• /************************************** * Au: Hany01 * Date: Feb 8th, 2018 * Prob: Geometry Template ...**************************************/ #include&amp;lt;bits/stdc++.h&...
/**************************************
* Au: Hany01
* Date: Feb 8th, 2018
* Prob: Geometry Template
* Email: hany01@foxmail.com
**************************************/

#include<bits/stdc++.h>

using namespace std;

typedef long long LL;
typedef pair<int, int> PII;
#define rep(i, j) for (register int i = 0, i##_end_ = (j); i < i##_end_; ++ i)
#define For(i, j, k) for (register int i = (j), i##_end_ = (k); i <= i##_end_; ++ i)
#define Fordown(i, j, k) for (register int i = (j), i##_end_ = (k); i >= i##_end_; -- i)
#define Set(a, b) memset(a, b, sizeof(a))
#define Cpy(a, b) memcpy(a, b, sizeof(a))
#define fir first
#define sec second
#define pb(a) push_back(a)
#define mp(a, b) make_pair(a, b)
#define ALL(a) (a).begin(), (a).end()
#define SZ(a) ((int)(a).size())
#define INF (0x3f3f3f3f)
#define INF1 (2139062143)
#define Mod (1000000007)
#define debug(...) fprintf(stderr, __VA_ARGS__)

template <typename T> inline bool chkmax(T &a, T b) { return a < b ? a = b, 1 : 0; }
template <typename T> inline bool chkmin(T &a, T b) { return b < a ? a = b, 1 : 0; }

{
register int _, __; register char c_;
for (_ = 0, __ = 1, c_ = getchar(); c_ < '0' || c_ > '9'; c_ = getchar()) if (c_ == '-') __ = -1;
for ( ; c_ >= '0' && c_ <= '9'; c_ = getchar()) _ = (_ << 1) + (_ << 3) + (c_ ^ 48);
return _ * __;
}

const double eps = 1e-10, Pi = acos(-1.0);

struct Point
{
double x, y;
Point(double x = 0, double y = 0): x(x), y(y) {}
};
typedef Point Vector;

struct Circle
{
Point O; double r;
Circle(Point O, double r): O(O), r(r) {}
Point point(double a) { return Point(O.x + cos(a) * r, O.y + sin(a) * r); }
};

int dcmp(double x) { if (fabs(x) < eps) return 0; else return x < 0 ? -1 : 1; }
Vector operator + (Vector A, Vector B) { return Vector(A.x + B.x, A.y + B.y); }
Vector operator - (Vector A, Vector 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); }
bool operator == (const Point& a, const Point& b) { return !dcmp(a.x - b.x) && !dcmp(a.y - b.y); }
double Dot(Vector A, Vector B) { return A.x * B.x + A.y * B.y; }
double Length(Vector A) { return sqrt(Dot(A, A)); }
double Angle(Vector A, Vector B) { return acos(Dot(A, B) / Length(A) / Length(B)); }
double Angle(Vector A) { return atan2(A.y, A.x); }
double Cross(Vector A, Vector B) { return A.x * B.y - A.y * B.x; } //B在A的左边为正、右边为负
double Area2(Point A, Point B, Point C) { return Cross(B - A, C - A); }
Vector Normal(Vector A) { double L = Length(A); return Vector(-A.y / L, A.x / L); } //向左旋转90度且长度变成1
Point GetLineProjection(Point P, Point A, Point B) {
Vector v = B - A;
return A + v * (Dot(v, P - A) / Dot(v, v));
}
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;
}
double DistanceToLine(Point P, Point A, Point B) {
Vector v1 = B - A, v2 = P - A;
return fabs(Cross(v1, v2)) / Length(v1);
}
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(v2);
if (dcmp(Dot(v1, v3)) > 0) return Length(v3);
return fabs(Cross(v1, v2)) / Length(v1);
}
int GetLineCircleIntersection(Point A, Point B, Circle C, Point& S1, Point& S2) {
double d = DistanceToLine(C.O, A, B);
register int tmp = dcmp(d - C.r);
if (tmp > 0) return 0;
Point P = GetLineProjection(C.O, A, B);
if (!tmp) { S1 = S2 = P; return 1; }
double L = sqrt(C.r * C.r - d * d);
Vector v = (B - A) / Length(B - A);
S1 = P - v * L, S2 = P + v * L;
return 2;
}
int GetCircleIntersection(Circle C1, Circle C2, Point& P1, Point& P2) {
double d = Length(C1.O - C2.O);
if (!dcmp(d)) { if (!dcmp(C1.r - C2.r)) return -1; return 0; }
if (dcmp(C1.r + C2.r - d) < 0 || dcmp(fabs(C1.r - C2.r) - d) > 0) return 0;
double a = Angle(C2.O - C1.O), da = acos((C1.r * C1.r + d * d - C2.r * C2.r) / (2 * C1.r * d));
P1 = C1.point(a - da), P2 = C1.point(a + da);
if (P1 == P2) return 1;
return 2;
}
int GetTangents(Point P, Circle C, Point& P1, Point& P2) {
double d = Length(C.O - P);
if (dcmp(d - C.r) < 0) return 0;
double aPC = Angle(P - C.O), theta = acos(C.r / d);
P1 = C.point(aPC + theta), P2 = C.point(aPC - theta);
if (P1 == P2) return 1;
return 2;
}
int GetTangents(Circle C1, Circle C2, Point* a, Point* b) {
int cnt = 0;
if (C1.r < C2.r) swap(C1, C2);
double d = Length(C1.O - C2.O), dif = C1.r - C2.r, sum = C1.r + C2.r;
if (d < dif) return 0;//内含
double base = Angle(C2.O - C1.O);
if (!dcmp(d) && !dcmp(C1.r - C2.r)) return -1;//重合
if (!dcmp(d - dif)) {//内切
a[++ cnt] = C1.point(base), b[cnt] = a[cnt];
return 1;
}
//外公切线
double ang = acos((C1.r - C2.r) / d);
a[++ cnt] = C1.point(base + ang), b[cnt] = C2.point(base + ang);
a[++ cnt] = C1.point(base - ang), b[cnt] = C2.point(base - ang);
if (!dcmp(d - sum)) a[++ cnt] = C1.point(base), b[cnt] = a[cnt];//一条内公切线
else if (dcmp(d - sum) > 0) {
ang = acos((C1.r + C2.r) / d);
a[++ cnt] = C1.point(base + ang), b[cnt] = C2.point(Pi + base + ang);
a[++ cnt] = C1.point(base - ang), b[cnt] = C2.point(Pi + base - ang);
}
return cnt;
}
bool IsPointOnSegment(Point P, Point A, Point B) {
if (A.x > B.x) swap(A, B);
if (P.x < A.x || P.x > B.x || dcmp((B.y - A.y) / (B.x - A.x) - (B.y - P.y) / (B.x - P.x))) return 0;
return 1;
}
int IsPointInPolygon(Point P, Point* p, int n) {
int wn = 0;
Point A, B;
For(i, 1, n) {
A = p[i], B = p[i % n + 1];
if (!IsPointOnSegment(P, A, B)) return -1;
int k = dcmp(Cross(B - A, P - A)), d1 = dcmp(A.y - P.y), d2 = dcmp(B.y - P.y);
if (k > 0 && d1 <= 0 && d2 > 0) ++ wn;
if (k < 0 && d2 <= 0 && d1 > 0) -- wn;
}
if (wn) return 1;
return 0;
}

int main()
{
return 0;
}
展开全文
• 2018 world final 金牌 吉老师几何板子 struct point{ db x,y; point operator + (const point &k1;) const{return (point){k1.x+x,k1.y+y};} point operator - (const point &k1;) const{return (point){x-k1.x...
• #include <iostream> #include <stdio.h> #include <string.h> #include <algorithm> #include <queue> #include <map> #include <vector>...set>...


#include <iostream>
#include <stdio.h>
#include <string.h>
#include <algorithm>
#include <queue>
#include <map>
#include <vector>
#include <set>
#include <string>
#include <math.h>
#define ll long long
using namespace std;
const int maxn=1e6+10;
const double pi=acos(-1.0);
const double D_MAX=1e100;
const double D_MIN=-1e100;
const double eps=1e-9;
int sgn(double x){ if(fabs(x) < eps)return 0;  if(x >0) return 1; return -1; }
int dcmp(double x, double y){ if(fabs(x - y) < eps) return 0; if(x > y) return 1;return -1;}
void usehanshu(){double x;}//floor(x)向下取整函数ceil(x)向上取整函数round(x)四舍五入函数
struct Point  { double x,y; Point(double x=0,double y=0) { x=x;y=y; };   };
struct Segment{ Point a,b;  Segment(Point x,Point y)     { a=x;b=y; };   };
struct Line   { Point a,b;  Line(Point x,Point y)        { a=x;b=y; };   };
typedef Point Vector;
Vector operator + (Vector A, Vector B){ return Vector(A.x+B.x, A.y+B.y); } // 向量相加
Vector operator - (Point  A, Point  B){ return Vector(A.x-B.x, A.y-B.y); } // 向量生成
double operator * (Vector A, Vector B){ return A.x*B.x-A.y*B.y;          } // 点积
double operator ^ (Vector A, Vector B){ return A.x*B.y-A.y*B.x;          } // 叉积
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));}           // 向量长度
double Angle(Vector A, Vector B){ return acos(Dot(A, B)/Length(A)/Length(B));}  // 角度
double Area2(Point A, Point B, Point C) { return Cross(B-A, C-A); }  // 四边形面积
Vector Normal(Vector A)            {double L = Length(A);return Vector(-A.y/L, A.x/L);}//向量A左转90°的单位法向量
bool ToLeftTest(Point a, Point b, Point c){return Cross(b - a, c - b) > 0;}
bool xx(Line A,point a){if( sgn( Cross(A.a-a,A.b-a) )==0 ) return 1;return 0;}  // 直线和点
void XX(Line A,Line B)  // 直线和直线的情况
{
Point a=A.a; Point b=A.b;Point c=B.a; Point d=B.b;
double A1=b.y-a.y,B1=-(b.x-a.x),C1=b.y*a.x-b.x*a.y;
double A2=d.y-c.y,B2=-(d.x-c.x),C2=d.y*c.x-d.x*c.y;
double k=A1*B2-A2*B1;
if(fabs(k)<eps)
{
if( fabs( C2*A1-C1*A2)<eps && fabs(B1*C2-C1*B2)<eps ) printf("LINE\n");
else                   printf("NONE\n");
}
else
{
double x=-(B1*C2-C1*B2)*1.000000000/k;
double y=(A1*C2-C1*A2)*1.00000000/k;
printf("POINT %.2f %.2f\n",x,y);
}
}

int main()
{
int T;scanf("%d",&T);
printf("INTERSECTING LINES OUTPUT\n");
while(T--)
{
Point a,b,c,d;
scanf("%lf %lf %lf %lf",&a.x,&a.y,&b.x,&b.y);
scanf("%lf %lf %lf %lf",&c.x,&c.y,&d.x,&d.y);
XX(Line(a,b),Line(c,d));
}
printf("END OF OUTPUT\n");
return 0;
}

