3D(ANIMATE(seq(F(n)rn=0.R)zAXES(FRAME)ZORIENTATION(5,55);图1

21、1点元链伸式踪迹（变元族，变步长）动画图111215线的进程动画利用对象函数制作线元的进程动画with(plots):with(plottools):N:=8:F:=i->curve(-1,0,0z1,0z0,0,0z0r0,2*cos(i*2*Pi/N)z2*sin(i*2*Pi/N),0z2*cos(i*2*Pi/N)+5,0z0,2*cos(i*2*Pi/N)+8z0,l/2,2*cos(i*2*Pi/N)+7,0,-1/2,2*cos(i*2*Pi/N)+7z0,0z2*cos(i*2*Pi/N)+8z0,color=blackzthickness=4):display(seq(

22、F(i)Ai=0.N)zinsequence=truezaxes=frame):;PLOT3D(ANIMATE(seq(F(i)zi=0.N),AXESSTYLE(FRAME):;图12用对象函数制作的线元进程动画(曲柄连杆机构的运动)利用画图函数制作线元的进程动画with(plots):N:=27:r:=25:F:=i->spacecurve(r*cos(i*2*Pi/N)+t*cos(t),r*sin(i*2*Pi/N)+t*sin(t)zt,t=0.6*Pi,color=black,thickness=3):display(seq(F(i)fi=0.N),insequence=tr

23、uezaxes=normal):PLOT3D(ANIMATE(seq(F(i)ri=0.N),AXESSTYLE(FRAME):图13用画图函数制作的线元进程动画利用变换函数制作线元的进程动画with(plots):with(plottools):N:=8:r:=15:spacecurve(t*cos(t),t*sin(t),-t/7,t=06*Pi,color=red,thickness=3):L:=%:F:=i->translate(Lzr*cos(i*2*Pi/N),r*sin(i*2*Pi/N),0):display(seq(F(i)fi=0.N)rinsequence=true

24、,axes=box):dl:=%:spacecurve(r*cos(t),r*sin(t),0,t=02*Pi,color=blue):d0:=%:display(dO,dlrorientation=5073);Q:=%:国14用变换函数制作的线元进程动画with(plots):M:=36:R:=24:a:=0:b:=4*Pi:c:=0:d:=2*Pi:f:=(i,n)->cos(a+(b-a)*i/M)+sin(c+(d-c)*n/R):g:=(izn)->(a+(b-a)*i/M)/6+(c+(d-c)*n/R):h:=(izn)->sin(a+(b-a)*i/M):F:

25、=n->CURVES(seq(evalf(f(i,n),g(izn)zh(ifn)fi=0.M).THICKNESS(3),COLOR(RGB,1,0,0):display(seq(F(n)fn=0.R),insequence=true,axes=NORMAL,labels=x,y,z,orientation=63r73);G-O-O->一©-40图15一个内层线元进程动画的连环画图11-1623线的的踪迹动画利用对象函数制作线元的踪迹动画(匀步长)with(plots):with(plottools):R:=16:G:=n->seq(curve(0,0z0zcos

26、(i*2*Pi/R),sin(i*2*Pi/R),5,*cos(i*2*Pi/R),*sin(i*2*Pi/R),5,cos(i*2*Pi/R),sin(i*2*Pi/R),2,cos(i*2*Pi/R),sin(i*2*Pi/R),3.linestyle=l,thickness=3,color=red),i=0.n):display(ANIMATE(seq(G(n)fn=0.R)zinsequence=true,axes=frame):图16用对象函数制作线元的踪迹动画利用画图函数制作线元的踪迹动画(匀步长)with(plots):R:=24:r:=8:G:=n->seq(spacec

27、urve(0+2*Pi*i/R,t*cos(t)-30+r*sin(2*Pi*i/R),t*sin(t)zt=0.6*Pizcolor=black,thickness=l),i=0.n):display(seq(display(G(n),n=0.R)insequence=true,axes=frame);图17用画图函数制作线元的踪迹动画利用变换函数制作线元的踪迹动画（匀步长）restart:with(plottools):with(plots):R:=24:r:=2:t:=4*Pi*i/R:L:=curve(0,0,0,1/3,-1/7,1,0,1,l/3zl/7,0,0,0zthickne

28、ss=3,color=COLOR(RGB,r:f:=i->translate(rotate(Lz0f0f-t),r*cos(t),r*sin(t),t/l)，spacecurve(r*cos(tl),r*sin(tl),tl/1,tl=0.4*Pi,color=brown,thickness=2):d0:=%:G:=n->seq(f(i),i=0.n):display(ANIMATE(seq(G(n),n=0.R)rinsequence=true,orientation=54,61zaxes=frame):;dh:=%:display(dhzd0);图18用变换函数制作线元的踪迹动

