//
// main.cpp
// demo
//
// Created by Yanbin GONG on 14/4/2018.
// Copyright © 2018 Yanbin GONG. All rights reserved.
//
//向量的基本运算
#include <cmath>
#include <vector>
using namespace std;
//基本定义
struct Point{
double x,y;
Point(double x=0, double y=0):x(x),y(y){}//构造函数方便代码编写
};
typedef Point Vector; //程序实现上, Vector只是Point的别名（因为起点挪到了原点）
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);}
// const &的作用是直接引用但是不改变，会节约内存
bool operator < (const Point& a, const Point& b){
return a.x<b.x || (a.x==b.x&&a.y<b.y);
}
const double eps = 1e-10; //设置精度在小数点后十位
//如果两个数的差距小于这个数字就当做他们相等
//判断这个数是为0，还是小于0，还是大于0
int dcmp(double x){
//fabs为绝对值函数
if(fabs(x)<eps)return 0; //fabs在cmath里
else return x<0? -1:1;
}
bool operator == (const Point& a, const Point& b){
return (dcmp(a.x-b.x)==0&&dcmp(a.y-b.y)==0);
}
//向量基本运算
double Dot(Vector A, Vector B) {return A.x*B.x + A.y*B.y;}//点积
double Length(Vector A) {return sqrt(Dot(A,A));}//自身乘积再开根号保证绝对值稳定性
double Angle(Vector A, Vector B) {return acos(Dot(A,B)/Length(A)/Length(B));}
//叉乘
double Cross(Vector A, Vector B) {return A.x*B.y-A.y*B.x;}
double Area2(Point A, Point B, Point C) {return Cross(B-A, C-A);}//相当于上面的为原点，为面积的两倍
//角度转弧度
double torad(double deg)
{
return deg/180*acos(-1);
}
//旋转
Vector Rotate(Vector A, double rad) {
return Vector(A.x*cos(rad)-A.y*sin(rad),A.x*sin(rad)+A.y*cos(rad));
}
//单位法线
Vector Normal(Vector A){
double L = Length(A);
return Vector(-A.y/L,A.x/L);
}
//点和直线
//两条直线的交点
//一条直线可以写成一个点和一个向量（方向）
//
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;
}
//点到直线的距离
double DistanceToLine(Point P, Point A, Point B){
Vector v1=B-A, v2=P-A;
return fabs(Cross(v1,v2))/Length(v1); //如果不取绝对值，得到的是有向距离
}
//点到线段的距离
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;
//投影不在线段上的情况
if(dcmp(Dot(v1,v2))<0) return Length(v2); //P在靠A侧
else if(dcmp(Dot(v1,v3))>0) return Length(v3); //在靠近B的一侧
else return fabs(Cross(v1,v2))/Length(v1);
}
//点在直线上的投影
Point GetLineProjection(Point P, Point A, Point B){
Vector v=B-A;
return A+v*(Dot(v,P-A)/Dot(v,v)); //从A移动到投影
}
//线段相交判定 相交为1 （交点不为任何一线段的端点）
bool SegmentProperIntersection(Point a1, Point a2, Point b1, Point b2){
double c1 = Cross(a2-a1,b1-a1);
double c2 = Cross(a2-a1,b2-a1);
double c3 = Cross(b2-b1,a1-b1);
double c4 = Cross(b2-b1,a2-b1);
return dcmp(c1)*dcmp(c2)<0 && dcmp(c3)*dcmp(c4)<0;
}
//判断一个点是否在一条线段上（用于判断一个端点是否在另一个线段上）
//如果c1 c2窦唯0，则线段共线
bool OnSegment(Point p, Point a1, Point a2){
return dcmp(Cross(a1-p, a2-p))==0 && dcmp(Dot(a1-p, a2-p))<0;
}
//与圆和球有关的计算问题
struct Line{
Point p;//线上一点
Vector v;//方向向量
double ang; //极角，从x正半轴旋转到v所需要的角（弧度）
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{ //排序用的比较运算符
return ang < L.ang;
}
};
struct Circle{
Point c;
double r;
Circle(Point c, double r):c(c),r(r){}
Point point(double a){ //通过圆心角求坐标的函数
return Point(c.x+cos(a)*r,c.y+sin(a)*r);
}
};
//直线与圆的交点
//sol存放的是交点本身，代码没有清空sol，就很方便：可以反复调用把所有交点放在一个sol里
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;//判别式
if(dcmp(delta)<0) return 0; //相离
if(dcmp(delta)==0){
t1=t2=-f/(2*e);
sol.push_back(L.point(t1));
return 1;
}
//相交
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;
}
//计算向量极角
double angle(Vector v){return atan2(v.y,v.x);}
//两圆相交
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; //内含