转载于:https://www.cnblogs.com/Andromeda-Galaxy/p/10451643.html
展开全文
• 板子
板子题链接
1_A：点积求投影
1_B：点积求投影后得到所需向量
1_C：叉积判一下
2_A：点积和叉积
2_B：判点在线段上和线段相交
2_C：判一下线段是否相交，然后用直线相交方法求交点
2_D：端点之间的距离，点到直线距离，都判一下。
3_A：叉积算一下
3_B：叉积判一下
3_C：点向平行于x轴正方向搞一个射线看一下和多边形的交点奇偶性，对于平行于x轴的边，需要一些细节判一下，详见jls板子
4_A：维护上下凸壳
4_B：利用凸多边形的凸函数性质
5_A：分治
6_A：离散化+扫描线+pbds
7_A：根据圆心之间的距离及两个圆的半径判断
7_B：角平分线求交点，点到直线距离
7_C：垂直平分线求交点，点到直线距离
7_D：投影后勾股定理
7_E：余弦定理
7_F：相似三角形
7_G：相似三角形
7_H：转成三角形和圆求有向面积交
7_I：扇形减去四边形面积
#define vec point
typedef long double db ; //卡精度就换long double
const db eps = 1e-6 ; //调参
const db pi = acos(-1.0) ;
int sgn(db x)
{
if(x < -eps) return -1 ;
else if (x > eps)  return 1 ;
else return 0 ;
}
int cmp(db x , db y)
{
return sgn(x - y) ;
}
void print(int num , db x)
{
cout << fixed << setprecision(num) << x << '\n' ;
}
struct point
{
db x , y ;
point(){}
point(db x2 , db y2)
{
x = x2 , y = y2 ;
}
point operator + (const point& s)const{return (point){x + s.x , y + s.y} ;}
point operator - (const point& s)const{return (point){x - s.x , y - s.y} ;}
point operator * (const db& k)const{return (point){x * k , y * k} ;}
point operator / (const db& k)const{return (point){x / k , y / k} ;}
bool operator < (point b) const
{
return sgn(x - b.x) == 0 ? sgn(y - b.y) < 0 : x < b.x ;
}
bool equal(point p2)
{
return cmp(x , p2.x) == 0 && cmp(y , p2.y) == 0 ;
}
db get_angle(){return atan2(y , x) ;} //极角
db sq(db x) {return x * x ;}
db dis(point p) //很吃精度，可能需要1e-10或更小的eps
{
return sqrtl(sq(x - p.x) + sq(y - p.y)) ; //check sqrt or sqrtl
}
db len()
{
return sqrtl(sq(x) + sq(y)) ;
}
db len2()
{
return sq(x) + sq(y) ;
}
point unit()
{
db w = len() ;
return (point){x / w , y / w} ;
}
vec rotate_left()  //向量逆时针旋转90度
{
return vec(-y , x) ;
}
vec rotate_right()  //向量顺时针旋转90度
{
return vec(y , -x) ;
}
point move(db r) //原点沿该点代表的向量方向移动r
{
db l = len() ;
if(sgn(l) == 0)  return *this ;
else  return point(x * r / l , y * r / l) ;
}
} ;
db cross(vec s , vec t){return s.x * t.y - s.y * t.x ;} // 叉积
db dot(vec s , vec t){return s.x * t.x + s.y * t.y ;} // 点积
db rad(vec p1 , vec p2){ return atan2(cross(p1 , p2) , dot(p1 , p2)) ;} // 有向弧度
int in_mid(db k1 , db k2 , db k3) // k3 在 [k1,k2] 内
{
return cmp(k1 , k3) * cmp(k2 , k3) <= 0 ;
}
int in_mid(point p1 , point p2 , point p3)
{
return in_mid(p1.x , p2.x , p3.x) && in_mid(p1.y , p2.y , p3.y) ;
}
int counter_clockwise(point p1 , point p2 , point p3) // p1 p2 p3 逆时针 1 顺时针 -1 否则 0
{
return sgn(cross(p2 - p1 , p3 - p1)) ;
}
struct line
{
point s , e ; // 点s在直线上，方向向量是e ，半平面在e左侧
db ang ; //在运算时注意设置ang的值！
line() {}
line(point s2 , point e2)
{
s = s2 , e = e2 ;
ang = e.get_angle() ;
}
bool operator < (const line& l1)const{return ang < l1.ang ;}
int onRight(point p){return sgn(cross(e , p - s)) < 0 ;} //点p在直线右侧
point projection(point p) //p在该直线上的投影点
{
db t = dot(e , p - s) / e.len() ;
return s + e.move(t) ;
}
point reflection(point p) //p关于直线的对称点
{
point p2 = projection(p) ;
vec v = p2 - p ;
return p + v * 2 ;
}
db dis_point(point p) //点到直线的距离
{
db t = dot(e , p - s) / e.len() ;
point pp = s + e.move(t) ;
return p.dis(pp) ;
}
db dis_segment_point(point p) //点到线段的距离
{
point p1 = s ; //线段p1->p2
point p2 = s + e ;
point p3 = projection(p) ;
if(in_mid(p1 , p2 , p3))  return p.dis(p3) ;
else  return min(p.dis(p1) , p.dis(p2)) ;
}
bool on_segment(point p) //p在线段s->s+e上
{
return sgn((p.dis(s) + p.dis(s + e)) - s.dis(s + e)) == 0 ;//
}
int direction(point p2) //p0 = s , p1 = s + e 判断p2与向量p0->p1的位置关系
{
point p0 = s ;
point p1 = s + e ;
db res = cross(p1 - p0 , p2 - p0) ;
if(sgn(res) > 0)  return 1 ; //p0->p1逆时针旋转得到p0->p2的方向
else if(sgn(res) < 0)  return 2 ; //p0->p1顺时针旋转得到p0->p2的方向
else
{
db res2 = dot(p1 - p0 , p2 - p0) ;
if(on_segment(p2))  return 5 ; //p2在线段p0->p1上
else if(sgn(res2) < 0)  return 3 ; //p0->p1与p0->p2方向相反
else  return 4 ; //p0->p1与p0->p2方向相同
}
}
int Parallel_Orthogonal(line l2) //判断两条直线平行、正交
{
if(sgn(cross(e , l2.e)) == 0)  return 2 ; //平行
else if(sgn(dot(e , l2.e)) == 0)  return 1 ; //正交
else  return 0 ;
}
point line_intersect(point s1 , point t1 , point s2 , point t2) // 直线交点（点+向量表示直线）
{
db x = cross(t2 , s1 - s2) / cross(t1 , t2) ;
return s1 + t1 * x ;
}
point line_intersect(line l2) //直线交点（点+向量表示直线）
{
assert(sgn(cross(e , l2.e)) != 0) ;
return line_intersect(s , e , l2.s , l2.e) ;
}
point segment_intersect(line l2) //线段s->s+e与线段l2.s->l2.s+l2.e的交点
{
assert(sgn(cross(e , l2.e)) != 0) ;
return line_intersect(s , e , l2.s , l2.e) ;
}
bool can_segment_intersect(line l2) //线段s->s+e与线段l2.s->l2.s+l2.e
{
int t1 = sgn(cross(e , l2.s - s)) ;
int t2 = sgn(cross(e , (l2.s + l2.e) - s)) ;
int t3 = sgn(cross(l2.e , s - l2.s)) ;
int t4 = sgn(cross(l2.e , (s + e) - l2.s)) ;
if(on_segment(l2.s) || on_segment(l2.s + l2.e)
|| l2.on_segment(s) || l2.on_segment(s + e))  //端点在另一条线段上的情况
return true ;
else if(t1 * t2 < 0 && t3 * t4 < 0)  return true ;
else  return false ;
}
db segment_segment_dis(line l2) //线段s->s+e与线段l2.s->l2.s+l2.e
{
if(can_segment_intersect(l2))  return 0.0 ;
else
{
db res = min({s.dis(l2.s) , s.dis(l2.s + l2.e) , (s + e).dis(l2.s) , (s + e).dis(l2.s + l2.e)}) ;
point t = l2.projection(s) ;
if(l2.on_segment(t))  res = min(res , s.dis(t)) ;
point t2 = l2.projection(s + e) ;
if(l2.on_segment(t2))  res = min(res , (s + e).dis(t2)) ;
point t3 = projection(l2.s) ;
if(on_segment(t3))  res = min(res , l2.s.dis(t3)) ;
point t4 = projection(l2.s + l2.e) ;
if(on_segment(t4))  res = min(res , (l2.s + l2.e).dis(t4)) ;
return res ;
}
}
line trans(db r) //半平面u向内平移r
{
line v ;
v.s = s + e.rotate_left().move(r) ;
v.e = e ;
v.ang = e.get_angle() ; //注意设置ang的值
return v ;
}
} ;
struct Polygon
{
vector<point> pp ;
Polygon() {}
Polygon(int n)
{
pp.resize(n) ;
}
int size() const
{
return pp.size() ;
}
void resize(int n)
{
pp.resize(n) ;
}
point operator [](int id) const
{
if(id < 0 || id >= size())  assert(false) ;
else  return pp[id] ;
}
point &operator [](int id)
{
return pp[id] ;
}
// 凸包
// Andrew算法
// 按照x优先的顺序排序（坐标从小到大）
// 从第一个点开始遍历，如果下一个点在栈顶的两个元素所连成直线的左边，那么就入栈
// 否则如果在右边，说明凸包有更优的方案，上次的点出栈，并直到新点在栈顶两个点所在的直线的左边为止
// 这样求得下凸包，类似方法求得上凸包。
void Convex_Hull(vector<point> P , vector<point> &res , int flag = 1) // flag = 0 不严格（存在三点共线） flag = 1 严格（不存在三点共线）
{
int n = P.size() ;
res.resize(0) ;
res.resize(n * 2) ;
sort(P.begin() , P.end()) ;
int now = -1 ;
for(int i = 0 ; i < (int)P.size() ; i ++)
{
while(now > 0 && sgn(cross(res[now] - res[now - 1] , P[i] - res[now - 1])) < flag)  now -- ;
res[++ now] = P[i] ;
}
int pre = now ;
for(int i = n - 2 ; i >= 0 ; i --)
{
while(now > pre && sgn(cross(res[now] - res[now - 1] , P[i] - res[now - 1])) < flag)  now -- ;
res[++ now] = P[i] ;
}
res.resize(now) ;
}
// 半平面交
// 排序增量算法
// 1.以逆时针为正方向，建边。(输入方向不确定时，可用叉乘求面积看正负得知输入的顺逆方向。)
// 2.对线段根据极角排序。
// 3.去除极角相同的情况下，位置在右边的边。
// 4.用双端队列储存线段集合L，遍历所有线段。
// 5.判断该线段加入后对半平面交的影响，(对双端队列的头部和尾部进行判断，因为线段加入是有序的。)。
// 6.如果某条线段对于新的半平面交没有影响，则从队列中剔除掉。
// 7.最后剩下的线段集合L，即使最后要求的半平面交。
int Half_Plane_Intersection(vector<line>L , vector<point> &res) 	//L.push_back({p[i] , p[(i + 1) % m] - p[i]}) ;
{
int n = L.size() ;
sort(L.begin() , L.end()) ;
int l = 0 , r = 0 ;
vector<point> p(n) ;
vector<line> q(n) ;
q[0] = L[0] ;
for(int i = 1 ; i <= n - 1 ; i ++)
{
while(l < r && L[i].onRight(p[r - 1]))  r -- ;
while(l < r && L[i].onRight(p[l]))  l ++ ;
q[++ r] = L[i] ;
if (sgn(cross(q[r].e , q[r - 1].e)) == 0)
{
r -- ;
if(!q[r].onRight(L[i].s))  q[r] = L[i] ;
}
if(l < r)  p[r - 1] = q[r - 1].line_intersect(q[r]) ;
}
while(l < r && q[l].onRight(p[r - 1]))  r -- ;
if(r - l <= 1)  return 0 ; // 交不存在
res.clear() ;
p[r] = q[l].line_intersect(q[r]) ;
for(int i = l ; i <= r ; i ++)  res.push_back(p[i]) ;
return 1 ;
}
db perimeter(vector<point> A) // 多边形用 vector<point> 表示 , 逆时针
{
db ans = 0 ;
for(int i = 0 ; i < (int)A.size() ; i ++)  ans += A[i].dis(A[(i + 1) % (int)A.size()]) ;
return ans ;
}
db area() // 多边形用 vector<point> 表示 , 逆时针
{
db ans = 0 ;
for(int i = 0 ; i < (int)pp.size() ; i ++)  ans += cross(pp[i] , pp[(i + 1) % (int)pp.size()]) ;
return fabs(ans) / 2 ;
}
bool is_convex(vector<point> A) // 多边形用 vector<point> 表示 , 逆时针
{
int n = A.size() ;
assert(n >= 3) ;
bool flag = true ;
for(int i = 0 ; i < n ; i ++)
{
int lst = (i - 1 + n) % n ;
int nxt = (i + 1) % n ;
if(sgn(cross(A[lst] - A[i] , A[nxt] - A[i])) > 0)  flag = false ;
}
return flag ;
}
int contain_point(point p0)  // 不一定是凸的 2 内部 1 边界 0 外部
{
int n = pp.size() ;
int res = 0 ;
for(int i = 0 ; i < n ; i ++)
{
point u = pp[(i - 1 + n) % n] ;
point v = pp[i] ;
line l ;
l.s = u ;
l.e = v - u ;
if(l.on_segment(p0))  return 1 ;
if(cmp(u.y , v.y) > 0)  swap(u , v) ;
if(cmp(u.y , p0.y) >= 0 || cmp(v.y , p0.y) < 0)  continue ;
if(sgn(cross(u - v , p0 - v)) < 0)  res ^= 1 ;
}
return res << 1 ;
}
db Convex_Diameter()
{
//原理是凸多边形的凸函数性质
int now = 0 ;
int n = pp.size() ;
db res = 0 ;
for(int i = 0 ; i < n ; i ++)
{
now = max(now , i) ;
while(true)
{
db k1 = pp[i].dis(pp[now % n]) ;
db k2 = pp[i].dis(pp[(now + 1) % n]) ;
res = max({res , k1 , k2}) ;
if(cmp(k2 , k1) > 0)  now ++ ;
else  break ;
}
}
return res ;
}
Polygon line_cut_convex(point p1 , point p2)
{
//在已有凸多边形基础上新加一个半平面p1->p2
//问之后的凸多边形
int n = pp.size() ;
line l ;
l.s = p1 ;
l.e = p2 - p1 ;
Polygon res ;
for(int i = 0 ; i < n ; i ++)
{
int w1 = counter_clockwise(p1 , p2 , pp[i]) ;
int w2 = counter_clockwise(p1 , p2 , pp[(i + 1) % n]) ;
if(w1 >= 0)  res.pp.push_back(pp[i]) ;
line l2 ;
l2.s = pp[i] ;
l2.e = pp[(i + 1) % n] - pp[i] ;
if(w1 * w2 < 0)  res.pp.push_back(l.line_intersect(l2)) ;
}
return res ;
}
} ;
struct circle
{
point o ;
db r ;
db area()
{
return pi * r * r ;
}
int inside(point p2)
{
return cmp(r , o.dis(p2)) ;
}
circle get_circle(point p1 , point p2 , point p3) //三点确定一个圆
{
db a1 = p2.x - p1.x , b1 = p2.y - p1.y , c1 = (a1 * a1 + b1 * b1) / 2 ;
db a2 = p3.x - p1.x , b2 = p3.y - p1.y , c2 = (a2 * a2 + b2 * b2) / 2 ;
db d = a1 * b2 - a2 * b1 ;
point o = (point){p1.x + (c1 * b2 - c2 * b1) / d , p1.y + (a1 * c2 - a2 * c1) / d} ;
return (circle){o , p1.dis(o)} ;
}
vector<point> Tangent_Point_Circle(point p) //沿圆的逆时针方向给出切点，点在圆上也给出两个
{
//相似三角形
assert(cmp(p.dis(o) , r) >= 0) ;
db a = (p - o).len() ;
assert(sgn(a) >= 0) ;
db b = r * r / a ;
db c = sqrt(max(db(0.0) , r * r - b * b)) ;
point k = (p - o).unit() ;
point m = o + k * b ;
point v = k.rotate_left() * c ;
return {m - v , m + v} ;
}
int check_pos_circle_circle(circle c1 , circle c2) //返回两个圆的公切线数量，即位置关系
{
//根据圆心之间的距离及两个圆的半径判断
if(cmp(c1.r , c2.r) < 0)  swap(c1 , c2) ;
db d = c1.o.dis(c2.o) ;
int w1 = cmp(d , c1.r + c2.r) ;
int w2 = cmp(d , c1.r - c2.r) ;
if(w1 > 0)  return 4 ; //相离
else if(w1 == 0)  return 3 ; //外切
else if(w2 > 0)  return 2 ; //相交
else if(w2 == 0)  return 1 ; //内切
else  return 0 ; //内含
}
vector<point> get_circle_line(line l) //圆与直线求交点，沿l.e方向的顺序给出交点，相切给出两个
{
//投影后勾股定理
point k = l.projection(o) ;
db d = r * r - (k - o).len2() ;
if(sgn(d) < 0)  return {} ;
vec v = l.e.unit() * sqrtl(max((db)0.0 , d)) ;
return {k - v , k + v} ;
}
vector<point> get_circle_circle(circle c1 , circle c2) //两个圆的交点，沿圆c1逆时针给出，相切给出两个
{
//余弦定理
int pos = check_pos_circle_circle(c1 , c2) ;
if(pos == 0 || pos == 4)  return {} ;
db a = (c2.o - c1.o).len2() ;
db cosA = (c1.r * c1.r + a - c2.r * c2.r) / (2 * c1.r * sqrtl(max(a , (db)0.0))) ;
db b = c1.r * cosA ;
db c = sqrtl(max((db)0.0 , c1.r * c1.r - b * b)) ;
point k = (c2.o - c1.o).unit() ;
point m = c1.o + k * b ;
vec v = k.rotate_left() * c ;
return {m - v , m + v} ;
}
circle Incircle_of_a_Triangle(point p1 , point p2 , point p3) //三角形p1p2p3内接圆
{
//角平分线求交点，点到直线距离
line l1 ;
l1.s = p1 ;
l1.e = (p2 - p1).unit() + (p3 - p1).unit() ;
line l2 ;
l2.s = p2 ;
l2.e = (p1 - p2).unit() + (p3 - p2).unit() ;
point p = l1.line_intersect(l2) ;
line l3 ;
l3.s = p1 ;
l3.e = p2 - p1 ;
circle cc ;
cc.o = p ;
cc.r = l3.dis_point(p) ;
return cc ;
}
circle Circumscribed_Circle_of_a_Triangle(point p1 , point p2 , point p3) //三角形p1p2p3外接圆
{
//垂直平分线求交点，点到直线距离
line l1 ;
l1.s = p1 + (p2 - p1) / 2 ;
l1.e = (p2 - p1).rotate_left() ; //顺时针或逆时针旋转90度均可，因为是直线求交
line l2 ;
l2.s = p2 + (p3 - p2) / 2 ;
l2.e = (p3 - p2).rotate_left() ;
point p = l1.line_intersect(l2) ;
circle cc ;
cc.o = p ;
cc.r = p.dis(p1) ;
return cc ;
}
vector<line> Tangent_out_Circle_Circle(circle c1 , circle c2) //外切线
{
//相似三角形
int pos = check_pos_circle_circle(c1 , c2) ;
if(pos == 0)  return {} ;
if(pos == 1)
{
point k = get_circle_circle(c1 , c2)[0] ;
vec v = k - c1.o ;
v = v.rotate_left() ;
line l ;
l.s = k ;
l.e = v ;
return {l} ;
}
if(cmp(c1.r , c2.r) == 0)
{
vec v = (c2.o - c1.o).rotate_left().unit() * c1.r ;
vec p1 = c1.o + v ;
line l1 ;
l1.s = p1 ;
l1.e = c2.o - c1.o ;
vec p2 = c1.o - v ;
line l2 ;
l2.s = p2 ;
l2.e = c2.o - c1.o ;
return {l1 , l2} ;
}
else
{
vec v = c2.o - c1.o ;
v = v.unit() * (c1.o.dis(c2.o) * c1.r / (c1.r - c2.r)) ;
point H = c1.o + v ;
vector<point> pp1 = c1.Tangent_Point_Circle(H) ;
line l1 ;
l1.s = H ;
l1.e = pp1[0] - H ;
line l2 ;
l2.s = H ;
l2.e = pp1[1] - H ;
return {l1 , l2} ;
}
}
vector<line> Tangent_in_Circle_Circle(circle c1 , circle c2) //内切线
{
int pos = check_pos_circle_circle(c1 , c2) ;
if(pos <= 2)  return {} ;
if(pos == 3)
{
point k = get_circle_circle(c1 , c2)[0] ;
vec v = k - c1.o ;
v = v.rotate_left() ;
line l ;
l.s = k ;
l.e = v ;
return {l} ;
}
db t = c1.r / (c1.r + c2.r) * c1.o.dis(c2.o) ;
point p = c1.o + (c2.o - c1.o).unit() * t ;
vector<point> pp1 = c1.Tangent_Point_Circle(p) ;
line l1 ;
l1.s = p ;
l1.e = pp1[0] - p ;
line l2 ;
l2.s = p ;
l2.e = pp1[1] - p ;
return {l1 , l2} ;
}
vector<line> Tangent_Circle_Circle(circle c1 , circle c2) //两个圆的公切线
{
//外切线和内切线一起考虑
if(cmp(c1.r , c2.r) < 0)  swap(c1 , c2) ;
vector<line> A = Tangent_out_Circle_Circle(c1 , c2) ;
vector<line> B = Tangent_in_Circle_Circle(c1 , c2) ;
for(auto u : B)  A.push_back(u) ;
return A ;
}
db get_area_of_Circle_Triangle(circle c1 , point p2 , point p3)
{
//圆c1与三角形c1 p2 p3的交
point k = c1.o ;
c1.o = c1.o - k ;
p2 = p2 - k ;
p3 = p3 - k ;
int pos1 = c1.inside(p2) ;
int pos2 = c1.inside(p3) ;
line l ;
l.s = p2 ;
l.e = p3 - p2 ;
vector<point> pp = c1.get_circle_line(l) ;
if(pos1 >= 0)
{
if(pos2 >= 0)  return cross(p2 , p3) / 2 ;
return c1.r * c1.r * rad(pp[1] , p3) / 2 + cross(p2 , pp[1]) / 2 ;
}
else if(pos2 >= 0)
{
return c1.r * c1.r * rad(p2 , pp[0]) / 2 + cross(pp[0] , p3) / 2 ;
}
else
{
line l ;
l.s = p2 ;
l.e = p3 - p2 ;
int pos = cmp(c1.r , l.dis_segment_point(c1.o)) ;
if(pos <= 0)  return c1.r * c1.r * rad(p2 , p3) / 2 ;
return cross(pp[0] , pp[1]) / 2 + c1.r * c1.r * (rad(p2 , pp[0]) + rad(pp[1] , p3)) / 2 ;
}
}
db get_area_of_Circle_Polygon(circle c1 , vector<point> pp) //pp按照逆时针给出
{
//转成三角形和圆求有向面积交
int n = pp.size() ;
db res = 0 ;
for(int i = 0 ; i < n ; i ++)
{
point now = pp[i] ; //三角形o pp[i] pp[(i + 1) % n] 需要按照顺序给出
point nxt = pp[(i + 1) % n] ;
res += get_area_of_Circle_Triangle(c1 , now , nxt) ; //已经除2
}
return fabs(res) ;
}
db get_area_of_Circle_Circle(circle c1 , circle c2) //两个圆的相交面积
{
//扇形减去四边形面积
if(cmp(c1.r , c2.r) < 0)  swap(c1 , c2) ;
int pos = check_pos_circle_circle(c1 , c2) ;
if(pos >= 3)  return 0.0 ;
if(pos <= 1)  return c2.area() ;
vector<point> pp = get_circle_circle(c1 , c2) ;
db S1 = fabs(2.0 * rad(pp[0] - c1.o , c2.o - c1.o)) * c1.r * c1.r / 2 ;
db S2 = fabs(2.0 * rad(pp[0] - c2.o , c1.o - c2.o)) * c2.r * c2.r / 2 ;
db S3 = fabs(cross(pp[0] - c1.o , c2.o - c1.o)) ;
return S1 + S2 - S3 ;
}
} c ;
db closest_point(vector<point> &A , int l , int r)
{
if(r - l <= 5)
{
db res = 1e20 ;
for(int i = l ; i <= r ; i ++)
for(int j = i + 1 ; j <= r ; j ++)
res = min(res , A[i].dis(A[j])) ;
return res ;
}
int mid = (l + r) / 2 ;
db res = min(closest_point(A , l , mid) , closest_point(A , mid + 1 , r)) ;
vector<point> B ;
for(int i = l ; i <= r ; i ++)
if(cmp(fabs(A[i].x - A[mid].x) , res) <= 0)
B.push_back(A[i]) ;
sort(B.begin() , B.end() , [&](point p1 , point p2){ return cmp(p1.y , p2.y) < 0 ; }) ;
for(int i = 0 ; i < B.size() ; i ++)
for(int j = i + 1 ; j < B.size() && B[j].y - B[i].y < res ; j ++)
res = min(res , B[i].dis(B[j])) ;
return res ;
}

