计算几何
二维几何：
点与向量
const double eps=1e-10;
const double PI=acos(-1.0);
struct Point{
double x,y;
Point(double x=0,double y=0):x(x),y(y){}
};
typedef Point Vector;
Vector operator -(Point a,Point b){
return Vector(a.x-b.x,a.y-b.y);
}
Vector operator +(Point a,Point b){
return Vector(a.x+b.x,a.y+b.y);
}
Vector operator *(Vector a,double p){
return Vector(a.x*p,a.y*p);
}
Vector operator /(Vector a,double p){
return Vector(a.x/p,a.y/p);
}
bool operator <(const Point& a,const Point& b){
return a.x<b.x||(a.x==b.x&&a.y<b.y);//在有精度需求，比如使用lower_bound的时候，加上dcmp()
}
int dcmp(double x){
if(fabs(x)<eps)return 0;
if(x<0)return -1;
return 1;
}
bool operator ==(const Point& a,const Point& b){
return dcmp(a.x-b.x)==0&&dcmp(a.y-b.y)==0;
}
double Dot(Vector a,Vector b){
return a.x*b.x+a.y*b.y;
}//点积
double Cross(Vector a,Vector b){
return a.x*b.y-a.y*b.x;
}//叉积
double Length(Vector a){
return sqrt(Dot(a,a));
}//长度
//返回逆时针旋转90度的单位法向量；
Vector Normal(Vector a){
double l=Length(a);
return Vector(-a.y/l,a.x/l);
}
//返回向量夹角，无方向
double Angle(Vector a,Vector b){
return acos(Dot(a,b)/Length(a)/Length(b));
}
//逆时针旋转向量
Vector Rotate(Vector a,double rad){
return Vector(a.x*cos(rad)-a.y*sin(rad),a.x*sin(rad)+a.y*cos(rad));
}
//求p+v*t与q+w*t的交点，使用时确保Cross(v,w)不等于0
Point GetlineIntersection(Point p,Vector v,Point q,Vector w){
Vector u=p-q;
double t=Cross(w,u)/Cross(v,w);
return p+v*t;
}
//求p到直线ab的距离
double DistanceToline(Point p,Point a,Point b){
Vector v1=p-a,v2=b-a;
return fabs(Cross(v1,v2)/Length(v2));
}
//求p到线段ab的距离
double DistanceToSegment(Point p,Point a,Point b){
if(a==b)return Length(p-a);
Vector v1=b-a,v2=p-a,v3=p-b;
if(dcmp(Dot(v1,v2)<0))return Length(p-a);
else if(dcmp(Dot(v1,v3))>0)return Length(p-b);
else return fabs(Cross(v1,v2)/Length(v1));
}
//线段a1a2与线段b1b2规范相交返回真
bool SegmenProperIntersection(Point a1,Point a2,Point b1,Point b2){
double c1=Cross(a2-a1,b1-a1),c2=Cross(a2-a1,b2-a1);
double c3=Cross(b2-b1,a1-b1),c4=Cross(b2-b1,a2-b1);
return dcmp(c1)*dcmp(c2)<0&&dcmp(c3)*dcmp(c4)<0;
}
//点p在线段a1a2上返回真
bool OnSegment(Point p,Point a1,Point a2){
return dcmp(Cross(a1-p,a2-p))==0&&dcmp(Dot(a1-p,a2-p))<0;
}
//点p在ab上的投影
Point GetLineProjection(Point P,Point A,Point B)
{
Vector v=B-A;
return A+v*(Dot(v,P-A)/Dot(v,v));
}
//与 x 轴的夹角，取值范围为 (-π，π]
double angle(Vector v){
return atan2(v.y,v.x);
}
//求线段a1,a2到线段b1,b2的最短距离
double disSegmenttoSegment(Point a1,Point a2,Point b1,Point b2)
{
double ans=DistanceToSegment(a1,b1,b2);
ans=min(ans,DistanceToSegment(a2,b1,b2));