if(dcmp(fabs(C1.r-C2.r)-d)>0) return 0; //外离
double a = angle(C2.c-C1.c); //向量C1C2的极角
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;
}
//过定点作圆切线,v[i]是第i条切线，返回切线条数
int getTangents(Point p, Circle C, Vector* v){
Vector u = C.c-p;
double dist = Length(u);
if(dist<C.r) return 0;
else if(dcmp(dist-C.r)==0){ //p在圆上，只有一条切线
v[0]=Rotate(u,M_PI/2);
return 1;
}
else{
double ang = asin(C.r/dist);
v[0] = Rotate(u, -ang);
v[1] = Rotate(u, +ang);
return 2;
}
}
//两圆的公切线
int getTangents(Circle A, Circle B, Point* a, Point* b){
int cnt=0;
if(A.r<B.r){
swap(A,B);
swap(a,b);
}
int d2=(A.c.x-B.c.x)*(A.c.x-B.c.x)+(A.c.y-B.c.y)*(A.c.y-B.c.y);
int rdiff=A.r-B.r;
int rsum=A.r+B.r;
if(d2<rdiff*rdiff) return 0; //内含
double base = atan2(B.c.y-A.c.y,B.c.x-A.c.x);
if(d2==0&&A.r==B.r) return -1; //无限多条切线
if(d2==rdiff*rdiff){//内切，一条切线
a[cnt]=A.point(base);
b[cnt]=B.point(base);
cnt++;
return 1;
}
//有外共切线
double ang = acos(A.r-B.r)/sqrt(d2);
a[cnt] = A.point(base+ang);
b[cnt] = B.point(base+ang);
cnt++;
a[cnt] = A.point(base+ang);
b[cnt] = B.point(base-ang);
cnt++;
if(d2==rsum*rsum){
a[cnt]=A.point(base);
b[cnt]=B.point(M_PI+base);
cnt++;
}
else if(d2>rsum*rsum){
double ang=acos((A.r+B.r)/sqrt(d2));
a[cnt]=A.point(base+ang);
b[cnt]=B.point(M_PI+base+ang);
cnt++;
a[cnt]=A.point(base-ang);
b[cnt]=B.point(M_PI+base-ang);
cnt++;
}
return cnt;
}
//三角形外接圆（三点保证不共线）
Circle CircumscribedCircle(Point p1, Point p2, Point p3){
double Bx = p2.x-p1.x, By = p2.y-p1.y;
double Cx = p3.x-p1.x, Cy = p3.y-p1.y;
double D = 2*(Bx*Cy-By*Cx);
double cx = (Cy*(Bx*Bx+By*By)-By*(Cx*Cx+Cy*Cy))/D+p1.x;
double cy = (Bx*(Cx*Cx+Cy*Cy)-Cx*(Bx*Bx+By*By))/D+p1.y;
Point p = Point(cx,cy);
return Circle(p,Length(p1-p));
}
//三角形内切圆
Circle InscribedCircle(Point p1, Point p2, Point p3){
double a = Length(p2-p3);
double b = Length(p3-p1);
double c = Length(p1-p2);
Point p = (p1*a+p2*b+p3*c)/(a+b+c);
return Circle(p, DistanceToLine(p, p1, p2));
}
//二维几何常用算法
typedef vector<Point> Polygon;
//多边形的有向面积
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;
}
//点在多边形内判定
int isPointInPolygon(Point p, Polygon poly){
int wn = 0; //绕数
int n = poly.size();
for(int i=0;i<n;i++){
if(OnSegment(p,poly[i],poly[(i+1)%n])) return -1;//边界上
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
}
//凸包
//Andrew算法
bool myCmp(Point A, Point B)
{
if(A.x != B.x) return A.x < B.x;
else return A.y < B.y;
}
int ConvexHall (Point* p, int n, Point* ch){
sort(p,p+n,myCmp); //先比较x坐标，再比较y坐标
int m = 0;
for(int i=0;i<n;i++){
while(m>1&&Cross(ch[m-1]-ch[m-2], p[i]-ch[m-2])<=0) m--;
ch[m++] = p[i];
}
if(n>1)m--;
return m;
}
//凸包(Andrew算法)
//如果不希望在凸包的边上有输入点，把两个 <= 改成 <
//如果不介意点集被修改，可以改成传递引用
Polygon ConvexHull(vector<Point> p) {
//预处理，删除重复点
sort(p.begin(), p.end());
p.erase(unique(p.begin(), p.end()), p.end());
int n = p.size(), m = 0;
Polygon res(n+1);
for(int i = 0; i < n; i++) {
while(m > 1 && Cross(res[m-1]-res[m-2], p[i]-res[m-2]) <= 0) m--;
res[m++] = p[i];
}
int k = m;
for(int i = n-2; i >= 0; i--) {
while(m > k && Cross(res[m-1]-res[m-2], p[i]-res[m-2]) <= 0) m--;
res[m++] = p[i];
}
m -= n > 1;
res.resize(m);
return res;
}
1 //
2 // main.cpp
3 // demo
4 //
5 // Created by Yanbin GONG on 14/4/2018.
