计算几何模板
二维凸包
const int maxn=1e5+100;
const double eps=1e-8;
int sgn(double x){///判断x是否等于0
if(fabs(x)<eps) return 0;
if(x<0) return -1;
else return 1;
}
struct point{
double x,y;
point(){}
point(double x,double y):x(x),y(y){}
point operator+(point b){
return point(x+b.x,y+b.y);
}
point operator-(point b){
return point(x-b.x,y-b.y);
}
bool operator == (point b){
return sgn(x-b.x)==0&&sgn(y-b.y)==0;
}
bool operator<(point b)const{
if(sgn(x-b.x)==0) return sgn(y-b.y)<0;
return sgn(x-b.x)<0;
}
};
point s[maxn],g[maxn],h[maxn];
double dot(point a,point b){return a.x*b.x+a.y*b.y;}
double cross(point a,point b){return a.x*b.y-a.y*b.x;}
struct line{
point p1,p2;
line(){}
line(point p1,point p2):p1(p1),p2(p2){}
};
double mult(point a,point b,point o){
///计算叉乘ao和bo
return (a.x-o.x)*(b.y-o.y)>=(b.x-o.x)*(a.y-o.y);
}
int Graham(int n){
int idx=1;
sort(s,s+n);
if(!n) return 0;
h[0]=s[0];
if(n==1) return 0;
h[1]=s[1];
if(n==2) return 0;
h[2]=s[2];
///求凸包的上半部分
for(int i=2;i<n;i++){
while(idx&&(mult(s[i],h[idx],h[idx-1]))) idx--;
h[++idx]=s[i];
}
int tmp=idx;
h[++idx]=s[n-2];
for(int i=n-3;i>=0;i--){
while(idx!=tmp&&(mult(s[i],h[idx],h[idx-1]))) idx--;
h[++idx]=s[i];
}
return idx;
}
struct polygon{
int n;point p[1010];line v[1010];
};
///点和线段的关系 0不在1在
bool point_on_seg(point p,line v){
return sgn(cross(p-v.p1,v.p2-v.p1))==0&&sgn(dot(p-v.p1,v.p2-v.p1))==0;
}
///判断点和任意多边形的关系:3在点上 2在边上 1在内部 0在外部
int Point_in_polygon(point pt,point* p,int n){
int i;
for(i=0;i<n;i++)
if(p[i]==pt) return 3;
for(i=0;i<n;i++){
line v=line(p[i],p[(i+1)%n]);
if(point_on_seg(pt,v)) return 2;
}
int num=0;
for(int i=0;i<n;i++){
int j=(i+1)%n;
int c=sgn(cross(pt-p[j],p[i]-p[j]));
int u=sgn(p[i].y-pt.y);
int v=sgn(p[j].y-pt.y);
if(c>0&&u<0&&v>=0) num++;
if(c<0&&u>=0&&v<0) num--;
}
return num!=0;
}
double culdis(point a,point b){
return sqrt((a.x-b.x)*(a.x-b.x)+(a.y-b.y)*(a.y-b.y));
}
//说明 返回值为凸包点的个数,h数组存储的是凸包中的点。
三维凸包
const int MAXN=505;
const double EPS=1e-8;
struct Point{
double x,y,z;
Point(){}
Point(double xx,double yy,double zz):x(xx),y(yy),z(zz){}
Point operator -(const Point p1){ //两向量之差
return Point(x-p1.x,y-p1.y,z-p1.z);
}
Point operator *(Point p){ //叉乘
return Point(y*p.z-z*p.y,z*p.x-x*p.z,x*p.y-y*p.x);
}
double operator ^(Point p){ //点乘
return (x*p.x+y*p.y+z*p.z);
}
};
struct CH3D{
struct face{
int a,b,c; //表示凸包一个面上三个点的编号
bool ok; //表示该面是否属于最终凸包中的面
};
int n; //初始顶点数
Point P[MAXN]; //初始顶点
int num; //凸包表面的三角形数
face F[8*MAXN];
int g[MAXN][MAXN]; //凸包表面的三角形
double vlen(Point a){ //向量长度
return sqrt(a.x*a.x+a.y*a.y+a.z*a.z);
}
Point cross(const Point &a, const Point &b, const Point &c){ //叉乘
