hdu 3629 Convex

题意：给你N个点，让你选四个点组成凸多边形，求总的方法数

详细解释：http://blog.sina.com.cn/s/blog_64675f540100ksug.html

  1 #include<iostream>
  2 #include<vector>
  3 #include<cstring>
  4 #include<cstdio>
  5 #include<cmath>
  6 #include<stdlib.h>
  7 #include<queue>
  8 #include<map>
  9 #include<algorithm>
 10 using namespace std;
 11 const double eps = 1e-12;
 12 const double pi = acos(-1.0);
 13 const double INF = 10001000;
 14 double sqr(double x){
 15     return x*x;
 16 }
 17 int cmp(double x){
 18     if(fabs(x)<eps) return 0;
 19     if(x>0) return 1;
 20     return -1;
 21 }
 22 struct point{
 23     double x,y;
 24     int index;
 25     point(){}
 26     point(double a,double b):x(a),y(b){}
 27     void input(){
 28         scanf("%lf%lf",&x,&y);
 29     }
 30     double angle() {
 31         return atan2(y, x);
 32     }
 33     friend point operator + (const point &a,const point &b){
 34         return point(a.x+b.x,a.y+b.y);
 35     }
 36     friend point operator - (const point &a,const point &b){
 37         return point(a.x-b.x,a.y-b.y);
 38     }
 39     friend bool operator == (const point &a,const point &b){
 40         return (cmp(a.x-b.x)==0)&&(cmp(a.y-b.y))==0;
 41     }
 42     friend point operator * (const point &a,double b){
 43         return point(a.x*b,a.y*b);
 44     }
 45     friend point operator / (const point &a,double b){
 46         return point(a.x/b,a.y/b);
 47     }
 48     friend double operator ^ (const point &a,const point &b)
 49     {
 50         return a.x*b.y - a.y*b.x;
 51     }
 52      double operator *(const point &b)const
 53     {
 54         return x*b.x + y*b.y;
 55     }
 56     double norm(){                              //向量的模长
 57         return sqrt(sqr(x)+sqr(y));
 58     }
 59 };
 60
 61 double det(const point &a,const point &b){      //向量的叉集
 62     return a.x*b.y-a.y*b.x;
 63 }
 64 double dot(const point &a,const point &b){      //向量的点集
 65     return a.x*b.x+a.y*b.y;
 66 }
 67 double dot(const point &a,const point &b,const point &c){      //向量的点集ba 到bc
 68     return dot(a-b,c-b);
 69 }
 70 double dist(const point &a,const point &b){     //两点间的距离
 71     return (a-b).norm();
 72 }
 73 point rotate_point(const point &p,double A){    // 绕原点逆时针旋转 A（弧度）
 74     double tx=p.x,ty=p.y;
 75     return point(tx*cos(A)-ty*sin(A),tx*sin(A)+ty*cos(A));
 76 }
 77 //向量的旋转 //底边线段ab 绕a逆时针旋转角度A，b->b1，sinl是sinA的值。
 78 point rotate_point(double cosl,double sinl,point a, point b){
 79     b.x -= a.x; b.y -= a.y;
 80     point c;
 81     c.x = b.x * cosl - b.y * sinl + a.x;
 82     c.y = b.x * sinl + b.y * cosl + a.y;
 83    return c;
 84 }
 85 double xml(point x,point t1,point t2){    // 如果值为正，（t1-x）在（t2-x）的瞬时间方向
 86     return det((t1-x),(t2-x));
 87 }
 88 double area(point x,point y,point z){
 89     return (det(y-x,z-x));
 90 }
 91 struct line {
 92     point a,b;
 93     line(){}
 94     line(point x,point y):a(x),b(y){}
 95 };
 96 point P_chuizhi_line(point a,point l1,point l2)    // 求一个点，使得ac垂直于l1l2
 97 {
 98     point c;
 99     l2.x -= l1.x; l2.y -= l1.y;
100     c.x = a.x - l1.x - l2.y + l1.x;
101     c.y = a.y - l1.y + l2.x + l1.y;
102     return c;
103 }
104 point P_To_seg(point P,line L)                  //点到线段 最近的一个点
105 {
106     point result;
107     double t = ((P-L.a)*(L.b-L.a))/((L.b-L.a)*(L.b-L.a));
108     if(t >= 0 && t <= 1)
