18-自由网格曲面的生成

1自由网格曲面的绘制内容大纲1双线性曲面2单线性曲面3Bezier曲面4B样条曲面已知任意4个角点的坐标值x(u,v)=x00(1-u)(1-v)+x01(1-u)v+x10(1-v)u+x11uv

y(u,v)=y00

(1-u)(1-v)+y01(1-u)v+y10(1-v)u+y11uvz(u,v)=z00

(1-u)(1-v)+z01(1-u)v+z10(1-v)u+y11uv

(x01,y01)(x00,y00)(x10,y10)(x11,y11)(0≤u≤1,0≤v≤1)u=0,v=0～1

u=0,v=0u=1,v=0u=0,v=1u=1,v=1u=0.2,v=0～1

……u=1,v=0～1

v=0,u=0～1

v=0.2,u=0～1

……v=1,u=0～1

1双线性曲面双线性曲面就是两簇直线段的叠加曲面1双线性曲面voidDoubleLinear(CDC*pDC,intX[2][2],intY[2][2],intZ[2][2],floatdu,floatdv,floatcx,floatcy,floatcz,intdx,intdy,intdz，COLORREFcolor){floatU[4],V[4],x[5],y[5],z[5],xx[5],yy[5],zz[5];for(floatu=0;u<=1.0001;u=u+du)for(floatv=0;v<=1.0001;v=v+dv){U[0]=u,V[0]=v,U[1]=u+du,V[1]=v,U[2]=u+du,V[2]=v+dv,U[3]=u,V[3]=v+dv;for(inti=0;i<4;i++){x[i]=X[0][0]*(1-U[i])*(1-U[i])+X[0][1]*(1-U[i])*U[i]+X[1][0]*(1-U[i])*U[i]+X[1][1]*U[i]*U[i];y[i]=Y[0][0]*(1-U[i])*(1-U[i])+Y[0][1]*(1-U[i])*U[i]+Y[1][0]*(1-U[i])*U[i]+Y[1][1]*U[i]*U[i];z[i]=Z[0][0]*(1-U[i])*(1-U[i])+Z[0][1]*(1-U[i])*U[i]+Z[1][0]*(1-U[i])*U[i]+Z[1][1]*U[i]*U[i];RevolveX(cx,x[i],y[i],z[i],xx[i],yy[i],zz[i]);//绕X轴旋转

RevolveY(cy,xx[i],yy[i],zz[i],x[i],y[i],z[i]);//绕Y轴旋转

RevolveZ(cz,x[i],y[i],z[i],xx[i],yy[i],zz[i]);//绕Z轴旋转

xx[i]=xx[i]+dx,yy[i]=yy[i]+dy,zz[i]=zz[i]+dz;//加上平移量}xx[4]=xx[0],yy[4]=yy[0];//封闭小四边形

for(i=0;i<4;i++) Line_DDA(pDC,xx[i],yy[i],xx[i+1],yy[i+1],color);//绘制小四边形

}}1双线性曲面x(u,v)=(1-v)fx(u)+vgx(u)y(u,v)=(1-v)fy(u)+vgy(u)z(u,v)=(1-v)fz(u)+vgz(u)v:

[0，1]u:

[u1，u2]关于v是线性，关于u是非线性。f(u)、g(u)是空间中任意的两条曲线。x(u,v)=(1-v)(400u+30)+

v300uy(u,v)=(1-v)(40sin(3

u)+100)+v30sin(4

u+3)z(u,v)=0f(u)g(u)2单线性曲面v=0.5也称直纹面v=1v=0v=0.25v=0.75u=0u=0.2u=0.4u=0.6u=0.8u=12单线性曲面voidSingerLinear(CDC*pDC,float(*fx)(floatu),float(*fy)(float),float(*fz)(float),float(*gx)(float),float(*gy)(float),float(*gz)(float),floatdu,floatdv,floatcx,floatcy,floatcz,intdx,intdy,intdz，COLORREFcolor){floatU[4],V[4],x[5],y[5],z[5],xx[5],yy[5],zz[5];for(floatu=0;u<=1.0001;u=u+du)for(floatv=0;v<=1.0001;v=v+dv){U[0]=u,V[0]=v,U[1]=u+du,V[1]=v,U[2]=u+du,V[2]=v+dv,U[3]=u,V[3]=v+dv;for(inti=0;i<4;i++){x[i]=