ans=min(ans,DistanceToSegment(b1,a1,a2));
ans=min(ans,DistanceToSegment(b2,a1,a2));
return ans;
}
线
struct Line{
Point p;//点
Vector v;//向量
double ang;
Line(){}
Line(Point p,Vector v):p(p),v(v){ang=atan2(v.y,v.x);}
Point point(double t){return p+v*t;}
bool operator <(const Line& L)const{
return ang<L.ang;
}
};
//点在直线的左边
bool OnLeft(Line l,Point p)
{
return Cross(l.v,p-l.p)>0;
}
//两直线交点
Point GetIntersection(Line a,Line b)
{
Vector u=a.p-b.p;
double t=Cross(b.v,u)/Cross(a.v,b.v);
return a.p+a.v*t;
}
多边形
typedef vector<Point> Polygon;
//多边形的有向面积，逆时针为正
double PolygonArea(Polygon po) {
int n = po.size();
double area = 0.0;
for(int i = 1; i < n-1; i++) {
area += Cross(po[i]-po[0], po[i+1]-po[0]);
}
return area * 0.5;
}
//点是否在多边形内
int isPointInPolygon(Point p,Point poly[],int n)
{
int wn=0;
for(int i=0;i<n;i++){
if(poly[i]==p||poly[(i+1)%n]==p||OnSegment(p,poly[i],poly[(i+1)%n]))return -1;
int k=dcmp(Cross(poly[(i+1)%n]-poly[i],p-poly[i]));
int d1=dcmp(poly[i].y-p.y);
int d2=dcmp(poly[(i+1)%n].y-p.y);
if(k>0&&d1<=0&&d2>0)
wn++;
if(k<0&&d2<=0&&d1>0)
wn--;
}
if(wn!=0)return 1;//内部
return 0;//外部
}
//求凸包，ch为返回凸包，m为凸包内点的数目，<=不允许点在边上，
int ConvecHull(Point* p,int n,Point* ch)
{
sort(p,p+n);
int m=0;
for(int i=0;i<n;i++){
while(m>1&&dcmp(Cross(ch[m-1]-ch[m-2],p[i]-ch[m-2])<=0))m--;//注意<=与<的区别
ch[m++]=p[i];
}
int k=m;
for(int i=n-2;i>=0;i--){
while(m>k&&dcmp(Cross(ch[m-1]-ch[m-2],p[i]-ch[m-2])<=0))m--;//注意<=与<的区别
ch[m++]=p[i];
}
if(n>1)m--;
return m;
}
圆
struct Circle{
Point c;
double r;
Circle(){};
Circle(Point c,double r=0):c(c),r(r){}
Point point(double a){
return Point(c.x+cos(a)*r,c.y+sin(a)*r);
}
};
//求圆与直线的交点,t1,t2为(at+b)^2+(ct+d)^2=r^2的解，交点放入sol
int getLineCircleIntersection(Line L,Circle C,double& t1,double& t2,vector<Point>& sol)
{
double a=L.v.x,b=L.p.x-C.c.x,c=L.v.y,d=L.p.y-C.c.y;
double e=a*a+c*c,f=2*(a*b+c*d),g=b*b+d*d-C.r*C.r;
double delta=f*f-4*e*g;
if(dcmp(delta)<0)return 0;
if(dcmp(delta)==0){
t1=t2=-f/(2*e);
sol.push_back(L.point(t1));
return 1;
}
t1=(-f-sqrt(delta))/(2*e);
sol.push_back(L.point(t1));
t2=(-f+sqrt(delta))/(2*e);
sol.push_back(L.point(t2));
return 2;
}
//求圆与圆的交点，交点放入sol
int getCircleCircleIntersection(Circle C1,Circle C2,vector<Point>& sol)
{
double d=Length(C1.c-C2.c);
if(dcmp(d)==0){
if(dcmp(C1.r-C2.r)==0)return -1;//两圆重合
return 0;
}
if(dcmp(C1.r+C2.r-d)<0)return 0;
if(dcmp(fabs(C1.r-C2.r)-d)>0)return 0;
double a=angle(C2.c-C1.c);//直线c1c2
double da=acos((C1.r*C1.r+d*d-C2.r*C2.r)/(2*C1.r*d));//c1c2到c1p1的角
