[模板] 计算几何2: 自适应Simpson/凸包/半平面交/旋转卡壳/闵可夫斯基和

//to update

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

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.

原文地址：https://www.cnblogs.com/ubospica/p/10828219.html