6 // Copyright © 2018 Yanbin GONG. All rights reserved.
7 //
8
9 //向量的基本运算
10
11 #include <cmath>
12 #include <vector>
13
14 using namespace std;
15
16
17 //基本定义
18 struct Point{
19 double x,y;
20 Point(double x=0, double y=0):x(x),y(y){}//构造函数方便代码编写
21 };
22 typedef Point Vector; //程序实现上, Vector只是Point的别名（因为起点挪到了原点）
23
24 Vector operator + (Vector A, Vector B) {return Vector(A.x+B.x,A.y+B.y);}
25 Vector operator - (Vector A, Vector B) {return Vector(A.x-B.x,A.y-B.y);}
26 Vector operator * (Vector A, double p) {return Vector(A.x*p,A.y*p);}
27 Vector operator / (Vector A, double p) {return Vector(A.x/p,A.y/p);}
28
29 // const &的作用是直接引用但是不改变，会节约内存
30 bool operator < (const Point& a, const Point& b){
31 return a.x<b.x || (a.x==b.x&&a.y<b.y);
32 }
33
34 const double eps = 1e-10; //设置精度在小数点后十位
35 //如果两个数的差距小于这个数字就当做他们相等
36
37 //判断这个数是为0，还是小于0，还是大于0
38 int dcmp(double x){
39 //fabs为绝对值函数
40 if(fabs(x)<eps)return 0; //fabs在cmath里
41 else return x<0? -1:1;
42 }
43
44 bool operator == (const Point& a, const Point& b){
45 return (dcmp(a.x-b.x)==0&&dcmp(a.y-b.y)==0);
46 }
47
48 //向量基本运算
49 double Dot(Vector A, Vector B) {return A.x*B.x + A.y*B.y;}//点积
50 double Length(Vector A) {return sqrt(Dot(A,A));}//自身乘积再开根号保证绝对值稳定性
51 double Angle(Vector A, Vector B) {return acos(Dot(A,B)/Length(A)/Length(B));}
52
53 //叉乘
54 double Cross(Vector A, Vector B) {return A.x*B.y-A.y*B.x;}
55 double Area2(Point A, Point B, Point C) {return Cross(B-A, C-A);}//相当于上面的为原点，为面积的两倍
56
57 //角度转弧度
58 double torad(double deg)
59 {
60 return deg/180*acos(-1);
61 }
62 //旋转
63 Vector Rotate(Vector A, double rad) {
64 return Vector(A.x*cos(rad)-A.y*sin(rad),A.x*sin(rad)+A.y*cos(rad));
65 }
66
67 //单位法线
68 Vector Normal(Vector A){
69 double L = Length(A);
70 return Vector(-A.y/L,A.x/L);
71 }
72
73 //点和直线
74
75 //两条直线的交点
76 //一条直线可以写成一个点和一个向量（方向）
77 //
78 Point GetLineIntersection(Point P, Vector v, Point Q, Vector w){
79 Vector u = P-Q;
80 double t = Cross(w,u)/Cross(v,w);
81 return P+v*t;
82 }
83
84 //点到直线的距离
85 double DistanceToLine(Point P, Point A, Point B){
86 Vector v1=B-A, v2=P-A;
87 return fabs(Cross(v1,v2))/Length(v1); //如果不取绝对值，得到的是有向距离
88 }
89
90 //点到线段的距离
91 double DistanceToSegment(Point P, Point A, Point B){
92 if(A==B) return Length(P-A);
93 Vector v1=B-A, v2=P-A, v3=P-B;
94 //投影不在线段上的情况
95 if(dcmp(Dot(v1,v2))<0) return Length(v2); //P在靠A侧
96 else if(dcmp(Dot(v1,v3))>0) return Length(v3); //在靠近B的一侧
97 else return fabs(Cross(v1,v2))/Length(v1);
98 }
99
100 //点在直线上的投影
101 Point GetLineProjection(Point P, Point A, Point B){
102 Vector v=B-A;
103 return A+v*(Dot(v,P-A)/Dot(v,v)); //从A移动到投影
104 }
105
106 //线段相交判定 相交为1 （交点不为端点）
107 bool SegmentProperIntersection(Point a1, Point a2, Point b1, Point b2){
