1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
| struct Point{
double x,y;
Point(double x=0, double y=0):x(x),y(y) {}
void input() { scanf("%lf%lf",&x,&y); }
};
typedef Point Vector;
struct Circle{
Point c;
double r;
Circle(){}
Circle(Point c,double r):c(c),r(r) {}
Point point(double a) { return Point(c.x + cos(a)*r, c.y + sin(a)*r); }
void input() { scanf("%lf%lf%lf",&c.x,&c.y,&r); }
};
struct Line{
Point p;
Vector v;
double ang;
Line(){}
Line(Point p, Vector v):p(p),v(v) { ang = atan2(v.y,v.x); }
Point point(double t) { return Point(p.x + t*v.x, p.y + t*v.y); }
bool operator < (const Line &L)const { return ang < L.ang; }
};
int dcmp(double x) {
if(x < -eps) return -1;
if(x > eps) return 1;
return 0;
}
template <class T> T sqr(T x) { return x * x;}
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 a.x >= b.x && a.y >= b.y; }
bool operator <= (const Point& a, const Point& b) { return a.x <= b.x && a.y <= b.y; }
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; }
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 Cross(Vector A, Vector B) { return A.x*B.y - A.y*B.x; }
Vector VectorUnit(Vector x){ return x / Length(x);}
Vector Normal(Vector x) { return Point(-x.y, x.x) / Length(x);}
double angle(Vector v) { return atan2(v.y, v.x); }
bool OnSegment(Point P, Point A, Point B) { //端点也算
return dcmp(Cross(A-P,B-P)) == 0 && dcmp(Dot(A-P,B-P)) <= 0;
}
double DistanceToSeg(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);
}
double DistanceToLine(Point P, Point A, Point B) {
Vector v1 = B-A, v2 = P-A;
return fabs(Cross(v1,v2)) / Length(v1);
}
Point GetLineIntersection(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;
}
double DisP(Point A,Point B) {
return Length(B-A);
}
bool SegmentIntersection(Point A,Point B,Point C,Point D) {
return max(A.x,B.x) >= min(C.x,D.x) &&
max(C.x,D.x) >= min(A.x,B.x) &&
max(A.y,B.y) >= min(C.y,D.y) &&
max(C.y,D.y) >= min(A.y,B.y) &&
dcmp(Cross(C-A,B-A)*Cross(D-A,B-A)) <= 0 &&
dcmp(Cross(A-C,D-C)*Cross(B-C,D-C)) <= 0;
}
bool LineSegmentIntersection(Point A,Point B,Point C,Point D) {
return dcmp(Cross(C-A,B-A)*Cross(D-A,B-A)) <= 0;
}
void SegIntersectionPoint(Point& P,Point a,Point b,Point c,Point d) { //需保证ab,cd相交
P.x = (Cross(d-a,b-a)*c.x - Cross(c-a,b-a)*d.x)/(Cross(d-a,b-a)-Cross(c-a,b-a));
P.y = (Cross(d-a,b-a)*c.y - Cross(c-a,b-a)*d.y)/(Cross(d-a,b-a)-Cross(c-a,b-a));
}
Vector Rotate(Point P,Vector A,double rad){ //以P为基准点把向量A旋转rad
return Vector(P.x+A.x*cos(rad)-A.y*sin(rad),P.y+A.x*sin(rad)+A.y*cos(rad));
}
//点是否在多边形内部（环顾法）
int CheckPointInPolygon(Point A,Point* p,int n){
double TotalAngle = 0.0;
for(int i=0;i<n;i++) {
if(dcmp(Cross(p[i]-A,p[(i+1)%n]-A)) >= 0) TotalAngle += Angle(p[i]-A,p[(i+1)%n]-A);
else TotalAngle -= Angle(p[i]-A,p[(i+1)%n]-A);
}
if(dcmp(TotalAngle) == 0) return 0; //外部
else if(dcmp(fabs(TotalAngle)-2*pi) == 0) return 1; //完全内部
else if(dcmp(fabs(TotalAngle)-pi) == 0) return 2; //边界上
else return 3; //多边形顶点
}