(1-U[i])*(*)fx(U[i])+V[i]*(*)gx(U[i]);y[i]=(1-U[i])*(*)fy(U[i])+V[i]*(*)gy(U[i]);z[i]=(1-U[i])*(*)fz(U[i])+V[i]*(*)gz(U[i]);//计算曲面上的点RevolveX(cx,x[i],y[i],z[i],xx[i],yy[i],zz[i]);//绕X轴旋转

RevolveY(cy,xx[i],yy[i],zz[i],x[i],y[i],z[i]);//绕Y轴旋转

RevolveZ(cz,x[i],y[i],z[i],xx[i],yy[i],zz[i]);//绕Z轴旋转

xx[i]=xx[i]+dx,yy[i]=yy[i]+dy,zz[i]=zz[i]+dz;}//加上平移量xx[4]=xx[0],yy[4]=yy[0];//封闭小四边形

for(i=0;i<4;i++) Line_DDA(pDC,xx[i],yy[i],xx[i+1],yy[i+1],color);//绘制小四边形

}}2单线性曲面

x(u,v)=400u+100v+100y(u,v)=30sin(4

u)+100v+100z(u,v)=

02单线性曲面柱面Bezier曲面由控制多面体控制p(u，v)=

u，v∈[0，1]3Bezier曲面

在双三次Bezier曲面中，

需给定的16个顶点。

4个角点p00、p03、p30

和p33在曲面上。

其余顶点均不在曲面上。3Bezier曲面floatB(inti,intn,floatt){intk;floatt1=1,t2=1,h;for(k=i+1;k<=n;k++)t1=t1*k;for(k=1;k<=n-i;k++)t2=t2*k;h=t1/t2;for(k=1;k<=i;k++)h=h*t;for(k=1;k<=n-i;k++)h=h*(1-t);returnh;}基函数的程序设计3Bezier曲面voidBezierFace(CDC*pDC,intX[10][10],intY[10][10],intZ[10][10],intm,intn,floatdu,floatdv,fintdx,intdy,COLORREFcolor){floatU[4],V[4],x[5],y[5],z[5],b1,b2;for(floatu=0;u<=1;u=u+du)

for(floatv=0;v<=1.;v=v+dv)

{U[0]=u,V[0]=v,U[1]=u+du,V[1]=v,U[2]=u+du,V[2]=v+dv,U[3]=u,V[3]=v+dv;

for(intk=0;k<4;k++){x[k]=0,y[k]=0,z[k]=0;for(inti=0;i<=m;i++)for(intj=0;j<=n;j++) {b1=B(i,m,U[k]),b2=B(j,n,V[k]);

x[k]=x[k]+X[i][j]*b1*b2;y[k]=y[k]+Y[i][j]*b1*b2;z[k]=z[k]+Z[i][j]*b1*b2;}}x[4]=x[0],y[4]=y[0];

for(k=0;k<4;k++)Line_DDA(pDC,x[k]+dx,y[k]+dy,x[k+1]+dx,y[k+1]+dy,color);}}3Bezier曲面3Bezier曲面intx[5][4]={60,150,230,300,80,180,260,330,120,230,280,350,180,280,330,380,220,320,350,440},y[5][4]={180,140,140,190,120,50,60,110,50,20,30,70,80,40,50,80,40,10,20,50},z[5][4]={0,0,0,0,50,50,50,50,100,100,100,100,200,200,200,200,250,250,250,250};CDC*p=GetDC();BezierFace(p,x,y,z,4,3,0.05,0.05,100,100,RGB(0,0,0));3Bezier曲面4B样条曲面（1）均匀周期二次B样条曲面N0(t)=(1-t)2/2N1(t)=

