基本原理就是分成2部分和一点，然后递归。
比如4点曲线的基本原理：
图1的蓝线=图3的绿线+曲线点【红绿交点】+图3的红线；
图3的绿线=>图1的蓝线;
图3的红线=>图1的蓝线；

 

 

 

 

struct xyf;
struct xy{int x;int y;xy(){}xy(int m){x=m;y=m;}xy(int x,int y){this->x=x;this->y=y;}xy(xyf m);
xy operator  *(xy&m){return xy(x *m.x,y *m.y);}
xy operator  /(xy&m){return xy(x /m.x,y /m.y);}
xy operator  %(xy&m){return xy(x %m.x,y %m.y);}
xy operator  +(xy&m){return xy(x +m.x,y +m.y);}
xy operator  -(xy&m){return xy(x -m.x,y -m.y);}
xy operator <<(xy&m){return xy(x<<m.x,y<<m.y);}
xy operator >>(xy&m){return xy(x>>m.x,y>>m.y);}
xy operator  &(xy&m){return xy(x &m.x,y &m.y);}
xy operator  |(xy&m){return xy(x |m.x,y |m.y);}
xy operator  ^(xy&m){return xy(x ^m.x,y ^m.y);}
xy operator  *=(xy&m){x *=m.x;y *=m.y;return *this;}
xy operator  /=(xy&m){x /=m.x;y /=m.y;return *this;}
xy operator  %=(xy&m){x %=m.x;y %=m.y;return *this;}
xy operator  +=(xy&m){x +=m.x;y +=m.y;return *this;}
xy operator  -=(xy&m){x -=m.x;y -=m.y;return *this;}
xy operator <<=(xy&m){x<<=m.x;y<<=m.y;return *this;}
xy operator >>=(xy&m){x>>=m.x;y>>=m.y;return *this;}
xy operator  &=(xy&m){x &=m.x;y &=m.y;return *this;}
xy operator  |=(xy&m){x |=m.x;y |=m.y;return *this;}
xy operator  ^=(xy&m){x ^=m.x;y ^=m.y;return *this;}
xy operator  *(int m){return xy(x *m,y *m);}
xy operator  /(int m){return xy(x /m,y /m);}
xy operator  %(int m){return xy(x %m,y %m);}
xy operator  +(int m){return xy(x +m,y +m);}
xy operator  -(int m){return xy(x -m,y -m);}
xy operator <<(int m){return xy(x<<m,y<<m);}
xy operator >>(int m){return xy(x>>m,y>>m);}
xy operator  &(int m){return xy(x &m,y &m);}
xy operator  |(int m){return xy(x |m,y |m);}
xy operator  ^(int m){return xy(x ^m,y ^m);}
xy operator  *=(int m){x *=m;y *=m;return *this;}
xy operator  /=(int m){x /=m;y /=m;return *this;}
xy operator  %=(int m){x %=m;y %=m;return *this;}
xy operator  +=(int m){x +=m;y +=m;return *this;}
xy operator  -=(int m){x -=m;y -=m;return *this;}
xy operator <<=(int m){x<<=m;y<<=m;return *this;}
xy operator >>=(int m){x>>=m;y>>=m;return *this;}
xy operator  &=(int m){x &=m;y &=m;return *this;}
xy operator  |=(int m){x |=m;y |=m;return *this;}
xy operator  ^=(int m){x ^=m;y ^=m;return *this;}
bool operator  <(xy&m){return((unsigned(x) <unsigned(m.x))&&(unsigned(y) <unsigned(m.y)));}
bool operator  >(xy&m){return((unsigned(x) >unsigned(m.x))&&(unsigned(y) >unsigned(m.y)));}
bool operator <=(xy&m){return((unsigned(x)<=unsigned(m.x))&&(unsigned(y)<=unsigned(m.y)));}
