计算几何基础（模板）

1.多边形面积计算

1     double S(Point p[],int n)
2     {
3         double ans = 0;
4         p[n] = p[0];
5         for(int i=1;i<n;i++)
6            ans += cross(p[0],p[i],p[i+1]);
7         if(ans < 0) ans = -ans;
8         return ans / 2.0;
9     }  

2.求凸包

 1     bool cmp(Point A,Point B)
 2     {
 3         double k = cross(MinA,A,B);
 4         if(k<0) return 0;
 5         if(k>0) return 1;
 6         return dist(MinA,A)<dist(MinA,B);
 7     }
 8
 9     void Graham(Point p[],int n)
10     {
11         for(int i=1;i<n;i++)
12            if(p[i].y<p[0].y || (p[i].y == p[0].y && p[i].x < p[0].x))
13                 swap(p[i],p[0]);
14         MinA = p[0];
15         p[n] = p[0];
16         sort(p+1,p+n,cmp);
17         stack[0] = p[0];
18         stack[1] = p[1];
19         top = 2;
20         for(int i=2;i<n;i++)
21         {
22             while(top >= 2 && cross(stack[top-2],stack[top-1],p[i])<=0) top--;
23             stack[top++] = p[i];
24         }
25     }  

3.任意多边形求重心

 1     Point Gravity(Point p[],int n)
 2     {
 3         Point O,t;
 4         O.x = O.y = 0;
 5         t.x = t.y = 0;
 6         p[n] = p[0];
 7         double A = 0;
 8         for(int i=0; i<n; i++)
 9             A += cross(O,p[i],p[i+1]);
10         A /= 2.0;
11         for(int i=0; i<n; i++)
12         {
13             t.x += (p[i].x + p[i+1].x) * cross(O,p[i],p[i+1]);
14             t.y += (p[i].y + p[i+1].y) * cross(O,p[i],p[i+1]);
15         }
16         t.x /= 6*A;
17         t.y /= 6*A;
18         return t;
19     }  

4.求线段交点的坐标

 1     bool Segment_crossing(Segment u,Segment v)   /** 判断线段是否相交 */
 2     {
 3              return((max(u.a.x,u.b.x)>=min(v.a.x,v.b.x))&&
 4                (max(v.a.x,v.b.x)>=min(u.a.x,u.b.x))&&
 5                (max(u.a.y,u.b.y)>=min(v.a.y,v.b.y))&&
 6                (max(v.a.y,v.b.y)>=min(u.a.y,u.b.y))&&
 7                (cross(v.a,u.b,u.a)*cross(u.b,v.b,u.a)>=0)&&
 8                (cross(u.a,v.b,v.a)*cross(v.b,u.b,v.a)>=0));
 9     }
10
11     /**求直线交点的坐标，如果没有交点返回NULL，否则返回交点p的地址*/
12     Point* CrossPoint(Segment u,Segment v)
13     {
14         Point p;
15         if(Segment_crossing(u,v))
16         {
17             p.x=(cross(v.b,u.b,u.a)*v.a.x-cross(v.a,u.b,u.a)*v.b.x)/(cross(v.b,u.b,u.a)-cross(v.a,u.b,u.a));
18             p.y=(cross(v.b,u.b,u.a)*v.a.y-cross(v.a,u.b,u.a)*v.b.y)/(cross(v.b,u.b,u.a)-cross(v.a,u.b,u.a));
19             return &p;
20         }
21         return NULL;
22     }  

5.三角形外接圆的半径与圆心

 1     Point Circle_Point(Point A,Point B,Point C)
 2     {
 3         double a=dist(B,C);
 4         double b=dist(A,C);
 5         double c=dist(A,B);
 6         double p=(a+b+c)/2.0;
 7         double S=sqrt(p*(p-a)*(p-b)*(p-c));
 8         R=(a*b*c)/(4*S);    //三角形外接圆的半径为R
 9
10         double t1=(A.x*A.x+A.y*A.y-B.x*B.x-B.y*B.y)/2;
11         double t2=(A.x*A.x+A.y*A.y-C.x*C.x-C.y*C.y)/2;
12         Point center;
13         center.x=(t1*(A.y-C.y)-t2*(A.y-B.y))/((A.x-B.x)*(A.y-C.y)-(A.x-C.x)*(A.y-B.y));
14         center.y=(t1*(A.x-C.x)-t2*(A.x-B.x))/((A.y-B.y)*(A.x-C.x)-(A.y-C.y)*(A.x-B.x));
15         return center;
16     }  

6.旋转卡壳求凸包的直径，也就是平面最远点对，p[]为凸包的点集

 1     double rotating_calipers(Point p[],int n)
 2     {
 3         int k = 1;
 4         double ans = 0;
 5         p[n] = p[0];
 6         for(int i=0;i<n;i++)
 7         {
 8             while(fabs(cross(p[i],p[i+1],p[k])) < fabs(cross(p[i],p[i+1],p[k+1])))
 9                  k = (k+1) % n;
10             ans = max(ans, max(dist(p[i],p[k]),dist(p[i+1],p[k])));
11         }
12         return ans;
13     }  

7.求凸包的宽度

 1     double rotating_calipers(Point p[],int n)
 2     {
 3         int k = 1;
 4         double ans = 0x7FFFFFFF;
 5         p[n] = p[0];
 6         for(int i=0;i<n;i++)
 7         {
 8             while(fabs(cross(p[i],p[i+1],p[k])) < fabs(cross(p[i],p[i+1],p[k+1])))
 9                  k = (k+1) % n;
10             double tmp = fabs(cross(p[i],p[i+1],p[k]));
11             double d   = dist(p[i],p[i+1]);
12             ans = min(ans,tmp/d);
13         }
14         return ans;
15     }