展开全文
• 包含点、线、多边形、凸多边形、圆、半平面交等算法的板子
• 计算几何模板 if( (a.y2-a.y1)*(b.x2-b.x1)==(b.y2-b.y1)*(a.x2-a.x1) ) //向量判别是否平行 x1y2=x2y1 { if( (a.y1-b.y2)*(a.x2-b.x1)==(a.y2-b.y1)*(a.x1-b.x2) && (b.y1-a.y1)*(a.x2-b.x2)==(a.y2-b.y2)*(b.x1-a...
• 转自：... 二维几何 // 计算几何模板 const double eps = 1e-8; const double inf = 1e20; const double pi = acos(-1.0); const int maxp = 1010; //Compares a double ...
转自：https://kuangbin.github.io/2019/04/28/20190428/#more
二维几何
// 计算几何模板
const double eps = 1e-8;
const double pi = acos(-1.0);
const int maxp = 1010;
//Compares a double to zero
int sgn(double x){
if(fabs(x) < eps)return 0;
if(x < 0)return -1;
else return 1;
}
//square of a double
inline double sqr(double x){return x*x;}
/*
* Point
* Point()               - Empty constructor
* Point(double _x,double _y)  - constructor
* input()             - double input
* output()            - %.2f output
* operator ==         - compares x and y
* operator <          - compares first by x, then by y
* operator -          - return new Point after subtracting curresponging x and y
* operator ^          - cross product of 2d points
* operator *          - dot product
* len()               - gives length from origin
* len2()              - gives square of length from origin
* distance(Point p)   - gives distance from p
* operator + Point b  - returns new Point after adding curresponging x and y
* operator * double k - returns new Point after multiplieing x and y by k
* operator / double k - returns new Point after divideing x and y by k
* rad(Point a,Point b)- returns the angle of Point a and Point b from this Point
* trunc(double r)     - return Point that if truncated the distance from center to r
* rotleft()           - returns 90 degree ccw rotated point
* rotright()          - returns 90 degree cw rotated point
* rotate(Point p,double angle) - returns Point after rotateing the Point centering at p by angle radian ccw
*/
struct Point{
double x,y;
Point(){}
Point(double _x,double _y){
x = _x;
y = _y;
}
void input(){
scanf("%lf%lf",&x,&y);
}
void output(){
printf("%.2f %.2f\n",x,y);
}
bool operator == (Point b)const{
return sgn(x-b.x) == 0 && sgn(y-b.y) == 0;
}
bool operator < (Point b)const{
return sgn(x-b.x)== 0?sgn(y-b.y)<0:x<b.x;
}
Point operator -(const Point &b)const{
return Point(x-b.x,y-b.y);
}
//叉积
double operator ^(const Point &b)const{
return x*b.y - y*b.x;
}
//点积
double operator *(const Point &b)const{
return x*b.x + y*b.y;
}
//返回长度
double len(){
return hypot(x,y);//库函数
}
//返回长度的平方
double len2(){
return x*x + y*y;
}
//返回两点的距离
double distance(Point p){
return hypot(x-p.x,y-p.y);
}
Point operator +(const Point &b)const{
return Point(x+b.x,y+b.y);
}
Point operator *(const double &k)const{
return Point(x*k,y*k);
}
Point operator /(const double &k)const{
return Point(x/k,y/k);
}
//计算pa  和  pb 的夹角
//就是求这个点看a,b 所成的夹角
//测试 LightOJ1203
Point p = *this;
return fabs(atan2( fabs((a-p)^(b-p)),(a-p)*(b-p) ));
}
//化为长度为r的向量
Point trunc(double r){
double l = len();
if(!sgn(l))return *this;
r /= l;
return Point(x*r,y*r);
}
//逆时针旋转90度
Point rotleft(){
return Point(-y,x);
}
//顺时针旋转90度
Point rotright(){
return Point(y,-x);
}
//绕着p点逆时针旋转angle
Point rotate(Point p,double angle){
Point v = (*this) - p;
double c = cos(angle), s = sin(angle);
return Point(p.x + v.x*c - v.y*s,p.y + v.x*s + v.y*c);
}
};
/*
* Stores two points
* Line()                         - Empty constructor
* Line(Point _s,Point _e)        - Line through _s and _e
* operator ==                    - checks if two points are same
* Line(Point p,double angle)     - one end p , another end at angle degree
* Line(double a,double b,double c) - Line of equation ax + by + c = 0
* input()                        - inputs s and e
* adjust()                       - orders in such a way that s < e
* length()                       - distance of se
* angle()                        - return 0 <= angle < pi
* relation(Point p)              - 3 if point is on line
*                                  1 if point on the left of line
*                                  2 if point on the right of line
* pointonseg(double p)           - return true if point on segment
* parallel(Line v)               - return true if they are parallel
* segcrossseg(Line v)            - returns 0 if does not intersect
*                                  returns 1 if non-standard intersection
*                                  returns 2 if intersects
* linecrossseg(Line v)           - line and seg
* linecrossline(Line v)          - 0 if parallel
*                                  1 if coincides
*                                  2 if intersects
* crosspoint(Line v)             - returns intersection point
* dispointtoline(Point p)        - distance from point p to the line
* dispointtoseg(Point p)         - distance from p to the segment
* dissegtoseg(Line v)            - distance of two segment
* lineprog(Point p)              - returns projected point p on se line
* symmetrypoint(Point p)         - returns reflection point of p over se
*
*/
struct Line{
Point s,e;
Line(){}
Line(Point _s,Point _e){
s = _s;
e = _e;
}
bool operator ==(Line v){
return (s == v.s)&&(e == v.e);
}
//根据一个点和倾斜角angle确定直线,0<=angle<pi
Line(Point p,double angle){
s = p;
if(sgn(angle-pi/2) == 0){
e = (s + Point(0,1));
}
else{
e = (s + Point(1,tan(angle)));
}
}
//ax+by+c=0
Line(double a,double b,double c){
if(sgn(a) == 0){
s = Point(0,-c/b);
e = Point(1,-c/b);
}
else if(sgn(b) == 0){
s = Point(-c/a,0);
e = Point(-c/a,1);
}
else{
s = Point(0,-c/b);
e = Point(1,(-c-a)/b);
}
}
void input(){
s.input();
e.input();
}
if(e < s)swap(s,e);
}
//求线段长度
double length(){
return s.distance(e);
}
//返回直线倾斜角 0<=angle<pi
double angle(){
double k = atan2(e.y-s.y,e.x-s.x);
if(sgn(k) < 0)k += pi;
if(sgn(k-pi) == 0)k -= pi;
return k;
}
//点和直线关系
//1  在左侧
//2  在右侧
//3  在直线上
int relation(Point p){
int c = sgn((p-s)^(e-s));
if(c < 0)return 1;
else if(c > 0)return 2;
else return 3;
}
// 点在线段上的判断
bool pointonseg(Point p){
return sgn((p-s)^(e-s)) == 0 && sgn((p-s)*(p-e)) <= 0;
}
//两向量平行(对应直线平行或重合)
bool parallel(Line v){
return sgn((e-s)^(v.e-v.s)) == 0;
}
//两线段相交判断
//2 规范相交
//1 非规范相交
//0 不相交
int segcrossseg(Line v){
int d1 = sgn((e-s)^(v.s-s));
int d2 = sgn((e-s)^(v.e-s));
int d3 = sgn((v.e-v.s)^(s-v.s));
int d4 = sgn((v.e-v.s)^(e-v.s));
if( (d1^d2)==-2 && (d3^d4)==-2 )return 2;
return (d1==0 && sgn((v.s-s)*(v.s-e))<=0) ||
(d2==0 && sgn((v.e-s)*(v.e-e))<=0) ||
(d3==0 && sgn((s-v.s)*(s-v.e))<=0) ||
(d4==0 && sgn((e-v.s)*(e-v.e))<=0);
}
//直线和线段相交判断
//-*this line   -v seg
//2 规范相交
//1 非规范相交
//0 不相交
int linecrossseg(Line v){
int d1 = sgn((e-s)^(v.s-s));
int d2 = sgn((e-s)^(v.e-s));
if((d1^d2)==-2) return 2;
return (d1==0||d2==0);
}
//两直线关系
//0 平行
//1 重合
//2 相交
int linecrossline(Line v){
if((*this).parallel(v))
return v.relation(s)==3;
return 2;
}
//求两直线的交点
//要保证两直线不平行或重合
Point crosspoint(Line v){
double a1 = (v.e-v.s)^(s-v.s);
double a2 = (v.e-v.s)^(e-v.s);
return Point((s.x*a2-e.x*a1)/(a2-a1),(s.y*a2-e.y*a1)/(a2-a1));
}
//点到直线的距离
double dispointtoline(Point p){
return fabs((p-s)^(e-s))/length();
}
//点到线段的距离
double dispointtoseg(Point p){
if(sgn((p-s)*(e-s))<0 || sgn((p-e)*(s-e))<0)
return min(p.distance(s),p.distance(e));
return dispointtoline(p);
}
//返回线段到线段的距离
//前提是两线段不相交，相交距离就是0了
double dissegtoseg(Line v){
return min(min(dispointtoseg(v.s),dispointtoseg(v.e)),min(v.dispointtoseg(s),v.dispointtoseg(e)));
}
//返回点p在直线上的投影
Point lineprog(Point p){
return s + ( ((e-s)*((e-s)*(p-s)))/((e-s).len2()) );
}
//返回点p关于直线的对称点
Point symmetrypoint(Point p){
Point q = lineprog(p);
return Point(2*q.x-p.x,2*q.y-p.y);
}
};
//圆
struct circle{
Point p;//圆心
double r;//半径
circle(){}
circle(Point _p,double _r){
p = _p;
r = _r;
}
circle(double x,double y,double _r){
p = Point(x,y);
r = _r;
}
//三角形的外接圆
//需要Point的+ /  rotate()  以及Line的crosspoint()
//利用两条边的中垂线得到圆心
//测试：UVA12304
circle(Point a,Point b,Point c){
Line u = Line((a+b)/2,((a+b)/2)+((b-a).rotleft()));
Line v = Line((b+c)/2,((b+c)/2)+((c-b).rotleft()));
p = u.crosspoint(v);
r = p.distance(a);
}
//三角形的内切圆
//参数bool t没有作用，只是为了和上面外接圆函数区别
//测试：UVA12304
circle(Point a,Point b,Point c,bool t){
Line u,v;
double m = atan2(b.y-a.y,b.x-a.x), n = atan2(c.y-a.y,c.x-a.x);
u.s = a;
u.e = u.s + Point(cos((n+m)/2),sin((n+m)/2));
v.s = b;
m = atan2(a.y-b.y,a.x-b.x) , n = atan2(c.y-b.y,c.x-b.x);
v.e = v.s + Point(cos((n+m)/2),sin((n+m)/2));
p = u.crosspoint(v);
r = Line(a,b).dispointtoseg(p);
}
//输入
void input(){
p.input();
scanf("%lf",&r);
}
//输出
void output(){
printf("%.2lf %.2lf %.2lf\n",p.x,p.y,r);
}
bool operator == (circle v){
return (p==v.p) && sgn(r-v.r)==0;
}
bool operator < (circle v)const{
return ((p<v.p)||((p==v.p)&&sgn(r-v.r)<0));
}
//面积
double area(){
return pi*r*r;
}
//周长
double circumference(){
return 2*pi*r;
}
//点和圆的关系
//0 圆外
//1 圆上
//2 圆内
int relation(Point b){
double dst = b.distance(p);
if(sgn(dst-r) < 0)return 2;
else if(sgn(dst-r)==0)return 1;
return 0;
}
//线段和圆的关系
//比较的是圆心到线段的距离和半径的关系
int relationseg(Line v){
double dst = v.dispointtoseg(p);
if(sgn(dst-r) < 0)return 2;
else if(sgn(dst-r) == 0)return 1;
return 0;
}
//直线和圆的关系
//比较的是圆心到直线的距离和半径的关系
int relationline(Line v){
double dst = v.dispointtoline(p);
if(sgn(dst-r) < 0)return 2;
else if(sgn(dst-r) == 0)return 1;
return 0;
}
//两圆的关系
//5 相离
//4 外切
//3 相交
//2 内切
//1 内含
//需要Point的distance
//测试：UVA12304
int relationcircle(circle v){
double d = p.distance(v.p);
if(sgn(d-r-v.r) > 0)return 5;
if(sgn(d-r-v.r) == 0)return 4;
double l = fabs(r-v.r);
if(sgn(d-r-v.r)<0 && sgn(d-l)>0)return 3;
if(sgn(d-l)==0)return 2;
if(sgn(d-l)<0)return 1;
}
//求两个圆的交点，返回0表示没有交点，返回1是一个交点，2是两个交点
//需要relationcircle
//测试：UVA12304
int pointcrosscircle(circle v,Point &p1,Point &p2){
int rel = relationcircle(v);
if(rel == 1 || rel == 5)return 0;
double d = p.distance(v.p);
double l = (d*d+r*r-v.r*v.r)/(2*d);
double h = sqrt(r*r-l*l);
Point tmp = p + (v.p-p).trunc(l);
p1 = tmp + ((v.p-p).rotleft().trunc(h));
p2 = tmp + ((v.p-p).rotright().trunc(h));
if(rel == 2 || rel == 4)
return 1;
return 2;
}
//求直线和圆的交点，返回交点个数
int pointcrossline(Line v,Point &p1,Point &p2){
if(!(*this).relationline(v))return 0;
Point a = v.lineprog(p);
double d = v.dispointtoline(p);
d = sqrt(r*r-d*d);
if(sgn(d) == 0){
p1 = a;
p2 = a;
return 1;
}
p1 = a + (v.e-v.s).trunc(d);
p2 = a - (v.e-v.s).trunc(d);
return 2;
}
//得到过a,b两点，半径为r1的两个圆
int gercircle(Point a,Point b,double r1,circle &c1,circle &c2){
circle x(a,r1),y(b,r1);
int t = x.pointcrosscircle(y,c1.p,c2.p);
if(!t)return 0;
c1.r = c2.r = r1;
return t;
}
//得到与直线u相切，过点q,半径为r1的圆
//测试：UVA12304
int getcircle(Line u,Point q,double r1,circle &c1,circle &c2){
double dis = u.dispointtoline(q);
if(sgn(dis-r1*2)>0)return 0;
if(sgn(dis) == 0){
c1.p = q + ((u.e-u.s).rotleft().trunc(r1));
c2.p = q + ((u.e-u.s).rotright().trunc(r1));
c1.r = c2.r = r1;
return 2;
}
Line u1 = Line((u.s + (u.e-u.s).rotleft().trunc(r1)),(u.e + (u.e-u.s).rotleft().trunc(r1)));
Line u2 = Line((u.s + (u.e-u.s).rotright().trunc(r1)),(u.e + (u.e-u.s).rotright().trunc(r1)));
circle cc = circle(q,r1);
Point p1,p2;
if(!cc.pointcrossline(u1,p1,p2))cc.pointcrossline(u2,p1,p2);
c1 = circle(p1,r1);
if(p1 == p2){
c2 = c1;
return 1;
}
c2 = circle(p2,r1);
return 2;
}
//同时与直线u,v相切，半径为r1的圆
//测试：UVA12304
int getcircle(Line u,Line v,double r1,circle &c1,circle &c2,circle &c3,circle &c4){
if(u.parallel(v))return 0;//两直线平行
Line u1 = Line(u.s + (u.e-u.s).rotleft().trunc(r1),u.e + (u.e-u.s).rotleft().trunc(r1));
Line u2 = Line(u.s + (u.e-u.s).rotright().trunc(r1),u.e + (u.e-u.s).rotright().trunc(r1));
Line v1 = Line(v.s + (v.e-v.s).rotleft().trunc(r1),v.e + (v.e-v.s).rotleft().trunc(r1));
Line v2 = Line(v.s + (v.e-v.s).rotright().trunc(r1),v.e + (v.e-v.s).rotright().trunc(r1));
c1.r = c2.r = c3.r = c4.r = r1;
c1.p = u1.crosspoint(v1);
c2.p = u1.crosspoint(v2);
c3.p = u2.crosspoint(v1);
c4.p = u2.crosspoint(v2);
return 4;
}
//同时与不相交圆cx,cy相切，半径为r1的圆
//测试：UVA12304
int getcircle(circle cx,circle cy,double r1,circle &c1,circle &c2){
circle x(cx.p,r1+cx.r),y(cy.p,r1+cy.r);
int t = x.pointcrosscircle(y,c1.p,c2.p);
if(!t)return 0;
c1.r = c2.r = r1;
return t;
}