109     {
110         result.x = L.a.x + (L.b.x - L.a.x)*t;
111         result.y = L.a.y + (L.b.y - L.a.y)*t;
112     }
113     else
114     {
115         if(dist(P,L.a) < dist(P,L.b))
116             result = L.a;
117         else result = L.b;
118     }
119     return result;
120 }
121 double dis_p_to_line(point p,line l){         //点到直线的距离
122     return fabs(area(p,l.a,l.b))/dist(l.a,l.b);
123 }
124 double dis_p_to_seg(point p,line l)            //点到线段的距离
125 {
126    return dist(p,P_To_seg(p,l));
127 }
128 double dis_pall_seg(point p1, point p2, point p3, point p4)  //平行线段之间的最短距离
129 {
130     return min(min(dis_p_to_seg(p1,line(p3,p4)),
131            dis_p_to_seg(p2, line(p3, p4))),
132            min(dis_p_to_seg(p3,line(p1, p2)),
133            dis_p_to_seg(p4,line(p1, p2)))
134         );
135 }
136 bool intbr(line l1,line l2) {            //    线段相交
137     return
138     max(l1.a.x,l1.b.x) >= min(l2.a.x,l2.b.x) &&
139     max(l2.a.x,l2.b.x) >= min(l1.a.x,l1.b.x) &&
140     max(l1.a.y,l1.b.y) >= min(l2.a.y,l2.b.y) &&
141     max(l2.a.y,l2.b.y) >= min(l1.a.y,l1.b.y) &&
142     cmp((l2.a-l1.a)^(l1.b-l1.a))*cmp((l2.b-l1.a)^(l1.b-l1.a)) <= 0 &&
143     cmp((l1.a-l2.a)^(l2.b-l2.a))*cmp((l1.b-l2.a)^(l2.b-l2.a)) <= 0;
144 }
145 point line_inter(point A,point B,point C,point D){ //直线相交交点
146         point ans;
147         double a1=A.y-B.y;
148         double b1=B.x-A.x;
149         double c1=A.x*B.y-B.x*A.y;
150
151         double a2=C.y-D.y;
152         double b2=D.x-C.x;
153         double c2=C.x*D.y-D.x*C.y;
154
155         ans.x=(b1*c2-b2*c1)/(a1*b2-a2*b1);
156         ans.y=(a2*c1-a1*c2)/(a1*b2-a2*b1);
157         return ans;
158 }
159
160 int n;
161 point ttmp;
162 point pt1[1001000],pt2[1001000];
163 bool cmpx(point xx,point yy){
164     if(cmp(xx.y-yy.y)==0) return xx.x<yy.x;
165     return xx.y<yy.y;
166 }
167 bool cmpd(point xx, point yy){
168     double db=(xx-ttmp)^(yy-ttmp);
169     if(cmp(db)==0) return dist(xx,ttmp)<dist(yy,ttmp);
170     if(cmp(db)>0) return 1;
171     else return 0;
172 }
173 point grp1[1001000],grp2[1001000];
174 int Graham(point* grp,point *pt,int n){             //凸包
175     int top=1;
176     sort(pt,pt+n,cmpx);ttmp=pt[0];
177     sort(pt+1,pt+n,cmpd);
178     grp[0]=pt[0];
179     grp[1]=pt[1];
180     for(int i=2;i<n;i++){
181         while(top>0){
182             double db=(pt[i]-grp[top])^(grp[top]-grp[top-1]);
183             if(cmp(db)>=0) top--;
184             else break;
185         }
186         grp[++top]=pt[i];
187     }
188     return top+1;
189 }
190 double rotating_calipers(point* grp ,int len){ //旋转卡壳求凸包直径
191     int i=0,j=1;
192     double ans=0;
193     while(i<len){
194         while(area(grp[i],grp[i+1],grp[(j+1)%len])>
195               area(grp[i],grp[i+1],grp[j])) j=(j+1)%len;
196         ans=max(ans,max(dist(grp[i],grp[j]),dist(grp[i+1],grp[j])));
197         i++;
198     }
199    return ans;
200 }
201 double rotating_calipers2(point* grp1,int len1,point* grp2,int len2){             //旋转卡壳 求两个凸包的最远距离
202     int p=0,q=0;
203     for(int i=0;i<len1;i++) if(grp1[i].y<grp1[p].y) p=i;
204     for(int i=0;i<len2;i++) if(grp2[i].y>grp2[q].y) q=i;
205     double ans=1e99,tmp;
206     grp1[len1]=grp1[0];//避免取模
207     grp2[len2]=grp2[0];//避免取模
208     for(int i=0;i<len1;i++){
209         while(tmp=cmp(area(grp1[p],grp1[p+1],grp2[q+1])-