29、画线的步进式踪迹动画（匀步长）M:=24:R:=8:w:=2:a:=-2*Pi:b:=2*Pi:c:=0:d:=2*Pi:f:=(i,k)->(a+(b-a)*i/M)+3*sin(c+(d-c)*k/R/w):g:=(i,k)->(c+(d-c)*k/R/w):h:=(i,k)->cos(a+(b-a)*i/M):F:=n->CURVES(seq(seq(evalf(f(i1k),g(i1k),h(i,k),i=0.M),k=0.w*n),THICKNESS(2),COLOR(RGB,0,Ofl)：PLOT3D(ANIMATE(seq(F(n),n=O.R),AXES

30、(FRAME);Q:=%:图19线元匀步长步进式踪迹动画的内层设计线的步进式踪迹动画（变步长）M:=21:R:=8:u:=M/R:a:=-4:b:=4:c:=0:d:=2*Pi:f: =(i,k)->(a+(b-a)*i/M):g:=(i,k)->(c+(d-c)*k/R/u):h:=(izk)(a+(b-a)*i/M)A2+3*sin(c+(d-c)*k/R/u):F:=n->CURVES(seq(seq(evalf(f(i,k),g(i,k),h(i,k)0.M),k=0.M/R*n),THICKNESS(2),COLOR(RGB,1f0,0):PLOT3D(ANIMAT

31、E(seq(F(n)fn=0.R),AXES(NORMAL),ORIENTATION(30,75);图20线元变步长步进踪迹动画的内层设计线的链伸式踪迹动画（全元族）M:=15:K:=12:R:=8:a:=-1:b:=1:c:=0:d:=2*Pi:f:=(i,k,n)->(a+(b-a)*i/M)-2*sin(c+(d-c)*k/K*n/R):g:=(i,k,n)->(c+(d-c)*k/K*n/R):h:=(i,k,n)->1-(a+(b-a)*i/M)A3:F:=n->CURVES(seq(seq(evalf(f(i,k,n)zg(i,k,n),h(i,k,n),i

32、=0.M),k=0.K)zTHICKNESS(2)rCOLOR(RGB,1,0r0):PLOT3D(ANIMATE(seq(F(n)fn=0.R),AXES(FRAME),ORIENTATION(-23,74);图21线元全元族链伸动画的内层设计线的链伸式踪迹动画(变元族，匀步长)M:=18:R:=8:w:=2:a:=0:b:=2*Pi:c:=0:d:=2*Pi:m:=5:f:=(izk,n)->m*cos(a+(b-a)*i/M)+3*sin(c+(d-c)*k/R*n/R/w):g: =(i,kzn)->(c+(d-c)*k/R*n/R/w):h: =(i,kzn)->m

33、*sin(a+(b-a)*i/M):F:=n->seq(CURVES(seq(evalf(f(i,k,n)fg(i,k,n),h(i,kzn)ri=0.M)zCOLOR(HUEzk/16),THICKNESS(2)zk=0.w*n):PLOT3D(ANIMATE(seq(F(n)zn=0.R)zAXES(NORMAL);图22内层线元变元族匀步长链伸动画的设计线的链伸式踪迹动画(变元族，变步长)M:=5:K:=13:R:=8:a:=-l:b:=l:c:=0:d:=2*Pi:f:=(i,k,n)->(a+(b-a)*i/M)+l/2*sin(c+(d-c)*k/K*n/R):g:=(

34、i,kzn)->(c+(d-c)*k/K*n/R):h:=(i,k,n)->-1/5*(a+(b-a)*i/M)A2:F:=n->CURVES(seq(seq(evalf(f(i,k,n)zg(irk,n),h(i,k,n),i=0.M)zk=0.K/R*n),THICKNESS(2),COLOR(RGB,1r0,0)：PLOT3D(ANIMATE(seq(F(n),n=0.R),AXES(NORMAL),ORIENTATION(35,60);Q:=%:图23内层线元变元族变步长链伸动画的设计图11-2427面体的进程动画利用对象函数制作面体的进程动画with(plots):

35、with(plottools):N:=8:F:=i->cone(4*cos(i*2*Pi/N),4*sin(i*2*Pi/N),3z(2+sin(i)*,color=greenfgrid=8,5):display(seq(F(i)Ai=0.N),insequence=true,axes=box,scaling=unconstrainedorientation=36,65);图24面体的外层进程动画利用画图函数制作面体的进程动画restart:with(plots):with(plottools):N:=8:F:=i->plot3d(x,sin(x),y*5*izx=0.2*Pi*(