//过一点作圆的切线(先判断点和圆的关系)
//测试：UVA12304
int tangentline(Point q,Line &u,Line &v){
int x = relation(q);
if(x == 2)return 0;
if(x == 1){
u = Line(q,q + (q-p).rotleft());
v = u;
return 1;
}
double d = p.distance(q);
double l = r*r/d;
double h = sqrt(r*r-l*l);
u = Line(q,p + ((q-p).trunc(l) + (q-p).rotleft().trunc(h)));
v = Line(q,p + ((q-p).trunc(l) + (q-p).rotright().trunc(h)));
return 2;
}
//求两圆相交的面积
double areacircle(circle v){
int rel = relationcircle(v);
if(rel >= 4)return 0.0;
if(rel <= 2)return min(area(),v.area());
double d = p.distance(v.p);
double hf = (r+v.r+d)/2.0;
double ss = 2*sqrt(hf*(hf-r)*(hf-v.r)*(hf-d));
double a1 = acos((r*r+d*d-v.r*v.r)/(2.0*r*d));
a1 = a1*r*r;
double a2 = acos((v.r*v.r+d*d-r*r)/(2.0*v.r*d));
a2 = a2*v.r*v.r;
return a1+a2-ss;
}
//求两圆相交的面积(精度更高)(需要long double)
double areacircle2(circle v)
{
double a=hypot(p.x-v.p.x,p.y-v.p.y),b=r,c=v.r;
double s1=pi*r*r,s2=pi*v.r*v.r;
if(sgn(a-b-c)>=0)
return 0;
if(sgn(a+min(b,c)-max(b,c))<=0)
return min(s1,s2);
else
{
double cta1=2*acos((a*a+b*b-c*c)/(2*a*b));
double cta2=2*acos((a*a+c*c-b*b)/(2*a*c));
return cta1/(2*pi)*s1-0.5*sin(cta1)*b*b+cta2/(2*pi)*s2-0.5*sin(cta2)*c*c;
}
}
//求圆和三角形pab的相交面积
//测试：POJ3675 HDU3982 HDU2892
double areatriangle(Point a,Point b){
if(sgn((p-a)^(p-b)) == 0)return 0.0;
Point q[5];
int len = 0;
q[len++] = a;
Line l(a,b);
Point p1,p2;
if(pointcrossline(l,q[1],q[2])==2){
if(sgn((a-q[1])*(b-q[1]))<0)q[len++] = q[1];
if(sgn((a-q[2])*(b-q[2]))<0)q[len++] = q[2];
}
q[len++] = b;
if(len == 4 && sgn((q[0]-q[1])*(q[2]-q[1]))>0)swap(q[1],q[2]);
double res = 0;
for(int i = 0;i < len-1;i++){
if(relation(q[i])==0||relation(q[i+1])==0){
res += r*r*arg/2.0;
}
else{
res += fabs((q[i]-p)^(q[i+1]-p))/2.0;
}
}
return res;
}
};