(-2t2+2t+1)/2N2(t)=t2/20

<=t<=1,i=0,1,2,……m-2，j=0,1,2,……n-2，4B样条曲面在二次B样条曲面的其中一个面片中，面片的四个顶点分别是四个四边形的中心点。二次B样条基函数的程序设计floatN2(intn,floatt){if(n==0)return((1-t)*(1-t)/2);elseif(n==1)return((-2*t*t+2*t+1)/2);elseif(n==2)return(t*t/2);}4B样条曲面voidBFace2(CDC*pDC,intX[50][50],intY[50][50],intZ[50][50],intm,intn,floatdu,floatdv,intdx,intdy,COLORREFcolor){floatU[4],V[4],x[5],y[5],z[5];for(inti=0;i<m-1;i++)for(intj=0;j<n-1;j++)for(floatu=0;u<=1.0001;u=u+du)

for(floatv=0;v<=1.0001;v=v+dv)

{U[0]=u,V[0]=v,U[1]=u+du,V[1]=v,U[2]=u+du,V[2]=v+dv,U[3]=u,V[3]=v+dv;

for(intk=0;k<4;k++){x[k]=0,y[k]=0,z[k]=0;for(ints=0;s<=2;s++)for(intt=0;t<=2;t++) {x[k]=x[k]+X[i+s][j+t]*N2(s,U[k])*N2(t,V[k]),y[k]=y[k]+Y[i+s][j+t]*N2(s,U[k])*N2(t,V[k]),z[k]=z[k]+Z[i+s][j+t]*N2(s,U[k])*N2(t,V[k]);}}x[4]=x[0],y[4]=y[0];

for(k=0;k<4;k++)Line_DDA(pDC,x[k]+dx,y[k]+dy,x[k+1]+dx,y[k+1]+dy,color);}}4B样条曲面intx[5][4]={60,150,230,300,80,180,260,330,120,230,280,350,180,280,330,380,220,320,350,440},y[5][4]={180,140,140,190,120,50,60,110,50,20,30,70,80,40,50,80,40,10,20,50},z[5][4]={0,0,0,0,50,50,50,50,100,100,100,100,200,200,200,200,250,250,250,250};CDC*p=GetDC();BFace2(p,x,y,z,4,3,0.2,0.2,100,100,RGB(0,0,0));4B样条曲面4B样条曲面（2）均匀周期三次B样条曲面N0(t)=(-t3+3t2-3t+1)/6N1(t)=(3t3-6t2+4)/60

<=t<=1,i=0,1,2,……m-3，j=0,1,2,……n-3N2(t)=(-3t3+3t2+3t+1)/6N3(t)=t3/6三次B样条基函数的程序设计floatN3(intn,floatt){if(n==0)return((-t*t*t+3*t*t-3*t+1)/6);elseif(n==1)return((3*t*t*t-6*t*t+4)/6);elseif(n==2)return((-3*t*t*t+3*t*t+3*t+1)/6);elseif(n==3)return(t*t*t/6);}4B样条曲面voidBFace3(CDC*pDC,intX[50][50],intY[50][50],intZ[50][50],intm,intn,floatdu,floatdv,intdx,intdy,COLORREFcolor){floatU[4],V[4],x[5],y[5],z[5];

for(inti=0;i<m-2;i++)for(intj=0;j<n-2;j++)for(floatu=0;u<=1.0001;u=u+du)

for(floatv=0;v<=1.0001;v=v+dv)

{U[0]=u,V[0]=v,U[1]=u+du,V[1]=v,U[2]=u+du,V[2]=v+dv,U[3]=u,V[3]=v+dv;

for(intk=0;k<4;k++){x[k]=0,y[k]=0,z[k]=0;