Point p1=C1.point(a-da),p2=C1.point(a+da);
sol.push_back(p1);
if(p1==p2)return 1;
sol.push_back(p2);
return 2;
}
//求直线与圆的切线
int getTangents(Point p,Circle C,Vector* v){
Vector u=C.c-p;
double dist=Length(u);
if(dcmp(dist-C.r)<0)return 0;
else if(dcmp(dist-C.r)==0){
v[0]=Rotate(u,PI/2);
return 1;
}
else {
double ang=asin(C.r/dist);
v[0]=Rotate(u,-ang);
v[1]=Rotate(u,+ang);
return 2;
}
}
半平面交
int HalfplaneIntersection(Line* L,int n,Point* poly)
{
sort(L,L+n);
int first,last;
Point *p=new Point[n];//会在函数内开大容量数组，请在使用时注意开成全局！！！！！
Line *q=new Line[n];//会在函数内开大容量数组，请在使用时注意开成全局！！！！！
q[first=last=0]=L[0];
for(int i=1;i<n;i++){
while(first<last&&!OnLeft(L[i],p[last-1]))last--;
while(first<last&&!OnLeft(L[i],p[first]))first++;
q[++last]=L[i];
if(fabs(Cross(q[last].v,q[last-1].v))<eps){
last--;
if(OnLeft(q[last],L[i].p))q[last]=L[i];
}
if(first<last)p[last-1]=GetIntersection(q[last-1],q[last]);
}
while(first<last&&!OnLeft(q[first],p[last-1]))last--;
if(last-first<=1)return 0;
p[last]=GetIntersection(q[last],q[first]);
int m=0;
for(int i=first;i<=last;i++)poly[m++]=p[i];
return m;
}//会在函数内开大容量数组，请在使用时注意开成全局
平面直线图（PSGL）
struct Edge
{
int from, to; // 起点，终点，左边的面编号
double ang;
Edge(int f,int t,double a):from(f),to(t),ang(a) {}
};
// 平面直线图（PSGL）实现
struct PSLG {
int n, m, face_cnt;
double x[maxn], y[maxn];
vector<Edge> edges;
vector<int> G[maxn];
int vis[maxn*2]; // 每条边是否已经访问过
int left[maxn*2]; // 左面的编号(该边在哪个面内)
int prev[maxn*2]; // 相同起点的上一条边（即顺时针旋转碰到的下一条边）的编号
vector<Polygon> faces;
double area[maxn]; // 每个polygon的面积
void init(int n) {
this->n = n;
for(int i = 0; i < n; i++) G[i].clear();
edges.clear();
faces.clear();
}
// 有向线段from->to的极角
double getAngle(int from, int to) {
return atan2(y[to]-y[from], x[to]-x[from]);
}
void AddEdge(int from, int to) {
edges.push_back((Edge){from, to, getAngle(from, to)});
edges.push_back((Edge){to, from, getAngle(to, from)});
m = edges.size();
G[from].push_back(m-2);
G[to].push_back(m-1);
}
// 找出faces并计算面积
void Build() {
for(int u = 0; u < n; u++) {
// 给从u出发的各条边按极角排序
int d = G[u].size();
for(int i = 0; i < d; i++)
for(int j = i+1; j < d; j++) // 这里偷个懒，假设从每个点出发的线段不会太多
if(edges[G[u][i]].ang > edges[G[u][j]].ang) swap(G[u][i], G[u][j]);
for(int i = 0; i < d; i++)
prev[G[u][(i+1)%d]] = G[u][i]; //u点出发的第i条边顺时针转的第一条边是prev[i]
}
memset(vis, 0, sizeof(vis));
face_cnt = 0;
for(int u = 0; u < n; u++)
for(int i = 0; i < G[u].size(); i++) {
int e = G[u][i]; //逆时针转的第i条边
if(!vis[e]) { // 逆时针找圈
face_cnt++;
Polygon poly;
for(;;) {
vis[e] = 1; left[e] = face_cnt;