/*
* n,p  Line l for each side
* input(int _n)                        - inputs _n size polygon
* add(Point q)                         - adds a point at end of the list
* getline()                            - populates line array
* cmp                                  - comparision in convex_hull order
* norm()                               - sorting in convex_hull order
* getconvex(polygon &convex)           - returns convex hull in convex
* Graham(polygon &convex)              - returns convex hull in convex
* isconvex()                           - checks if convex
* relationpoint(Point q)               - returns 3 if q is a vertex
*                                                2 if on a side
*                                                1 if inside
*                                                0 if outside
* convexcut(Line u,polygon &po)        - left side of u in po
* gercircumference()                   - returns side length
* getarea()                            - returns area
* getdir()                             - returns 0 for cw, 1 for ccw
* getbarycentre()                      - returns barycenter
*
*/
struct polygon{
int n;
Point p[maxp];
Line l[maxp];
void input(int _n){
n = _n;
for(int i = 0;i < n;i++)
p[i].input();
}
p[n++] = q;
}
void getline(){
for(int i = 0;i < n;i++){
l[i] = Line(p[i],p[(i+1)%n]);
}
}
struct cmp{
Point p;
cmp(const Point &p0){p = p0;}
bool operator()(const Point &aa,const Point &bb){
Point a = aa, b = bb;
int d = sgn((a-p)^(b-p));
if(d == 0){
return sgn(a.distance(p)-b.distance(p)) < 0;
}
return d > 0;
}
};
//进行极角排序
//首先需要找到最左下角的点
//需要重载号好Point的 < 操作符(min函数要用) 
void norm(){
Point mi = p[0];
for(int i = 1;i < n;i++)mi = min(mi,p[i]);
sort(p,p+n,cmp(mi));
}
//得到凸包
//得到的凸包里面的点编号是0$\sim$n-1的
//两种凸包的方法
//注意如果有影响，要特判下所有点共点，或者共线的特殊情况
//测试 LightOJ1203  LightOJ1239
void getconvex(polygon &convex){
sort(p,p+n);
convex.n = n;
for(int i = 0;i < min(n,2);i++){
convex.p[i] = p[i];
}
if(convex.n == 2 && (convex.p[0] == convex.p[1]))convex.n--;//特判
if(n <= 2)return;
int &top = convex.n;
top = 1;
for(int i = 2;i < n;i++){
while(top && sgn((convex.p[top]-p[i])^(convex.p[top-1]-p[i])) <= 0)
top--;
convex.p[++top] = p[i];
}
int temp = top;
convex.p[++top] = p[n-2];
for(int i = n-3;i >= 0;i--){
while(top != temp && sgn((convex.p[top]-p[i])^(convex.p[top-1]-p[i])) <= 0)
top--;
convex.p[++top] = p[i];
}
if(convex.n == 2 && (convex.p[0] == convex.p[1]))convex.n--;//特判
convex.norm();//原来得到的是顺时针的点，排序后逆时针
}
//得到凸包的另外一种方法
//测试 LightOJ1203  LightOJ1239
void Graham(polygon &convex){
norm();
int &top = convex.n;
top = 0;
if(n == 1){
top = 1;
convex.p[0] = p[0];
return;
}
if(n == 2){
top = 2;
convex.p[0] = p[0];
convex.p[1] = p[1];
if(convex.p[0] == convex.p[1])top--;
return;
}
convex.p[0] = p[0];
convex.p[1] = p[1];
top = 2;
for(int i = 2;i < n;i++){
while( top > 1 && sgn((convex.p[top-1]-convex.p[top-2])^(p[i]-convex.p[top-2])) <= 0 )
top--;
convex.p[top++] = p[i];
}
if(convex.n == 2 && (convex.p[0] == convex.p[1]))convex.n--;//特判
}
//判断是不是凸的
bool isconvex(){
bool s[3];
memset(s,false,sizeof(s));
for(int i = 0;i < n;i++){
int j = (i+1)%n;
int k = (j+1)%n;
s[sgn((p[j]-p[i])^(p[k]-p[i]))+1] = true;
if(s[0] && s[2])return false;
}
return true;
}
//判断点和任意多边形的关系
// 3 点上
// 2 边上
// 1 内部
// 0 外部
int relationpoint(Point q){
for(int i = 0;i < n;i++){
if(p[i] == q)return 3;
}
getline();
for(int i = 0;i < n;i++){
if(l[i].pointonseg(q))return 2;
}
int cnt = 0;
for(int i = 0;i < n;i++){
int j = (i+1)%n;
int k = sgn((q-p[j])^(p[i]-p[j]));
int u = sgn(p[i].y-q.y);
int v = sgn(p[j].y-q.y);
if(k > 0 && u < 0 && v >= 0)cnt++;
if(k < 0 && v < 0 && u >= 0)cnt--;
}
return cnt != 0;
}
//直线u切割凸多边形左侧
//注意直线方向
//测试：HDU3982
void convexcut(Line u,polygon &po){
int &top = po.n;//注意引用
top = 0;
for(int i = 0;i < n;i++){
int d1 = sgn((u.e-u.s)^(p[i]-u.s));
int d2 = sgn((u.e-u.s)^(p[(i+1)%n]-u.s));
if(d1 >= 0)po.p[top++] = p[i];
if(d1*d2 < 0)po.p[top++] = u.crosspoint(Line(p[i],p[(i+1)%n]));
}
}
//得到周长
//测试 LightOJ1239
double getcircumference(){
double sum = 0;
for(int i = 0;i < n;i++){
sum += p[i].distance(p[(i+1)%n]);
}
return sum;
}
//得到面积
double getarea(){
double sum = 0;
for(int i = 0;i < n;i++){
sum += (p[i]^p[(i+1)%n]);
}
return fabs(sum)/2;
}
//得到方向
// 1 表示逆时针，0表示顺时针
bool getdir(){
double sum = 0;
for(int i = 0;i < n;i++)
sum += (p[i]^p[(i+1)%n]);
if(sgn(sum) > 0)return 1;
return 0;
}
//得到重心
Point getbarycentre(){
Point ret(0,0);
double area = 0;
for(int i = 1;i < n-1;i++){
double tmp = (p[i]-p[0])^(p[i+1]-p[0]);
if(sgn(tmp) == 0)continue;
area += tmp;
ret.x += (p[0].x+p[i].x+p[i+1].x)/3*tmp;
ret.y += (p[0].y+p[i].y+p[i+1].y)/3*tmp;
}
if(sgn(area)) ret = ret/area;
return ret;
}
//多边形和圆交的面积
//测试：POJ3675 HDU3982 HDU2892
double areacircle(circle c){
double ans = 0;
for(int i = 0;i < n;i++){
int j = (i+1)%n;
if(sgn( (p[j]-c.p)^(p[i]-c.p) ) >= 0)
ans += c.areatriangle(p[i],p[j]);
else ans -= c.areatriangle(p[i],p[j]);
}
return fabs(ans);
}
//多边形和圆关系
// 2 圆完全在多边形内
// 1 圆在多边形里面，碰到了多边形边界
// 0 其它
int relationcircle(circle c){
getline();
int x = 2;
if(relationpoint(c.p) != 1)return 0;//圆心不在内部
for(int i = 0;i < n;i++){
if(c.relationseg(l[i])==2)return 0;
if(c.relationseg(l[i])==1)x = 1;
}
return x;
}
};
//AB X AC
double cross(Point A,Point B,Point C){
return (B-A)^(C-A);
}
//AB*AC
double dot(Point A,Point B,Point C){
return (B-A)*(C-A);
}
//最小矩形面积覆盖
// A 必须是凸包(而且是逆时针顺序)
// 测试 UVA 10173
double minRectangleCover(polygon A){
//要特判A.n < 3的情况
if(A.n < 3)return 0.0;
A.p[A.n] = A.p[0];
double ans = -1;
int r = 1, p = 1, q;
for(int i = 0;i < A.n;i++){
//卡出离边A.p[i] - A.p[i+1]最远的点
while( sgn( cross(A.p[i],A.p[i+1],A.p[r+1]) - cross(A.p[i],A.p[i+1],A.p[r]) ) >= 0 )
r = (r+1)%A.n;
//卡出A.p[i] - A.p[i+1]方向上正向n最远的点
while(sgn( dot(A.p[i],A.p[i+1],A.p[p+1]) - dot(A.p[i],A.p[i+1],A.p[p]) ) >= 0 )
p = (p+1)%A.n;
if(i == 0)q = p;
//卡出A.p[i] - A.p[i+1]方向上负向最远的点
while(sgn(dot(A.p[i],A.p[i+1],A.p[q+1]) - dot(A.p[i],A.p[i+1],A.p[q])) <= 0)
q = (q+1)%A.n;
double d = (A.p[i] - A.p[i+1]).len2();
double tmp = cross(A.p[i],A.p[i+1],A.p[r]) *
(dot(A.p[i],A.p[i+1],A.p[p]) - dot(A.p[i],A.p[i+1],A.p[q]))/d;
if(ans < 0 || ans > tmp)ans = tmp;
}
return ans;
}

