UVA 11168 Airport(凸包)

 

UVA 11168 Airport(凸包)

分类： 计算几何 2013-08-27 09:14  245人阅读  评论(1)  收藏  举报

给出平面上的n个点，求一条直线，使得所有点在该直线的同一侧且所有点到该直线的距离和最小，输出该距离和。

要使所有点在该直线的同一侧，明显是直接利用凸包的边更优。所以枚举凸包的没条边，然后求距离和。直线一般式为Ax + By + C = 0.点(x0, y0)到直线的距离为

fabs(Ax0+By0+C)/sqrt(A*A+B*B).由于所有点在直线的同一侧，那么对于所有点，他们的(Ax0+By0+C)符号相同，显然可以累加出sumX和sumY，然后统一求和。

 #include<algorithm>
 #include<iostream>
 #include<cstring>
 #include<cstdlib>
 #include<fstream>
 #include<sstream>
 #include<bitset>
 #include<vector>
 #include<string>
 #include<cstdio>
 #include<cmath>
 #include<stack>
 #include<queue>
 #include<stack>
 #include<map>
 #include<set>
 #define FF(i, a, b) for(int i=a; i<b; i++)
 #define FD(i, a, b) for(int i=a; i>=b; i--)
 #define REP(i, n) for(int i=0; i<n; i++)
 #define CLR(a, b) memset(a, b, sizeof(a))
 #define debug puts("**debug**")
 #define LL long long
 #define PB push_back
 #define eps 1e-10
 using namespace std;

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

 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);
 }

 int dcmp(double x)
 {
     if(fabs(x) < eps) return 0;
     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;
 }

 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 Angel(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; }
 double Area2(Vector A, Vector B, Vector C) { return Cross(B-A, C-A); }


 //向量逆时针旋转
 Vector Rotate(Vector A, double rad)
 {
     return Vector(A.x*cos(rad)-A.y*sin(rad), A.x*sin(rad)+A.y*cos(rad));
 }

 //求直线p+tv和q+tw的交点 Cross(v, w) == 0无交点
 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p到直线ab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);//如果不带fabs 得到的是有向距离
 }

 //点p到线段ab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;
     if(dcmp(Dot(v1, v2) < 0)) return Length(v2);
     else if(dcmp(Dot(v1, v3)) > 0) return Length(v3);
     else return fabs(Cross(v1, v2)) / Length(v1);
 }

 //点p在直线ab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));
 }

 //点段相交判定
 bool SegmentItersection(Point a1, Point a2, Point b1, Point b2)
 {
     double c1 = Cross(a2-a1, b1-a1), c2 = Cross(a2-a1, b2-a1),
     c3 = Cross(b2-b1, a1-b1), c4 = Cross(b2-b1, a2-b1);
     return dcmp(c1)*dcmp(c2) < 0 && dcmp(c3)*dcmp(c4) < 0;
 }

 //点在线段上
 bool OnSegment(Point p, Point a1, Point a2)
 {
     return dcmp(Cross(a1-p, a2-p)) == 0 && dcmp(Dot(a1-p, a2-p)) < 0;
 }

 //多变形面积
 double PolygonArea(Point* p, int n)
 {
     double ret = 0;
     FF(i, 1, n-1) ret += Cross(p[i]-p[0], p[i+1]-p[0]);
     return ret/2;
 }

 Point read_point()
 {
     Point a;
     scanf("%lf%lf", &a.x, &a.y);
     return a;
 }

 double torad(double d)//角度转弧度
 {
     return d/180 *acos(-1);
 }

 int ConvexHull(Point *p, int n, Point* ch)//凸包
 {
     sort(p, p+n);
     int m = 0;
     REP(i, n)
     {
         while(m > 1 && Cross(ch[m-1]-ch[m-2], p[i]-ch[m-2]) <= 0) m--;
         ch[m++] = p[i];
     }
     int k = m;
     FD(i, n-2, 0)
     {
         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;
 }

 void getLineABC(Point A, Point B, double& a, double& b, double& c)//直线两点式转一般式
 {
     a = A.y-B.y, b = B.x-A.x, c = A.x*B.y-A.y*B.x;
 }

 const int maxn = 10001;
 int n, T;
 Point p[maxn], ch[maxn];

 int main()
 {
     scanf("%d", &T);
     FF(kase, 1, T+1)
     {
         scanf("%d", &n);
         double X = 0, Y = 0, ans = 1e10;
         REP(i, n) p[i] = read_point(), X += p[i].x, Y += p[i].y;
         int m = ConvexHull(p, n, ch);
         ch[m] = ch[0];
         REP(i, m)
         {
             double a, b, c;
             getLineABC(ch[i], ch[i+1], a, b, c);
             ans = min(ans, fabs(a*X+b*Y+c*n)/(sqrt(a*a+b*b)));
         }
         printf("Case #%d: %.3f\n", kase, n > 2 ? ans/n : 0);
     }
     return 0;
 }