一些基本的定义在这里: [模板] 计算几何1(基础): 点/向量/线/圆/多边形/其他运算
自适应Simpson
Simpson's Rule:
[int ^b_a f(x)dxapprox frac{b-a}6(f(a)+4f(frac{a+b}{2})+f(b))
]
这是对二次函数的积分估值, 对于一, 二次函数来说都是准确的.
但是对于其他函数来说, 这只是利用二次函数进行近似.
可以采用自适应精度的手段, 使得估值接近真实结果. 详见代码.
然后这是误差估计, 详见 adaptive.pdf:
[int_{a}^b f(x) mathrm{d}x = S(a, c) + S(c, b) + frac{1}{15}[S(a, c) + S(b, c) - S(a, b)] + O((b - a)^6)
]
一种实现:
db f(db x){
//returns f(x)
}
db simp(db l,db r){
db mid=(l+r)/2.0;
return (r-l)*(f(l)+4*f(mid)+f(r))/6.0;
}
db asr(db l,db r,db ans){
db mid=(l+r)/2.0;
db vl=simp(l,mid),vr=simp(mid,r),tmp=vl+vr-ans;
if(fabs(tmp)<=eps)return ans;
else return asr(l,mid,vl)+asr(mid,r,vr);
}
凸包
Andrew 算法, 即分别求上, 下凸包. 时间复杂度 (O(n log n)).
struct tvec{db x,y;};
il int dcmp(db a){return fabs(a)<=eps?0:(a>0?1:-1);}
il db p2(db a){return a*a;}
il db gougu1(db a,db b){return sqrt(p2(a)+p2(b));}
il tvec operator+(tvec a,tvec b){return (tvec){a.x+b.x,a.y+b.y};}
il tvec operator-(tvec a,tvec b){return (tvec){a.x-b.x,a.y-b.y};}
il tvec operator*(tvec a,db b){return (tvec){a.x*b,a.y*b};}
il tvec operator*(db a,tvec b){return b*a;}
il db operator*(tvec a,tvec b){return a.x*b.y-b.x*a.y;}
il db operator^(tvec a,tvec b){return a.x*b.x+a.y*b.y;}
il db len(tvec a){return gougu1(a.x,a.y);}
bool cmp(tvec a,tvec b){int tmp=dcmp(a.x-b.x);return tmp?tmp<0:dcmp(a.y-b.y)<0;}
tvec li[nsz],conv[nsz];
int pc=0;
void getconv(){
sort(li+1,li+n+1,cmp);
rep(i,1,n){
while(pc>1&&dcmp((conv[pc]-conv[pc-1])*(li[i]-conv[pc]))<=0)--pc;
conv[++pc]=li[i];
}
int tmp=pc;
repdo(i,n-1,1){
while(pc>tmp&&dcmp((conv[pc]-conv[pc-1])*(li[i]-conv[pc]))<=0)--pc;
conv[++pc]=li[i];
}
if(n>1)--pc;
}
半平面交
增量法, 时间复杂度 (O(n log n)) (排序的复杂度).
需要保证不是开放的半平面. 否则加上四个 (pm infty) 的平面即可.
细节较多. 详见代码...
const int psz=550;
const db eps=1e-9;
int n,m;
db dcmp(db v){return fabs(v)<=eps?0:(v>0?1:-1);}
db p2(db v){return v*v;}
struct tvec{db x,y;};
tvec operator+(tvec a,tvec b){return (tvec){a.x+b.x,a.y+b.y};}
tvec operator-(tvec a,tvec b){return (tvec){a.x-b.x,a.y-b.y};}
tvec operator*(tvec a,db b){return (tvec){a.x*b,a.y*b};}
tvec operator*(db a,tvec b){return b*a;}
db operator*(tvec a,tvec b){return a.x*b.y-a.y*b.x;}
db operator^(tvec a,tvec b){return a.x*b.x+a.y*b.y;}
db len(tvec a){return sqrt(p2(a.x)+p2(a.y));}
struct tl{
tvec p,v;
db d;
tl(){}
tl(tvec a,tvec b):p(a),v(b-a){d=atan2(v.y,v.x);}
}li[psz];
int pl=0;
bool operator<(tl a,tl b){return a.d<b.d;}
bool isleft(tl a,tvec b){return dcmp(a.v*(b-a.p))>0;}
il tvec inters(tl a,tl b){db v=(b.v*(a.p-b.p))/(a.v*b.v);return a.p+a.v*v;}
tvec poly[psz];
int ppo=0;
tl que[psz]; //queue
tvec qp[psz]; //intersect points
int qh=1,qt=0;
int hplane(){//0 fail, 1 success
sort(li+1,li+pl+1);
int pl1=1;//suppose that pl>=1
rep(i,2,pl){
if(li[i].d>li[pl1].d)li[++pl1]=li[i];
else if(isleft(li[pl1],li[i].p))li[pl1]=li[i];
}
pl=pl1;
qh=1,qt=0;
rep(i,1,pl){
while(qh<qt&&!isleft(li[i],qp[qt-1]))--qt;
while(qh<qt&&!isleft(li[i],qp[qh]))++qh;
que[++qt]=li[i];
if(qh<qt)qp[qt-1]=inters(que[qt-1],que[qt]);
}
while(qh<qt&&!isleft(que[qh],qp[qt-1]))--qt; //**
ppo=0;
if(qt-qh<=1)return 0; //no sol
qp[qt]=inters(que[qh],que[qt]);
rep(i,qh,qt)poly[++ppo]=qp[i];
return 1;
}
旋转卡壳
这是一种拥有 (4) 个多音字, (2^4 = 16) 种读音的优秀算法.