//直线切凸多边形
//多边形是逆时针的，在q1q2的左侧
//测试:HDU3982
vector<Point> convexCut(const vector<Point> &ps,Point q1,Point q2){
vector<Point>qs;
int n = ps.size();
for(int i = 0;i < n;i++){
Point p1 = ps[i], p2 = ps[(i+1)%n];
int d1 = sgn((q2-q1)^(p1-q1)), d2 = sgn((q2-q1)^(p2-q1));
if(d1 >= 0)
qs.push_back(p1);
if(d1 * d2 < 0)
qs.push_back(Line(p1,p2).crosspoint(Line(q1,q2)));
}
return qs;
}
//半平面交
//测试 POJ3335 POJ1474 POJ1279
//***************************
struct halfplane:public Line{
double angle;
halfplane(){}
//表示向量s->e逆时针(左侧)的半平面
halfplane(Point _s,Point _e){
s = _s;
e = _e;
}
halfplane(Line v){
s = v.s;
e = v.e;
}
void calcangle(){
angle = atan2(e.y-s.y,e.x-s.x);
}
bool operator <(const halfplane &b)const{
return angle < b.angle;
}
};
struct halfplanes{
int n;
halfplane hp[maxp];
Point p[maxp];
int que[maxp];
int st,ed;
void push(halfplane tmp){
hp[n++] = tmp;
}
//去重
void unique(){
int m = 1;
for(int i = 1;i < n;i++){
if(sgn(hp[i].angle-hp[i-1].angle) != 0)
hp[m++] = hp[i];
else if(sgn( (hp[m-1].e-hp[m-1].s)^(hp[i].s-hp[m-1].s) ) > 0)
hp[m-1] = hp[i];
}
n = m;
}
bool halfplaneinsert(){
for(int i = 0;i < n;i++)hp[i].calcangle();
sort(hp,hp+n);
unique();
que[st=0] = 0;
que[ed=1] = 1;
p[1] = hp[0].crosspoint(hp[1]);
for(int i = 2;i < n;i++){
while(st<ed && sgn((hp[i].e-hp[i].s)^(p[ed]-hp[i].s))<0)ed--;
while(st<ed && sgn((hp[i].e-hp[i].s)^(p[st+1]-hp[i].s))<0)st++;
que[++ed] = i;
if(hp[i].parallel(hp[que[ed-1]]))return false;
p[ed]=hp[i].crosspoint(hp[que[ed-1]]);
}
while(st<ed && sgn((hp[que[st]].e-hp[que[st]].s)^(p[ed]-hp[que[st]].s))<0)ed--;
while(st<ed && sgn((hp[que[ed]].e-hp[que[ed]].s)^(p[st+1]-hp[que[ed]].s))<0)st++;
if(st+1>=ed)return false;
return true;
}
//得到最后半平面交得到的凸多边形
//需要先调用halfplaneinsert() 且返回true
void getconvex(polygon &con){
p[st] = hp[que[st]].crosspoint(hp[que[ed]]);
con.n = ed-st+1;
for(int j = st,i = 0;j <= ed;i++,j++)
con.p[i] = p[j];
}
};
//***************************