int from = edges[e].from;
poly.push_back(Point(x[from], y[from]));
//cout<<x[from]<<" "<<y[from]<<" ";
e = prev[e^1]; //反向边顺时针第一条
if(e == G[u][i]) break; //回到原点
assert(vis[e] == 0);
}
//cout<<endl;
faces.push_back(poly);
}
}
for(int i = 0; i < faces.size(); i++) {
area[i] = PolygonArea(faces[i]);
}
}
};
旋转卡壳
//向量(b-a)与向量（c-a)的叉积，相当于三角形abc的有向面积的2倍
double cross(Point a,Point b,Point c)
{
return Cross(b-a,c-a);
}
//求凸包内最远点
long long getmaxdistance(Point a[],int n)
{
int j=2;
double ans=0;
a[n]=a[0];//方便写下一个点，避免取模
for(int i=0;i<n;i++){
while(fabs(cross(a[i],a[i+1],a[j+1]))>fabs(cross(a[i],a[i+1],a[j])))j=(j+1)%n;
//通过比较面积大小，比较到直线的距离
ans=max(ans,Length2(a[j]-a[i]));
}
return ans;
}
//求2个凸包间的最短距离
double getmindistance(Point p1[],Point p2[],int n1,int n2)
{
int i=0,j=0;
for(int k=0;k<n1;k++){
if(p1[k].y<p1[i].y)i=k;//找出p1中的y最小值的点
}
for(int k=0;k<n2;k++){
if(p2[k].y>p2[j].y)j=k;//找出p2中的y最大值的点
}
p1[n1]=p1[0];
p2[n2]=p2[0];
double ans=99999999999;
for(int k=0;k<n1;k++){
//循环n1次，相当于求p1中每一条边与p2的最近距离
while((cross(p1[i],p1[i+1],p2[j+1])-cross(p1[i],p1[i+1],p2[j]))>eps)
j=(j+1)%n2;
ans=min(ans,disSegmenttoSegment(p1[i],p1[i+1],p2[j],p2[j+1]));//求线段间的最短距离
i=(i+1)%n1;
}
return ans;
}
//求凸包的内4个点组成的最大四边形面积
double solve(Point a[],int n)
{
a[n]=a[0];
int p1,p2;
double ans=0;
for(int i=0;i<n;i++){
p1=(i+0)%n;
p2=(i+1)%n;
for(int j=i+1;j<n;j++){
while(cross(a[i],a[j],a[p1+1])<cross(a[i],a[j],a[p1]))p1=(p1+1)%n;
while(cross(a[i],a[j],a[p2+1])>cross(a[i],a[j],a[p2]))p2=(p2+1)%n;
ans=max(ans,cross(a[i],a[j],a[p2])-cross(a[i],a[j],a[p1]));
}
}
return ans;
}
三维几何
基础点面
struct Point3{
double x,y,z;
Point3(double x=0,double y=0,double z=0):x(x),y(y),z(z){}
};
typedef Point3 Vector3;
Vector3 operator +(Vector3 A,Vector3 B){
return Vector3(A.x+B.x,A.y+B.y,A.z+B.z);
}
Vector3 operator -(Vector3 A,Vector3 B){
return Vector3(A.x-B.x,A.y-B.y,A.z-B.z);
}
Vector3 operator *(Vector3 A,double p){
return Vector3(A.x*p,A.y*p,A.z*p);
}
Vector3 operator /(Vector3 A,double p){
return Vector3(A.x/p,A.y/p,A.z/p);
}
int dcmp(double a){
if(fabs(a)<eps)return 0;
else if(a>0)return 1;
return -1;
}
bool operator ==(Vector3 a,Vector3 b){
return dcmp(a.x-b.x)==0&&dcmp(a.y-b.y)==0&&dcmp(a.z-b.z)==0;
}
double Dot(Vector3 A,Vector3 B){
return A.x*B.x+A.y*B.y+A.z*B.z;
}
double Length(Vector3 A){
return sqrt(Dot(A,A));
}
double Angle(Vector3 A,Vector3 B){
return acos(Dot(A,B/Length(A)/Length(B)));
}
//点p到平面p0-n的距离。n必须为单位向量
double DistancetoPlane(const Point3& p,const Point3& p0,const Vector3& n){