36、i/(2*N)+.5),y=0.1,grid=18,4,color=cyan):display(seq(F(i)fi=0.N),insequence=true,axes=box);图25用画图函数制作面体的进程动画利用变换函数制作面体的进程动画with(plots):with(plottools):1:=2:a:=3:b:=2:L:=plot3d(u,vz-uA2-vA2,u=-l.1,v=-l.1,grid=6z6,color=magenta):N:=24:t:=2*Pi*i/N:LI:=i->translate(Lza*cos(t)fb*sin(t),0):;display(seq(

37、LI(i)zi=0.N),view=-(a+1).a+1f-(b+1).b+1,-2.0zaxes=noinnal,labels=xzy,z,insequence=true,orientation=55,55);PLOT3D(ANIMATE(seq(op(op(LI(i)ri=0.N),AXES(FRAME)图26用变换函数制作面体的进程动画restart:with(plots):M:=6:N:=4:R:=15:a:=-3:b:=3:c:=-4:d:=4:u:=0:v:=30:f:=(i,jzn)->(a+(b-a)*i/M):g: =(i,j,n)->(c+(d-c)*j/N):

38、h:=(i,j,n)->(a+(b-a)*i/M)A2+(c+(d-c)*j/N)A2)*(u+(v-u)*n/R):F:=n->MESH(evalf(seq(seq(f(i,jzn)zg(i,jrn),h(izj,n),i=0.M)zj=0.N),C0L0R(HUE,n/16):PL0T3D(ANIMATE(seq(F(n)zn=0.R),AXES(BOX)zORIENTATION(9,71);图27面元进程动画的内层设计图11-2835面体的踪迹动画利用对象函数制作面体的踪迹动画(匀步长)with(plots):with(plottools):R:=24:G:=n->se

39、q(polygon(0,0,0,cos(i*2*Pi/R)rsin(i*2*Pi/R),5,cos(i*2*Pi/R),sin(i*2*Pi/R)r2zthickness=3,color=green),i=0.n):PLOT3D(ANIMATE(seq(G(n)zn=0.R)zAXES(BOX)zORIENTATION(150,75);Q:=%:display(ANIMATE(seq(G(n)fn=0.R)finsequence=true,axes=frame):图28用对象函数制作的面体踪迹动画利用画图函数制作面体的踪迹动画(匀步长)with(plots):N:=8:a:=4:G:=i-&g

40、t;(u-i/Nrv+i/N,(uA2+vA2)+(a*i/N)A2,u=-2.2,v=-l.l,grid=5,5zcolor=cyan):GG:=n->display(seq(plot3d(G(i)zi=0.n):display(seq(GG(n),n=0.N),insequence=trueraxes=frame,orientation=-15,60);国29只用画图函数和display设计的曲面步进踪迹动画利用变换函数制作面体的踪迹动画(匀步长)with(plots):with(plottools):R:=9:L:=cuboid(2,2,2,1,1,1,style=patch):f

41、:=i->rotate(Lz0f0f2*Pi*i/R):;display(seq(f(i),i=0.R);GG:=n->display(seq(f(i)fi=0.n):display(seq(GG(n),n=0.R),insequence=truefaxes=frame,orientation=-13z53);图30用变换函数制作的面体踪迹动画体步进式踪迹动画(匀步长)restart:with(plots):M:=4:N:=6:R:=12:w:=l:a:=-2:b:=2:c:=-2:d:=2:u:=0:v:=2*Pi:f:=(i,j,k)->(a+(b-a)*i/M):g:=

42、(i,j,k)->(c+(d-c)*j/N)+5*cos(u+(v-u)*k/R):h:=(i,jzk)->(a+(b-a)*i/M)A2+(c+(d-c)*j/N)A2+5*sin(u+(v-u)*k/R):Fl:=n->seq(MESH(seq(seq(evalf(f(i,jzk),g(i,j,k),h(i,jzk)zi=0.M),j=0.N),COLOR(HUE,k/16)zk=0.w*n):PLOT3D(ANIMATE(seq(Fl(n),n=0.R),AXES(BOX),SCALING(CONSTRAINED),ORIENTATION(23,64);图31面元匀步长

43、步进踪迹动画的内层设计面体步进式踪迹动画（变步长）restart:M:=3:N:=7:R:=8:K:=21:r:=15:u:=K/R:al:=-3:bl:=1:a2:=-4:b2:=1:c:=0:d:=2*Pi:f:=(i,j,k)->(al+(bl-al)*i/M)+r*cos(c+(d-c)*k/R/u):g:=(i,jzk)->(a2+(b2-a2)*j/N)+r*sin(c+(d-c)*k/R/u):h:=(i,j,k)->12*(al+(bl-al)*i/M)(a2+(b2-a2)*j/N):Fl:=n->seq(MESH(seq(seq(evalf(f(i,jzk),g(i,jzk)fh(i,jzk)