const int maxn = 1010;
struct circles{
circle c[maxn];
double ans[maxn];//ans[i]表示被覆盖了i次的面积
double pre[maxn];
int n;
circles(){}
c[n++] = cc;
}
//x包含在y中
bool inner(circle x,circle y){
if(x.relationcircle(y) != 1)return 0;
return sgn(x.r-y.r)<=0?1:0;
}
//圆的面积并去掉内含的圆
void init_or(){
bool mark[maxn] = {0};
int i,j,k=0;
for(i = 0;i < n;i++){
for(j = 0;j < n;j++)
if(i != j && !mark[j]){
if( (c[i]==c[j])||inner(c[i],c[j]) )break;
}
if(j < n)mark[i] = 1;
}
for(i = 0;i < n;i++)
if(!mark[i])
c[k++] = c[i];
n = k;
}
//圆的面积交去掉内含的圆
int i,j,k;
bool mark[maxn] = {0};
for(i = 0;i < n;i++){
for(j = 0;j < n;j++)
if(i != j && !mark[j]){
if( (c[i]==c[j])||inner(c[j],c[i]) )break;
}
if(j < n)mark[i] = 1;
}
for(i = 0;i < n;i++)
if(!mark[i])
c[k++] = c[i];
n = k;
}
//半径为r的圆，弧度为th对应的弓形的面积
double areaarc(double th,double r){
return 0.5*r*r*(th-sin(th));
}
//测试SPOJVCIRCLES SPOJCIRUT
//SPOJVCIRCLES求n个圆并的面积，需要加上init\_or()去掉重复圆（否则WA）
//SPOJCIRUT 是求被覆盖k次的面积，不能加init\_or()
//对于求覆盖多少次面积的问题，不能解决相同圆，而且不能init\_or()
//求多圆面积并，需要init\_or,其中一个目的就是去掉相同圆
void getarea(){
memset(ans,0,sizeof(ans));
vector<pair<double,int> >v;
for(int i = 0;i < n;i++){
v.clear();
v.push_back(make_pair(-pi,1));
v.push_back(make_pair(pi,-1));
for(int j = 0;j < n;j++)
if(i != j){
Point q = (c[j].p - c[i].p);
double ab = q.len(),ac = c[i].r, bc = c[j].r;
if(sgn(ab+ac-bc)<=0){
v.push_back(make_pair(-pi,1));
v.push_back(make_pair(pi,-1));
continue;
}
if(sgn(ab+bc-ac)<=0)continue;
if(sgn(ab-ac-bc)>0)continue;
double th = atan2(q.y,q.x), fai = acos((ac*ac+ab*ab-bc*bc)/(2.0*ac*ab));
double a0 = th-fai;
if(sgn(a0+pi)<0)a0+=2*pi;
double a1 = th+fai;
if(sgn(a1-pi)>0)a1-=2*pi;
if(sgn(a0-a1)>0){
v.push_back(make_pair(a0,1));
v.push_back(make_pair(pi,-1));
v.push_back(make_pair(-pi,1));
v.push_back(make_pair(a1,-1));
}
else{
v.push_back(make_pair(a0,1));
v.push_back(make_pair(a1,-1));
}
}
sort(v.begin(),v.end());
int cur = 0;
for(int j = 0;j < v.size();j++){
if(cur && sgn(v[j].first-pre[cur])){
ans[cur] += areaarc(v[j].first-pre[cur],c[i].r);
ans[cur] += 0.5*(Point(c[i].p.x+c[i].r*cos(pre[cur]),c[i].p.y+c[i].r*sin(pre[cur]))^Point(c[i].p.x+c[i].r*cos(v[j].first),c[i].p.y+c[i].r*sin(v[j].first)));
}
cur += v[j].second;
pre[cur] = v[j].first;
}
}
for(int i = 1;i < n;i++)
ans[i] -= ans[i+1];
}
};
三维几何
const double eps = 1e-8;
int sgn(double x){
if(fabs(x) < eps)return 0;
if(x < 0)return -1;
else return 1;
}
struct Point3{
double x,y,z;
Point3(double _x = 0,double _y = 0,double _z = 0){
x = _x;
y = _y;
z = _z;
}
void input(){
scanf("%lf%lf%lf",&x,&y,&z);
}
void output(){
printf("%.2lf %.2lf %.2lf\n",x,y,z);
}
bool operator ==(const Point3 &b)const{
return sgn(x-b.x) == 0 && sgn(y-b.y) == 0 && sgn(z-b.z) == 0;
}
bool operator <(const Point3 &b)const{
return sgn(x-b.x)==0?(sgn(y-b.y)==0?sgn(z-b.z)<0:y<b.y):x<b.x;
}
double len(){
return sqrt(x*x+y*y+z*z);
}
double len2(){
return x*x+y*y+z*z;
}
double distance(const Point3 &b)const{
return sqrt((x-b.x)*(x-b.x)+(y-b.y)*(y-b.y)+(z-b.z)*(z-b.z));
}
Point3 operator -(const Point3 &b)const{
return Point3(x-b.x,y-b.y,z-b.z);
}
Point3 operator +(const Point3 &b)const{
return Point3(x+b.x,y+b.y,z+b.z);
}
Point3 operator *(const double &k)const{
return Point3(x*k,y*k,z*k);
}
Point3 operator /(const double &k)const{
return Point3(x/k,y/k,z/k);
}
//点乘
double operator *(const Point3 &b)const{
return x*b.x+y*b.y+z*b.z;
}
//叉乘
Point3 operator ^(const Point3 &b)const{
return Point3(y*b.z-z*b.y,z*b.x-x*b.z,x*b.y-y*b.x);
}
Point3 p = (*this);
return acos( ( (a-p)*(b-p) )/ (a.distance(p)*b.distance(p)) );
}
//变换长度
Point3 trunc(double r){
double l = len();
if(!sgn(l))return *this;
r /= l;
return Point3(x*r,y*r,z*r);
}
};
struct Line3
{
Point3 s,e;
Line3(){}
Line3(Point3 _s,Point3 _e)
{
s = _s;
e = _e;
}
bool operator ==(const Line3 v)
{
return (s==v.s)&&(e==v.e);
}
void input()
{
s.input();
e.input();
}
double length()
{
return s.distance(e);
}
//点到直线距离
double dispointtoline(Point3 p)
{
return ((e-s)^(p-s)).len()/s.distance(e);
}
//点到线段距离
double dispointtoseg(Point3 p)
{
if(sgn((p-s)*(e-s)) < 0 || sgn((p-e)*(s-e)) < 0)
return min(p.distance(s),e.distance(p));
return dispointtoline(p);
}
//返回点p在直线上的投影
Point3 lineprog(Point3 p)
{
return s + ( ((e-s)*((e-s)*(p-s)))/((e-s).len2()) );
}
//p绕此向量逆时针arg角度
Point3 rotate(Point3 p,double ang)
{
if(sgn(((s-p)^(e-p)).len()) == 0)return p;
Point3 f1 = (e-s)^(p-s);
Point3 f2 = (e-s)^(f1);
double len = ((s-p)^(e-p)).len()/s.distance(e);
f1 = f1.trunc(len); f2 = f2.trunc(len);
Point3 h = p+f2;
Point3 pp = h+f1;
return h + ((p-h)*cos(ang)) + ((pp-h)*sin(ang));
}
//点在直线上
bool pointonseg(Point3 p)
{
return sgn( ((s-p)^(e-p)).len() ) == 0 && sgn((s-p)*(e-p)) == 0;
}
};
struct Plane
{
Point3 a,b,c,o;//平面上的三个点，以及法向量
Plane(){}
Plane(Point3 _a,Point3 _b,Point3 _c)
{
a = _a;
b = _b;
c = _c;
o = pvec();
}
Point3 pvec()
{
return (b-a)^(c-a);
}
//ax+by+cz+d = 0
Plane(double _a,double _b,double _c,double _d)
{
o = Point3(_a,_b,_c);
if(sgn(_a) != 0)
a = Point3((-_d-_c-_b)/_a,1,1);
else if(sgn(_b) != 0)
a = Point3(1,(-_d-_c-_a)/_b,1);
else if(sgn(_c) != 0)
a = Point3(1,1,(-_d-_a-_b)/_c);
}
//点在平面上的判断
bool pointonplane(Point3 p)
{
return sgn((p-a)*o) == 0;
}
//两平面夹角
double angleplane(Plane f)
{
return acos(o*f.o)/(o.len()*f.o.len());
}
//平面和直线的交点，返回值是交点个数
int crossline(Line3 u,Point3 &p)
{
double x = o*(u.e-a);
double y = o*(u.s-a);
double d = x-y;
if(sgn(d) == 0)return 0;
p = ((u.s*x)-(u.e*y))/d;
return 1;
}
//点到平面最近点(也就是投影)
Point3 pointtoplane(Point3 p)
{
Line3 u = Line3(p,p+o);
crossline(u,p);
return p;
}
//平面和平面的交线
int crossplane(Plane f,Line3 &u)
{
Point3 oo = o^f.o;
Point3 v = o^oo;
double d = fabs(f.o*v);
if(sgn(d) == 0)return 0;
Point3 q = a + (v*(f.o*(f.a-a))/d);
u = Line3(q,q+oo);
return 1;
}
};
平面最近点对
const int MAXN = 100010;
const double eps = 1e-8;
const double INF = 1e20;
struct Point{
double x,y;
void input(){
scanf("%lf%lf",&x,&y);
}
};
double dist(Point a,Point b){
return sqrt((a.x-b.x)*(a.x-b.x) + (a.y-b.y)*(a.y-b.y));
}
Point p[MAXN];
Point tmpt[MAXN];
bool cmpx(Point a,Point b){
return a.x < b.x || (a.x == b.x && a.y < b.y);
}
bool cmpy(Point a,Point b){
return a.y < b.y || (a.y == b.y && a.x < b.x);
}
double Closest_Pair(int left,int right){
double d = INF;
if(left == right)return d;
if(left+1 == right)return dist(p[left],p[right]);
int mid = (left+right)/2;
double d1 = Closest_Pair(left,mid);
double d2 = Closest_Pair(mid+1,right);
d = min(d1,d2);
int cnt = 0;
for(int i = left;i <= right;i++){
if(fabs(p[mid].x - p[i].x) <= d)
tmpt[cnt++] = p[i];
}
sort(tmpt,tmpt+cnt,cmpy);
for(int i = 0;i < cnt;i++){
for(int j = i+1;j < cnt && tmpt[j].y - tmpt[i].y < d;j++)
d = min(d,dist(tmpt[i],tmpt[j]));
}
return d;
}
int main(){
int n;
while(scanf("%d",&n) == 1 && n){
for(int i = 0;i < n;i++)p[i].input();
sort(p,p+n,cmpx);
printf("%.2lf\n",Closest_Pair(0,n-1));
}
return 0;
}
三维凸包：HDU4273
const double eps = 1e-8;
const int MAXN = 550;
int sgn(double x){
if(fabs(x) < eps)return 0;
if(x < 0)return -1;
else return 1;
}
struct Point3{
double x,y,z;
Point3(double _x = 0, double _y = 0, double _z = 0){
x = _x;
y = _y;
z = _z;
}
void input(){
scanf("%lf%lf%lf",&x,&y,&z);
}
bool operator ==(const Point3 &b)const{
return sgn(x-b.x) == 0 && sgn(y-b.y) == 0 && sgn(z-b.z) == 0;
}
double len(){
return sqrt(x*x+y*y+z*z);
}
double len2(){
return x*x+y*y+z*z;
}
double distance(const Point3 &b)const{
return sqrt((x-b.x)*(x-b.x)+(y-b.y)*(y-b.y)+(z-b.z)*(z-b.z));
}
Point3 operator -(const Point3 &b)const{
return Point3(x-b.x,y-b.y,z-b.z);
}
Point3 operator +(const Point3 &b)const{
return Point3(x+b.x,y+b.y,z+b.z);
}
Point3 operator *(const double &k)const{
return Point3(x*k,y*k,z*k);
}
Point3 operator /(const double &k)const{
return Point3(x/k,y/k,z/k);
}
//点乘
double operator *(const Point3 &b)const{
return x*b.x + y*b.y + z*b.z;
}
//叉乘
Point3 operator ^(const Point3 &b)const{
return Point3(y*b.z-z*b.y,z*b.x-x*b.z,x*b.y-y*b.x);
}
};
struct CH3D{
struct face{
//表示凸包一个面上的三个点的编号
int a,b,c;
//表示该面是否属于最终的凸包上的面
bool ok;
};
//初始顶点数
int n;
Point3 P[MAXN];
//凸包表面的三角形数
int num;
//凸包表面的三角形
face F[8*MAXN];
int g[MAXN][MAXN];
//叉乘
Point3 cross(const Point3 &a,const Point3 &b,const Point3 &c){
return (b-a)^(c-a);
}
//三角形面积*2
double area(Point3 a,Point3 b,Point3 c){
return ((b-a)^(c-a)).len();
}
//四面体有向面积*6
double volume(Point3 a,Point3 b,Point3 c,Point3 d){
return ((b-a)^(c-a))*(d-a);
}
//正：点在面同向
double dblcmp(Point3 &p,face &f){
Point3 p1 = P[f.b] - P[f.a];
Point3 p2 = P[f.c] - P[f.a];
Point3 p3 = p - P[f.a];
return (p1^p2)*p3;
}
void deal(int p,int a,int b){
int f = g[a][b];
if(F[f].ok){
if(dblcmp(P[p],F[f]) > eps)
dfs(p,f);
else {
g[p][b] = g[a][p] = g[b][a] = num;
}
}
}
//递归搜索所有应该从凸包内删除的面
void dfs(int p,int now){
F[now].ok = false;
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){
Point3 &a = P[F[s].a];
Point3 &b = P[F[s].b];
Point3 &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;
}
//构建三维凸包
void create(){
num = 0;