//射线法
int Ray_PointInPolygon(Point A,Point* p,int n) {
int wn = 0;
for(int i=0;i<n;i++) {
//if(OnSegment(A,p[i],p[(i+1)%n])) return -1; //边界
int k = dcmp(Cross(p[(i+1)%n]-p[i], A-p[i]));
int d1 = dcmp(p[i].y-A.y);
int d2 = dcmp(p[(i+1)%n].y-A.y);
if(k > 0 && d1 <= 0 && d2 > 0) wn++;
if(k < 0 && d2 <= 0 && d1 > 0) wn--;
}
if(wn) return 1; //内部
return 0; //外部
}
//判断未知时针方向的多边形是否是凸包
bool CheckConvexHull(Point* p,int n){
int dir = 0; //旋转方向
for(int i=0;i<n;i++) {
int nowdir = dcmp(Cross(p[(i+1)%n]-p[i],p[(i+2)%n]-p[i]));
if(!dir) dir = nowdir;
if(dir*nowdir < 0) return false; //非凸包
}
return true;
}
//////凸包
int ConvexHull(Point* p, int n, Point* ch) {
sort(p,p+n);
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];
}
int k = m;
for(int i=n-2;i>=0;i--) {
while(m > k && Cross(ch[m-1]-ch[m-2], p[i]-ch[m-2]) <= 0) m--;
ch[m++] = p[i];
}
if(n > 1) m--;
return m;
}
double CalcConvexArea(Point* p,int n) { //凸包面积
double area = 0.0;
for(int i=1;i<n-1;i++)
area += Cross(p[i]-p[0],p[i+1]-p[0]);
return fabs(area*0.5);
}
double CalcConvexLength(Point* p,int n) {
double Len = 0.0;
for(int i=0;i<n;i++) Len += Length(p[(i+1)%n]-p[i]);
return Len;
}
//////////旋转卡壳求凸包最远两点
double RotatingCalipers(Point* ch,int n) { //旋转卡壳
int p,q = 1;
double ans = 0.0;
ch[n] = ch[0];
for(p=0;p<n;p++) {
while(dcmp(Cross(ch[p+1]-ch[p],ch[q+1]-ch[p])-Cross(ch[p+1]-ch[p],ch[q]-ch[p])) > 0)
q = (q+1)%n;
ans = max(ans,max(DisP(ch[p],ch[q]),DisP(ch[p+1],ch[q+1])));
}
return ans*ans;
}
double MinDisOfTwoConvexHull(Point P[],int n,Point Q[],int m) { //旋转卡壳求两个顺时针凸包的最近距离
int Pymin = 0, Qymax = 0, i,j;
for(i=0;i<n;i++) if(dcmp(P[i].y-P[Pymin].y) < 0) Pymin = i;
for(i=0;i<m;i++) if(dcmp(Q[i].y-Q[Qymax].y) > 0) Qymax = i;
P[n] = P[0], Q[m] = Q[0];
double Mindis = Mod, Tmp;
for(i=0;i<n;i++) {
while(dcmp(Tmp = Cross(P[Pymin+1]-P[Pymin],Q[Qymax+1]-P[Pymin])-Cross(P[Pymin+1]-P[Pymin],Q[Qymax]-P[Pymin])) > 0)
Qymax = (Qymax+1)%m;
if(dcmp(Tmp) < 0) Mindis = min(Mindis,DistanceToSeg(Q[Qymax],P[Pymin],P[Pymin+1]));
else Mindis = min(Mindis,SegDistancetoSeg(P[Pymin],P[Pymin+1],Q[Qymax],Q[Qymax+1]));
Pymin = (Pymin+1)%n;
}
return Mindis;
}
bool OnLeft(Line L, Point p) { return dcmp(Cross(L.v,p-L.p)) > 0; }
Point* p;
bool CmpPolarPoint(Point a,Point b) { //点极角排序
int d = dcmp(Cross(a-p[0],b-p[0]));
if(!d) return DisP(p[0],a) < DisP(p[0],b);
return d > 0;
}
bool CmpPolarLine(Line a,Line b) { //直线极角排序
return angle(a.v) < angle(b.v);
}
void GetL(bool counter,Point* p,int n,Line* L) { //多边形的边转为直线
if(counter) { for(int i=n-1;i>=0;i--) L[n-i-1] = Line(p[(i+1)%n],p[i]-p[(i+1)%n]); }
else { for(int i=0;i<n;i++) L[i] = Line(p[i],p[(i+1)%n]-p[i]); }
}
int HalfPlaneIntersection(Line* L, int n, Point* poly) { //半平面交点存入poly
sort(L,L+n,CmpPolarLine);
int first,last;
Point *p = new Point[n];
Line *q = new Line[n];
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(dcmp(Cross(q[last].v, q[last-1].v)) == 0) {
last--;
if(OnLeft(q[last], L[i].p)) q[last] = L[i];