bool operator >=(xy&m){return((unsigned(x)>=unsigned(m.x))&&(unsigned(y)>=unsigned(m.y)));}
bool operator ==(xy&m){return((unsigned(x)==unsigned(m.x))&&(unsigned(y)==unsigned(m.y)));}
bool operator  <(int m){return((unsigned(x) <unsigned(m))&&(unsigned(y) <unsigned(m)));}
bool operator  >(int m){return((unsigned(x) >unsigned(m))&&(unsigned(y) >unsigned(m)));}
bool operator <=(int m){return((unsigned(x)<=unsigned(m))&&(unsigned(y)<=unsigned(m)));}
bool operator >=(int m){return((unsigned(x)>=unsigned(m))&&(unsigned(y)>=unsigned(m)));}
bool operator ==(int m){return((unsigned(x)==unsigned(m))&&(unsigned(y)==unsigned(m)));}
xy operator  !(){return xy( !x, !y);}
xy operator  ~(){return xy( ~x, ~y);}
xy operator  +(){return xy( +x, +y);}
xy operator  -(){return xy( -x, -y);}xy get_number(){xy m=0;if(x>0)m.x=+x;if(x<0)m.x=-x;if(y>0)m.y=+y;if(y<0)m.y=-y;return m;}xy get_flag(){xy m=0;if(x>0)m.x=+1;if(x<0)m.x=-1;if(y>0)m.y=+1;if(y<0)m.y=-1;return m;}void x_y(){int v=x;x=y;y=v;}int xx(){return x*x;}int yy(){return y*y;}int xx_yy(){return (x*x+y*y);}xy operator =(xy&m){x=m.x;y=m.y;return *this;}xy operator =(int m){x=m;y=m;return *this;}xy operator ++(int){xy temp=*this;++x;++y;return temp;}xy operator ++(){x++;y++;return *this;}xy operator --(int){xy temp=*this;--x;--y;return temp;}xy operator --(){x--;y--;return *this;}};struct xyf{double x;double y;xyf(){}xyf(double m){x=m;y=m;}xyf(double x,double y){this->x=x;this->y=y;}xyf(xy m){this->x=m.x;this->y=m.y;}
xyf operator  *(xyf&m){return xyf(x *m.x,y *m.y);}
xyf operator  /(xyf&m){return xyf(x /m.x,y /m.y);}
xyf operator  +(xyf&m){return xyf(x +m.x,y +m.y);}
xyf operator  -(xyf&m){return xyf(x -m.x,y -m.y);}
xyf operator  *=(xyf&m){x *=m.x;y *=m.y;return *this;}
xyf operator  /=(xyf&m){x /=m.x;y /=m.y;return *this;}
xyf operator  +=(xyf&m){x +=m.x;y +=m.y;return *this;}
xyf operator  -=(xyf&m){x -=m.x;y -=m.y;return *this;}
xyf operator  *(double m){return xyf(x *m,y *m);}
xyf operator  /(double m){return xyf(x /m,y /m);}
xyf operator  +(double m){return xyf(x +m,y +m);}
xyf operator  -(double m){return xyf(x -m,y -m);}
xyf operator  *=(double m){x *=m;y *=m;return *this;}
xyf operator  /=(double m){x /=m;y /=m;return *this;}
xyf operator  +=(double m){x +=m;y +=m;return *this;}