210               area(grp1[p],grp1[p+1],grp2[q]))>0)   q=(q+1)%len2;
211         if(tmp==0) ans=min(ans,dis_pall_seg(grp1[p],grp1[p+1],grp2[q],grp2[q+1]));
212         else ans=min(ans,dis_p_to_seg(grp2[q],line(grp1[p],grp1[p+1])));
213         p=(p+1)%len1;
214     }
215
216     return ans;
217 }
218 double rotating_calipers3(point* grp ,int len){             //旋转卡壳求凸包宽度
219     int p=0,q=0;
220     int tmp;
221     for(int i=0;i<len;i++) if(grp[i].y<grp[p].y) p=i;
222     for(int j=0;j<len;j++) if(grp[j].y>grp[q].y) q=j;
223     double ans=1e30;
224     for(int i=0;i<len;i++){
225         while(tmp=cmp(area(grp[p],grp[p+1],grp[(q+1)%len])-
226               area(grp[p],grp[p+1],grp[q]))>0) q=(q+1)%len;
227         ans=min(ans,dis_p_to_line(grp[q],line(grp[p],grp[(p+1)%len])));
228         p=(p+1)%len;
229     }
230     return ans;
231 }
232 double rotating_calipers4(point* grp,int len){
233     double ans;
234     int xmin=0,xmax=0,ymin=0,ymax=0;
235     for(int i=0;i<len;i++) if(cmp(grp[xmin].x-grp[i].x)>0) xmin=i;
236     for(int i=0;i<len;i++) if(cmp(grp[xmax].x-grp[i].x)<0) xmax=i;
237     for(int i=0;i<len;i++) if(cmp(grp[ymin].y-grp[i].y)>0) ymin=i;
238     for(int i=0;i<len;i++) if(cmp(grp[ymax].y-grp[i].y)<0) ymax=i;
239     ans=(grp[ymax].y-grp[ymin].y)*(grp[xmax].x-grp[xmin].x);
240     grp[len]=grp[0];
241     for(int i=0;i<len;i++){
242         while(cmp(area(grp[ymin],grp[ymin+1],grp[ymax+1])-
243                   area(grp[ymin],grp[ymin+1],grp[ymax]))>=0) ymax=(ymax+1)%len;
244         while(cmp(dot(grp[xmax+1],grp[ymin],grp[ymin+1])-
245                   dot(grp[xmax],grp[ymin],grp[ymin+1]))>=0) xmax=(xmax+1)%len;
246         if(i==0) xmin=xmax;
247         while(cmp(dot(grp[xmin+1],grp[ymin+1],grp[ymin])-
248                   dot(grp[xmin],grp[ymin+1],grp[ymin]))>=0) xmin=(xmin+1)%len;
249         double L1=dis_p_to_line(grp[ymax],line(grp[ymin],grp[ymin+1]));
250         point a=P_chuizhi_line(grp[xmin],grp[ymin],grp[ymin+1]);
251         double L2=dis_p_to_line(grp[xmax],line(grp[xmin],a));
252         if(ans>L1*L2){
253             ans=L1*L2;
254
255         }
256         ymin=(ymin+1)%len;
257     }
258     return ans;
259 }
260 struct node{
261     double angle;
262 }fuck[1000];
263 bool nodecmp(node a,node b){
264     return a.angle<b.angle;
265 }
266 int main(){
267     #ifndef ONLINE_JUDGE
268     freopen("input.txt","r",stdin);
269     #endif // ONLINE_JUDGE
270     int t;cin>>t;
271     while(t--){
272         scanf("%d",&n);
273         for(int i=0;i<n;i++) pt1[i].input();
274         long long ans=1ll*n*(n-1)*(n-2)*(n-3)/24;
275         for(int i=0;i<n;i++){
276             int cnt=0;
277             for(int j=0;j<n;j++) if(i!=j){
278                 fuck[cnt++].angle=(pt1[j]-pt1[i]).angle()+pi;
279             }
280             sort(fuck,fuck+cnt,nodecmp);
281             for(int j=cnt;j<2*cnt;j++) fuck[j].angle=fuck[j-cnt].angle+2*pi;
282             int st=1;
283             long long tmp=0;
284             for(int j=0;j<cnt;j++){
285                 while(fuck[st].angle-fuck[j].angle<pi) st++;
286                 if(st-j-1<2) continue;
287                 tmp+=(st-j-1)*(st-j-2)/2;
288             }
289             ans-=(1ll*(n-1)*(n-2)*(n-3)/6-tmp);
290         }
291         printf("%lld\n",ans);
292     }
293 }