108 double c1 = Cross(a2-a1,b1-a1);
109 double c2 = Cross(a2-a1,b2-a1);
110 double c3 = Cross(b2-b1,a1-b1);
111 double c4 = Cross(b2-b1,a2-b1);
112 return dcmp(c1)*dcmp(c2)<0 && dcmp(c3)*dcmp(c4)<0;
113 }
114 //判断一个点是否在一条线段上（用于判断一个端点是否在另一个线段上）
115 //如果c1 c2窦唯0，则线段共线
116 bool OnSegment(Point p, Point a1, Point a2){
117 return dcmp(Cross(a1-p, a2-p))==0 && dcmp(Dot(a1-p, a2-p))<0;
118 }
119
120 //与圆和球有关的计算问题
121
122 struct Line{
123 Point p;//线上一点
124 Vector v;//方向向量
125 double ang; //极角，从x正半轴旋转到v所需要的角（弧度）
126 Line(Point p, Vector v):p(p),v(v){ang = atan2(v.y,v.x);}
127 Point point(double t){return p+v*t;};
128 bool operator < (const Line& L) const{ //排序用的比较运算符
129 return ang < L.ang;
130 }
131 };
132
133 struct Circle{
134 Point c;
135 double r;
136 Circle(Point c, double r):c(c),r(r){}
137 Point point(double a){ //通过圆心角求坐标的函数
138 return Point(c.x+cos(a)*r,c.y+sin(a)*r);
139 }
140 };
141
142 //直线与圆的交点
143 //sol存放的是交点本身，代码没有清空sol，就很方便：可以反复调用把所有交点放在一个sol里
144 int getLineCircleIntersection(Line L, Circle C, double& t1, double& t2, vector<Point>& sol){
145 double a=L.v.x, b=L.p.x-C.c.x, c=L.v.y, d=L.p.y-C.c.y;
146 double e=a*a+c*c, f=2*(a*b+c*d), g=b*b+d*d-C.r*C.r;
147 double delta = f*f - 4*e*g;//判别式
148 if(dcmp(delta)<0) return 0; //相离
149 if(dcmp(delta)==0){
150 t1=t2=-f/(2*e);
151 sol.push_back(L.point(t1));
152 return 1;
153 }
154 //相交
155 t1 = (-f-sqrt(delta))/(2*e);
156 sol.push_back(L.point(t1));
157 t2 = (-f+sqrt(delta))/(2*e);
158 sol.push_back(L.point(t2));
159 return 2;
160 }
161
162 //计算向量极角
163 double angle(Vector v){return atan2(v.y,v.x);}
164
165 //两圆相交
166 int getCircleCircleIntersection(Circle C1, Circle C2, vector<Point>& sol){
167 double d=Length(C1.c-C2.c);
168 if(dcmp(d)==0){
169 if(dcmp(C1.r-C2.r)==0) return -1; //两圆重合
170 return 0;
171 }
172 if(dcmp(C1.r+C2.r-d)<0) return 0; //内含
173 if(dcmp(fabs(C1.r-C2.r)-d)>0) return 0; //外离
174
175 double a = angle(C2.c-C1.c); //向量C1C2的极角
176 double da = acos((C1.r*C1.r+d*d-C2.r*C2.r)/(2*C1.r*d));
177 //C1C2到C1P1的角
178 Point p1 = C1.point(a-da), p2 = C1.point(a+da);
179
180 sol.push_back(p1);
181 if(p1==p2)return 1;
182 sol.push_back(p2);
183 return 2;
184 }
185
186 //过定点作圆切线,v[i]是第i条切线，返回切线条数
187 int getTangents(Point p, Circle C, Vector* v){
188 Vector u = C.c-p;
189 double dist = Length(u);
190 if(dist<C.r) return 0;
191 else if(dcmp(dist-C.r)==0){ //p在圆上，只有一条切线
192 v[0]=Rotate(u,M_PI/2);
193 return 1;
194 }
195 else{
196 double ang = asin(C.r/dist);
197 v[0] = Rotate(u, -ang);
198 v[1] = Rotate(u, +ang);
199 return 2;
200 }
201 }
202
203 //两圆的公切线
204 int getTangents(Circle A, Circle B, Point* a, Point* b){
205 int cnt=0;
206 if(A.r<B.r){
207 swap(A,B);
208 swap(a,b);