}
if(first < last) p[last-1] = GetLineIntersection(q[last-1],q[last]);
}
while(first < last && !OnLeft(q[first],p[last-1])) last--;
if(last-first <= 1) return 0; //点或线或无界平面，返回0
p[last] = GetLineIntersection(q[last],q[first]);
int m = 0;
for(int i=first;i<=last;i++) poly[m++] = p[i];
delete p; delete q;
return m;
}
int LineCrossPolygon(Point& L1,Point& L2,Point* p,int n,Point* poly) { //直线(L1,L2)切割多边形p，形成新的多边形poly
int m = 0;
for(int i=0,j;i<n;i++) {
if(dcmp(Cross(L1-p[i],L2-p[i])) >= 0) { poly[m++] = p[i]; continue; }
j = (i-1+n)%n;
if(dcmp(Cross(L1-p[j],L2-p[j])) > 0) poly[m++] = GetLineIntersection(Line(L1,L2-L1),Line(p[j],p[i]-p[j]));
j = (i+1+n)%n;
if(dcmp(Cross(L1-p[j],L2-p[j])) > 0) poly[m++] = GetLineIntersection(Line(L1,L2-L1),Line(p[j],p[i]-p[j]));
}
return m;
}
//////圆
bool InCircle(Point x, Circle c) { return dcmp(c.r - Length(c.c-x)) > 0; } //not in border
int GetCircleCircleIntersection(Circle C1, Circle C2, vector<Point>& sol) //return 交点个数
{
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); //向量C1C2的极角
double da = acos((sqr(C1.r) + sqr(d) - sqr(C2.r)) / (2*C1.r*d)); //C1C2到C1P1的极角
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;
}
int GetSegCircleIntersection(Line L, Circle C, Point* sol)
{
Vector Noml = Normal(L.v);
Line PL = Line(C.c, Noml);
Point IP = GetLineIntersection(PL, L); //弦的中点
double Dis = Length(IP - C.c);
if(dcmp(Dis-C.r) > 0) return 0; //在圆外
Vector HalfChord = VectorUnit(L.v)*sqrt(sqr(C.r)-sqr(Dis));
int ind = 0;
sol[ind] = IP + HalfChord;
if(OnSegment(sol[ind],L.p,L.point(1))) ind++;
sol[ind] = IP - HalfChord;
if(OnSegment(sol[ind],L.p,L.point(1))) ind++;
return ind;
}
Point Zero = Point(0,0);
double TriAngleCircleInsection(Circle C, Point A, Point B)
{
Vector OA = A-C.c, OB = B-C.c;
Vector BA = A-B, BC = C.c-B;
Vector AB = B-A, AC = C.c-A;
double DOA = Length(OA), DOB = Length(OB),DAB = Length(AB), r = C.r;
if(dcmp(Cross(OA,OB)) == 0) return 0;
if(dcmp(DOA-C.r) < 0 && dcmp(DOB-C.r) < 0) return Cross(OA,OB)*0.5;
else if(DOB < r && DOA >= r) {
double x = (Dot(BA,BC) + sqrt(r*r*DAB*DAB-Cross(BA,BC)*Cross(BA,BC)))/DAB;
double TS = Cross(OA,OB)*0.5;
return asin(TS*(1-x/DAB)*2/r/DOA)*r*r*0.5+TS*x/DAB;
}
else if(DOB >= r && DOA < r) {
double y = (Dot(AB,AC)+sqrt(r*r*DAB*DAB-Cross(AB,AC)*Cross(AB,AC)))/DAB;
double TS = Cross(OA,OB)*0.5;
return asin(TS*(1-y/DAB)*2/r/DOB)*r*r*0.5+TS*y/DAB;
}
else if(fabs(Cross(OA,OB)) >= r*DAB || Dot(AB,AC) <= 0 || Dot(BA,BC) <= 0) {
if(Dot(OA,OB) < 0) {
if(Cross(OA,OB) < 0) return (-acos(-1.0)-asin(Cross(OA,OB)/DOA/DOB))*r*r*0.5;
else return ( acos(-1.0)-asin(Cross(OA,OB)/DOA/DOB))*r*r*0.5;
}
else return asin(Cross(OA,OB)/DOA/DOB)*r*r*0.5;
}
else {
double x = (Dot(BA,BC)+sqrt(r*r*DAB*DAB-Cross(BA,BC)*Cross(BA,BC)))/DAB;
double y = (Dot(AB,AC)+sqrt(r*r*DAB*DAB-Cross(AB,AC)*Cross(AB,AC)))/DAB;
double TS = Cross(OA,OB)*0.5;
return (asin(TS*(1-x/DAB)*2/r/DOA)+asin(TS*(1-y/DAB)*2/r/DOB))*r*r*0.5 + TS*((x+y)/DAB-1);
}
}
|
