基本原理就是分成2部分和一點，然後遞迴。
比如4點曲線的基本原理：
圖1的藍線=圖3的綠線+曲線點【紅綠交點】+圖3的紅線；
圖3的綠線=>圖1的藍線;
圖3的紅線=>圖1的藍線；
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;
}