209 }
210 int d2=(A.c.x-B.c.x)*(A.c.x-B.c.x)+(A.c.y-B.c.y)*(A.c.y-B.c.y);
211 int rdiff=A.r-B.r;
212 int rsum=A.r+B.r;
213 if(d2<rdiff*rdiff) return 0; //内含
214 double base = atan2(B.c.y-A.c.y,B.c.x-A.c.x);
215 if(d2==0&&A.r==B.r) return -1; //无限多条切线
216 if(d2==rdiff*rdiff){//内切，一条切线
217 a[cnt]=A.point(base);
218 b[cnt]=B.point(base);
219 cnt++;
220 return 1;
221 }
222 //有外共切线
223 double ang = acos(A.r-B.r)/sqrt(d2);
224 a[cnt] = A.point(base+ang);
225 b[cnt] = B.point(base+ang);
226 cnt++;
227 a[cnt] = A.point(base+ang);
228 b[cnt] = B.point(base-ang);
229 cnt++;
230 if(d2==rsum*rsum){
231 a[cnt]=A.point(base);
232 b[cnt]=B.point(M_PI+base);
233 cnt++;
234 }
235 else if(d2>rsum*rsum){
236 double ang=acos((A.r+B.r)/sqrt(d2));
237 a[cnt]=A.point(base+ang);
238 b[cnt]=B.point(M_PI+base+ang);
239 cnt++;
240 a[cnt]=A.point(base-ang);
241 b[cnt]=B.point(M_PI+base-ang);
242 cnt++;
243 }
244 return cnt;
245 }
246
247 //三角形外接圆（三点保证不共线）
248 Circle CircumscribedCircle(Point p1, Point p2, Point p3){
249 double Bx = p2.x-p1.x, By = p2.y-p1.y;
250 double Cx = p3.x-p1.x, Cy = p3.y-p1.y;
251 double D = 2*(Bx*Cy-By*Cx);
252 double cx = (Cy*(Bx*Bx+By*By)-By*(Cx*Cx+Cy*Cy))/D+p1.x;
253 double cy = (Bx*(Cx*Cx+Cy*Cy)-Cx*(Bx*Bx+By*By))/D+p1.y;
254 Point p = Point(cx,cy);
255 return Circle(p,Length(p1-p));
256 }
257 //三角形内切圆
258 Circle InscribedCircle(Point p1, Point p2, Point p3){
259 double a = Length(p2-p3);
260 double b = Length(p3-p1);
261 double c = Length(p1-p2);
262 Point p = (p1*a+p2*b+p3*c)/(a+b+c);
263 return Circle(p, DistanceToLine(p, p1, p2));
264 }
265
266
267 //二维几何常用算法
268 typedef vector<Point> Polygon;
269 //多边形的有向面积
270 double PolygonArea(Polygon po) {
271 int n = po.size();
272 double area = 0.0;
273 for(int i = 1; i < n-1; i++) {
274 area += Cross(po[i]-po[0], po[i+1]-po[0]);
275 }
276 return area * 0.5;
277 }
278
279 //点在多边形内判定
280 int isPointInPolygon(Point p, Polygon poly){
281 int wn = 0; //绕数
282 int n = poly.size();
283 for(int i=0;i<n;i++){
284 if(OnSegment(p,poly[i],poly[(i+1)%n])) return -1;//边界上
285 int k = dcmp(Cross(poly[(i+1)%n]-poly[i], p-poly[i]));
286 int d1 = dcmp(poly[i].y-p.y);
287 int d2 = dcmp(poly[(i+1)%n].y-p.y);
288 if(k>0&&d1<=0&&d2>0) wn++;
289 if(k<0&&d2<=0&&d1>0) wn--;
290 }
291 if(wn!=0) return 1;//内部
292 return 0;//外部
293 }
294
295 //凸包
296 //Andrew算法
297 bool myCmp(Point A, Point B)
298 {
299 if(A.x != B.x) return A.x < B.x;
300 else return A.y < B.y;
301 }
302
303 int ConvexHall (Point* p, int n, Point* ch){
304 sort(p,p+n,myCmp); //先比较x坐标，再比较y坐标
305 int m = 0;
306 for(int i=0;i<n;i++){
307 while(m>1&&Cross(ch[m-1]-ch[m-2], p[i]-ch[m-2])<=0) m--;
308 ch[m++] = p[i];
309 }
310 if(n>1)m--;
311 return m;
312 }