return Point((b.y-a.y)*(c.z-a.z)-(b.z-a.z)*(c.y-a.y),-((b.x-a.x)*(c.z-a.z)
-(b.z-a.z)*(c.x-a.x)),(b.x-a.x)*(c.y-a.y)-(b.y-a.y)*(c.x-a.x));
}
double area(Point a,Point b,Point c){ //三角形面积*2
return vlen((b-a)*(c-a));
}
double volume(Point a,Point b,Point c,Point d){ //四面体有向体积*6
return (b-a)*(c-a)^(d-a);
}
double dblcmp(Point &p,face &f){ //正:点在面同向
Point m=P[f.b]-P[f.a];
Point n=P[f.c]-P[f.a];
Point t=p-P[f.a];
return (m*n)^t;
}
void deal(int p,int a,int b){
int f=g[a][b];
face add;
if(F[f].ok){
if(dblcmp(P[p],F[f])>EPS)
dfs(p,f);
else{
add.a=b;
add.b=a;
add.c=p;
add.ok=1;
g[p][b]=g[a][p]=g[b][a]=num;
F[num++]=add;
}
}
}
void dfs(int p,int now){
F[now].ok=0;
deal(p,F[now].b,F[now].a);
deal(p,F[now].c,F[now].b);
deal(p,F[now].a,F[now].c);
}
bool same(int s,int t){
Point &a=P[F[s].a];
Point &b=P[F[s].b];
Point &c=P[F[s].c];
return fabs(volume(a,b,c,P[F[t].a]))<EPS && fabs(volume(a,b,c,P[F[t].b]))<EPS
&& fabs(volume(a,b,c,P[F[t].c]))<EPS;
}
void solve(){ //构建三维凸包
int i,j,tmp;
face add;
bool flag=true;
num=0;
if(n<4)
return;
for(i=1;i<n;i++){ //此段是为了保证前四个点不共面,若以保证,则可去掉
if(vlen(P[0]-P[i])>EPS){
swap(P[1],P[i]);
flag=false;
break;
}
}
if(flag)
return;
flag=true;
for(i=2;i<n;i++){ //使前三点不共线
if(vlen((P[0]-P[1])*(P[1]-P[i]))>EPS){
swap(P[2],P[i]);
flag=false;
break;
}
}
if(flag)
return;
flag=true;
for(i=3;i<n;i++){ //使前四点不共面
if(fabs((P[0]-P[1])*(P[1]-P[2])^(P[0]-P[i]))>EPS){
swap(P[3],P[i]);
flag=false;
break;
}
}
if(flag)
return;
for(i=0;i<4;i++){
add.a=(i+1)%4;
add.b=(i+2)%4;
add.c=(i+3)%4;
add.ok=true;
if(dblcmp(P[i],add)>0)
swap(add.b,add.c);
g[add.a][add.b]=g[add.b][add.c]=g[add.c][add.a]=num;
F[num++]=add;
}
for(i=4;i<n;i++){
for(j=0;j<num;j++){
if(F[j].ok && dblcmp(P[i],F[j])>EPS){
dfs(i,j);
break;
}
}
}
tmp=num;
for(i=num=0;i<tmp;i++)
if(F[i].ok){
F[num++]=F[i];
}
}
double area(){ //表面积
double res=0.0;
if(n==3){
Point p=cross(P[0],P[1],P[2]);
res=vlen(p)/2.0;
return res;
}
for(int i=0;i<num;i++)
res+=area(P[F[i].a],P[F[i].b],P[F[i].c]);
return res/2.0;
}
double volume(){ //体积
double res=0.0;
Point tmp(0,0,0);
for(int i=0;i<num;i++)
res+=volume(tmp,P[F[i].a],P[F[i].b],P[F[i].c]);
return fabs(res/6.0);
}
int triangle(){ //表面三角形个数
return num;
}
int polygon(){ //表面多边形个数
int i,j,res,flag;
for(i=res=0;i<num;i++){
flag=1;
for(j=0;j<i;j++)
if(same(i,j)){
flag=0;
break;
}
res+=flag;
}
return res;
}
Point getcent(){ //求凸包质心
Point ans(0,0,0),temp=P[F[0].a];
double v = 0.0,t2;
for(int i=0;i<num;i++){
if(F[i].ok == true){
Point p1=P[F[i].a],p2=P[F[i].b],p3=P[F[i].c];
t2 = volume(temp,p1,p2,p3)/6.0; //体积大于0,也就是说,点 temp 不在这个面上
if(t2>0){