//***********************************
//此段是为了保证前四个点不共面
bool flag = true;
for(int i = 1;i < n;i++){
if(!(P[0] == P[i])){
swap(P[1],P[i]);
flag = false;
break;
}
}
if(flag)return;
flag = true;
for(int i = 2;i < n;i++){
if( ((P[1]-P[0])^(P[i]-P[0])).len() > eps ){
swap(P[2],P[i]);
flag = false;
break;
}
}
if(flag)return;
flag = true;
for(int i = 3;i < n;i++){
if(fabs( ((P[1]-P[0])^(P[2]-P[0]))*(P[i]-P[0]) ) > eps){
swap(P[3],P[i]);
flag = false;
break;
}
}
if(flag)return;
//**********************************

for(int i = 0;i < 4;i++){
}
for(int i = 4;i < n;i++)
for(int j = 0;j < num;j++)
if(F[j].ok && dblcmp(P[i],F[j]) > eps){
dfs(i,j);
break;
}
int tmp = num;
num = 0;
for(int i = 0;i < tmp;i++)
if(F[i].ok)
F[num++] = F[i];
}
//表面积
//测试：HDU3528
double area(){
double res = 0;
if(n == 3){
Point3 p = cross(P[0],P[1],P[2]);
return p.len()/2;
}
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(){
double res = 0;
Point3 tmp = Point3(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);
}
//表面三角形个数
int triangle(){
return num;
}
//表面多边形个数
//测试：HDU3662
int polygon(){
int res = 0;
for(int i = 0;i < num;i++){
bool flag = true;
for(int j = 0;j < i;j++)
if(same(i,j)){
flag = 0;
break;
}
res += flag;
}
return res;
}
//重心
//测试：HDU4273
Point3 barycenter(){
Point3 ans = Point3(0,0,0);
Point3 o = Point3(0,0,0);
double all = 0;
for(int i = 0;i < num;i++){
double vol = volume(o,P[F[i].a],P[F[i].b],P[F[i].c]);
ans = ans + (((o+P[F[i].a]+P[F[i].b]+P[F[i].c])/4.0)*vol);
all += vol;
}
ans = ans/all;
return ans;
}
//点到面的距离
//测试：HDU4273
double ptoface(Point3 p,int i){
double tmp1 = fabs(volume(P[F[i].a],P[F[i].b],P[F[i].c],p));
double tmp2 = ((P[F[i].b]-P[F[i].a])^(P[F[i].c]-P[F[i].a])).len();
return tmp1/tmp2;
}
};
CH3D hull;
int main()
{
while(scanf("%d",&hull.n) == 1){
for(int i = 0;i < hull.n;i++)hull.P[i].input();
hull.create();
Point3 p = hull.barycenter();
double ans = 1e20;
for(int i = 0;i < hull.num;i++)
ans = min(ans,hull.ptoface(p,i));
printf("%.3lf\n",ans);
}
return 0;
}
叉乘性质：

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• // 计算几何模板 const double eps = 1e-8; const double inf = 1e20; const double pi = acos(-1.0); const int maxp = 1010; //Compares a double to zero` int sgn(double x){ if(fabs(x) < eps)re...
• 不要过分相信的我抄的，我在用的时候也经常会遇到错，所以如果不对，还请自己改，或者自己找一份原kuangbin大佬的计算几何板子。 # include using namespace std ; # define me(x,y) memset(x,y,sizeof x)...
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• } 2.2叉积的运用： 2.2.1计算多边形的面积 2.2.2判断凹凸多边形 2.2.3折线的拐向的方向 2.2.4:判断点是否在线段上 代码模板 struct line//线段 { Point s; Point e; line(Point s, Point e){s = a, e = b;} line(){...

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