return fabs(Dot(p-p0,n));
}
//点p在平面p0-n上的投影。n必须为单位向量
Point3 GetPlaneProjection(const Point3& p,const Point3& p0,const Vector3& n){
return p-n*Dot(p-p0,n);
}
//直线p1-p2到平面p0-n的交点。假设交点唯一存在
Point3 LinePlaneIntersection(Point3 p1,Point3 p2,Point3 p0,Vector3 n){
Vector3 v=p2-p1;
double t=(Dot(n,p0-p1))/Dot(n,p2-p1);
return p1+v*t;
}
Vector3 Cross(Vector3 A,Vector3 B){
return Vector3(A.y*B.z-A.z*B.y,A.z*B.x-A.x*B.z,A.x*B.y-A.y*B.x);
}
double Area2(Point3 A,Point3 B,Point3 C){
return Length(Cross(B-A,C-A));
}
//点p在△p0p1p2中
bool PointInTri(Point3 p,Point3 p0,Point3 p1,Point3 p2){
double area1=Area2(p,p0,p1);
double area2=Area2(p,p1,p2);
double area3=Area2(p,p2,p0);
return dcmp(area1+area2+area3-Area2(p0,p1,p2))==0;
}
//△p0p1p2是否和线段ab相交
bool TriSegIntersection(Point3 p0,Point3 p1,Point3 p2,Point3 a,Point3 b,Point3& p)
{
Vector3 n=Cross(p1-p0,p2-p0);
if(dcmp(Dot(n,b-a))==0)return false;
else {
double t=Dot(n,p0-a)/Dot(n,b-a);
if(dcmp(t)<0||dcmp(t-1)>0)return false;
p=a+(b-a)*t;
return PointInTri(p,p0,p1,p2);
}
}
//点p到直线ab的距离
double DistancetoLine(Point3 p,Point3 a,Point3 b)
{
Vector3 v1=b-a,v2=p-a;
return Length(Cross(v1,v2)/Length(v1));
}
//点p到线段ab的距离
double DistanceToSegment(Point3 p,Point3 a,Point3 b){
if(a==b)return Length(p-a);
Vector3 v1=b-a,v2=p-a,v3=p-b;
if(dcmp(Dot(v1,v2)<0))return Length(v2);
else if(dcmp(Dot(v1,v3))>0)return Length(v3);
else return Length(Cross(v1,v2))/Length(v1);
}
//返回ab，ac，ad的混合积。它也等于四面体的有向体积的6倍
double Volume6(Point3 a,Point3 b,Point3 c,Point3 d)
{
return Dot(d-a,Cross(b-a,c-a));
}
凸包
struct Face{
int v[3];
Vector3 normal(Point3 *p)const{
return Cross(p[v[1]]-p[v[0]],p[v[2]]-p[v[0]]);
}
int cansee(Point3 *p,int i)const{
return Dot(p[i]-p[v[0]],normal(p))>0?1:0;
}
};
const int N=1e3;
int vis[N][N];
//三维凸包，记得使用前扰动点，以避免特殊情况
vector<Face> CH3D(Point3 *p,int n)
{
vector<Face> cur;
//记得提前扰动
cur.push_back((Face){{0,1,2}});
cur.push_back((Face){{2,1,0}});
for(int i=3;i<n;i++){
vector<Face> next;
for(int j=0;j<cur.size();j++){
Face& f=cur[j];
int res=f.cansee(p,i);
if(!res)next.push_back(f);
for(int k=0;k<3;k++)vis[f.v[k]][f.v[(k+1)%3]]=res;
}
for(int j=0;j<cur.size();j++){
for(int k=0;k<3;k++){
int a=cur[j].v[k],b=cur[j].v[(k+1)%3];
if(vis[a][b]!=vis[b][a]&&vis[a][b]){
next.push_back((Face){{a,b,i}});
}
}
}
cur=next;
}
return cur;
}
double rand01(){
return rand()/(double)RAND_MAX;
}
double randeps(){
return (rand01()-0.5)*eps;
}
Point3 add_noise(Point3 p)
{
return Point3(p.x+randeps(),p.y+randeps(),p.z+randeps());
}