ans.x += (p1.x+p2.x+p3.x+temp.x)*t2;
ans.y += (p1.y+p2.y+p3.y+temp.y)*t2;
ans.z += (p1.z+p2.z+p3.z+temp.z)*t2;
v += t2;
}
}
}
ans.x /= (4*v); ans.y /= (4*v); ans.z /= (4*v);
return ans;
}
double function(Point fuck){ //点到凸包上的最近距离(枚举每个面到这个点的距离)
double min=99999999;
for(int i=0;i<num;i++){
if(F[i].ok==true){
Point p1=P[F[i].a] , p2=P[F[i].b] , p3=P[F[i].c];
double a = ( (p2.y-p1.y)*(p3.z-p1.z)-(p2.z-p1.z)*(p3.y-p1.y) );
double b = ( (p2.z-p1.z)*(p3.x-p1.x)-(p2.x-p1.x)*(p3.z-p1.z) );
double c = ( (p2.x-p1.x)*(p3.y-p1.y)-(p2.y-p1.y)*(p3.x-p1.x) );
double d = ( 0-(a*p1.x+b*p1.y+c*p1.z) );
double temp = fabs(a*fuck.x+b*fuck.y+c*fuck.z+d)/sqrt(a*a+b*b+c*c);
if(temp<min)min = temp;
}
}
return min;
}
};
CH3D hull;
/*
* 用法:
* input: n: 点数
* 下标从0开始
* p[i].x p[i].y p[i].z: 坐标
* solve(): 构建三维凸包
* return : double area(): 凸包表面积
* double volume(): 体积
* int polygon(): 表面三角形数
* int polygon(): 表面多边形数
* Point getcent(): 凸包质心
* double function(Point fuck): 点到凸包上的距离
*
*/
旋转卡壳
模板题
跟凸包的板子不大一样。
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const int maxn=50000+100;
const double eps=1e-8;
int sgn(ll x){///判断x是否等于0
if(fabs(x)<eps) return 0;
if(x<0) return -1;
else return 1;
}
struct point{
ll x,y;
point(){}
point(ll x,ll y):x(x),y(y){}
point operator+(point b){
return point(x+b.x,y+b.y);
}
point operator-(point b){
return point(x-b.x,y-b.y);
}
bool operator == (point b){
return sgn(x-b.x)==0&&sgn(y-b.y)==0;
}
};
point s[maxn],g[maxn],h[maxn],p;
bool cmp(point a,point b){
double A=atan2(a.y-p.y,a.x-p.x);
double B=atan2(b.y-p.y,b.x-p.x);
if(A!=B) return A<B;
else return a.x<b.x;
}
int n;
ll dot(point a,point b){return a.x*b.x+a.y*b.y;}
ll cross(point a,point b){return a.x*b.y-a.y*b.x;}
ll compare(point a,point b,point c){
return cross({b.x-a.x,b.y-a.y},{c.x-a.x,c.y-a.y});
}
ll cul(point a,point b){
return ((a.x-b.x)*(a.x-b.x)+(a.y-b.y)*(a.y-b.y));
}
int idx=1;
int Graham(int n){
p={1000000000,1000000000};
int k=0;
for(int i=0;i<n;i++){
if(p.y>s[i].y||(p.y==s[i].y&&p.x>s[i].x)){
p=s[i],k=i;
}
}
swap(s[k],s[0]);
sort(&s[1],&s[n],cmp);
h[0]=s[0];
h[1]=s[1];
idx=1;
for(int i=2;i<n;){
if(idx&&(compare(h[idx-1],s[i],h[idx])>=0)) idx--;
else h[++idx]=s[i++];
}
return idx;
}
//说明 返回值为凸包点的个数,h数组存储的是凸包中的点。
ll getmax(){
ll res=0;
if(idx==1) return cul(h[0],h[1]);
h[++idx]=h[0];
int j=2;
for(int i=0;i<idx;i++){
while(compare(h[i],h[i+1],h[j])<compare(h[i],h[i+1],h[j+1]))
j=(j+1)%idx;
res=max(res,max(cul(h[i],h[j]),cul(h[i+1],h[j])));
}
return res;
}
int main(){
int n;cin>>n;
for(int i=0;i<n;i++)
cin>>s[i].x>>s[i].y;
int t=Graham(n);
cout<<getmax()<<endl;
return 0;
}