xyf operator  -=(double m){x -=m;y -=m;return *this;}xyf get_number(){xyf m=0;if(x>0)m.x=+x;if(x<0)m.x=-x;if(y>0)m.y=+y;if(y<0)m.y=-y;return m;}xyf get_flag(){xyf m=0;if(x>0)m.x=+1;if(x<0)m.x=-1;if(y>0)m.y=+1;if(y<0)m.y=-1;return m;}void x_y(){double v=x;x=y;y=v;}double xx(){return x*x;}double yy(){return y*y;}double xx_yy(){return (x*x+y*y);}xyf operator =(xy&m){x=m.x;y=m.y;return *this;}xyf operator =(xyf&m){x=m.x;y=m.y;return *this;}xyf operator =(double m){x=m;y=m;return *this;}xyf operator ++(int){xyf temp=*this;++x;++y;return temp;}xyf operator ++(){x++;y++;return *this;}xyf operator --(int){xyf temp=*this;--x;--y;return temp;}xyf operator --(){x--;y--;return *this;}};xy::xy(xyf m){x=m.x;y=m.y;}int paint_line(xy r,xy *da){// float->int (f+0.5)// [-0.5,+0.5)=0// [+0.5,+1.5)=1// [+1.5,+2.5)=2// x*a+b*y=0;// (x+v)*a=x*a+v*a;// (y+v)*a=y*a+v*a;xy n=r.get_number();xy f=r.get_flag();xy nn=n*2;xy d=0;xy*db=da;*db++=d;if(n.y<n.x){int v=n.x;while(d.x!=r.x){d.x+=f.x;v-=nn.y;if(v<0){v+=nn.x;d.y+=f.y;}*db++=d;}}else{int v=n.y;while(d.y!=r.y){d.y+=f.y;v-=nn.x;if(v<0){v+=nn.y;d.x+=f.x;}*db++=d;}}return (db-da);
}
int paint_line(xy a,xy b,xy *da){int dn=paint_line(b-a,da);for(int i=0;i<dn;i++)da[i]+=a;return dn;
}const int curve_max=10;
const int curve_bit=16;
const int curve_bit_add=1<<(curve_bit-1);
xy curve_get(xy m){m+=curve_bit_add;m>>=curve_bit;return m;
}
xy*curve_call3(xy s1,xy s2,xy s3,xy*da){xy y1=(s1+s2)>>1;xy y2=(s2+s3)>>1;  xy zz=(y1+y2)>>1;xy z=curve_get(zz);if(z==curve_get(s1))return da;if(z==curve_get(s3))return da;da=curve_call3(s1,y1,zz,da);*da++=z;da=curve_call3(s3,y2,zz,da);return da;
}
xy*curve_call4(xy s1,xy s2,xy s3,xy s4,xy*da){xy x1=(s1+s2)>>1;xy x2=(s2+s3)>>1;xy x3=(s3+s4)>>1;xy y1=(x1+x2)>>1;xy y2=(x2+x3)>>1;  xy zz=(y1+y2)>>1;xy z=curve_get(zz);if(z==curve_get(s1))return da;if(z==curve_get(s4))return da;da=curve_call4(s1,x1,y1,zz,da);*da++=z;da=curve_call4(s4,x3,y2,zz,da);return da;
}
xy*curve_call(xy*sa,int sn,xy*da){xy a[curve_max];xy b[curve_max];xy s[curve_max];for(int i=0;i<sn;i++)s[i]=sa[i];for(int n=sn,j=0;0<n;n--,j++){a[j]=s[0];b[j]=s[n-1];for(int i=1;i<n;i++)s[i-1]=(s[i-1]+s[i])>>1;;}xy z=curve_get(s[0]);if(z==curve_get(sa[0]))return da;if(z==curve_get(sa[sn-1]))return da;da=curve_call(a,sn,da);*da++=z;da=curve_call(b,sn,da);return da;
}int curve(xy*sa__,int sn,xy*da){if(sn==2)return paint_line(sa__[0],sa__[1],da);if(3<=sn&&sn<=curve_max){}else return 0;xy sa[curve_max];for(int i=0;i<sn;i++)sa[i]=sa__[i]<<curve_bit;xy*db=da;*db++=curve_get(sa[0]); db=curve_call(sa,sn,db);/*if(0){}else if(3==sn){db=curve_call3(sa[0],sa[1],sa[2],db);}else if(4==sn){db=curve_call4(sa[0],sa[1],sa[2],sa[3],db);}else return 0;*/*db++=curve_get(sa[sn-1]);return db-da;
}