HDU 4643 GSM 算术几何

当火车处在换基站的临界点时，它到某两基站的距离相等。因此换基站的位置一定在某两个基站的中垂线上，

我们预处理出任意两基站之间的中垂线，对于每次询问，求询问线段与所有中垂线的交点。

检验这些交点是否满足条件（详见代码），如果满足，那么它是一个交换点。

#include <cstdio>
#include <cmath>
#include <vector>
#include <algorithm>

using namespace std;

const int MAXN = ;

const double eps = 1e-;

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

typedef Point Vector;

struct Line
{
    Point s;
    Vector v;
    Line( Point s = Point(), Point v = Point() ):
        s(s), v(v) { }
};

int dcmp( double x )    //控制精度
{
    if ( fabs(x) < eps ) return ;
    else return x <  ? - : ;
}

Vector operator+( Vector A, Vector B )       //向量加
{
    return Vector( A.x + B.x, A.y + B.y );
}

Vector operator-( Vector A, Vector B )       //向量减
{
    return Vector( A.x - B.x, A.y - B.y );
}

Vector operator*( Vector A, double p )      //向量数乘
{
    return Vector( A.x * p, A.y * p );
}

Vector operator/( Vector A, double p )      //向量数除
{
    return Vector( A.x / p, A.y / p );
}

bool operator<( const Point& A, const Point& B )   //两点比较
{
    return dcmp( A.x - B.x ) <  || ( dcmp( A.x - B.x ) ==  && dcmp( A.y - B.y ) <  );
}

bool operator==( const Point& a, const Point& b )   //两点相等
{
    return dcmp( a.x - b.x ) ==  && dcmp( a.y - b.y ) == ;
}

double Dot( Vector A, Vector B )    //向量点乘
{
    return A.x * B.x + A.y * B.y;
}

double Length( Vector A )           //向量模
{
    return sqrt( Dot( A, A ) );
}

double Angle( Vector A, Vector B )    //向量夹角
{
    return acos( Dot(A, B) / Length(A) / Length(B) );
}

double Cross( Vector A, Vector B )   //向量叉积
{
    return A.x * B.y - A.y * B.x;
}

double Area2( Point A, Point B, Point C )    //向量有向面积
{
    return Cross( B - A, C - A );
}

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

Vector Normal( Vector A )    //向量单位法向量
{
    double L = Length(A);
    return Vector( -A.y / L, A.x / L );
}

Point GetLineIntersection( Point P, Vector v, Point Q, Vector w )   //两直线交点
{
    Vector u = P - Q;
    double t = Cross( w, u ) / Cross( v, w );
    return P + v * t;
}

double DistanceToLine( Point P, Point A, Point B )    //点到直线的距离
{
    Vector v1 = B - A, v2 = P - A;
    return fabs( Cross( v1, v2 ) ) / Length(v1);
}

double DistanceToSegment( Point P, Point A, Point B )   //点到线段的距离
{
    if ( A == B ) return Length( P - A );
    Vector v1 = B - A, v2 = P - A, v3 = P - B;
    if ( dcmp( Dot(v1, v2) ) <  ) return Length(v2);
    else if ( dcmp( Dot(v1, v3) ) >  ) return Length(v3);
    else return fabs( Cross( v1, v2 ) ) / Length(v1);
}

Point GetLineProjection( Point P, Point A, Point B )    // 点在直线上的投影
{
    Vector v = B - A;
    return A + v*( Dot(v, P - A) / Dot( v, v ) );
}

bool SegmentProperIntersection( Point a1, Point a2, Point b1, Point b2 )  //线段相交，交点不在端点
{
    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) <  && dcmp(c3) * dcmp(c4) < ;
}

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

/****************以上模板******************/

int N, M;
Point city[MAXN];      //城市
Point GSM[MAXN];       //基站
Line L[MAXN][MAXN];    //点[i][j]之间的中垂线

void init()
{
    for ( int i = ; i <= M; ++i )
        for ( int j = i + ; j <= M; ++j )
        {
            Point mid = Point( (GSM[i].x+GSM[j].x)/2.0, (GSM[i].y+GSM[j].y)/2.0 );
            L[i][j] = Line( mid, Normal( GSM[j] - GSM[i] ) );
            L[j][i] = L[i][j];
        }
    return;
}

//判断交点是否在线段上
bool check( Point st, Point ed, Point cp )
{
    return ( st < cp || st == cp ) && ( cp < ed || cp == ed );
}

//假设我在此交点交换基站
//那么交点到形成 该中垂线的线段的其中一端点 的距离 L 应该是最小的
//判断是否有点到交点的距离小于L，如果有，则不是在这一点交换的基站
bool check2( double limit, Point jiao )
{
    for ( int i = ; i <= M; ++i )
    {
        double dis = Length( GSM[i] - jiao );
        if ( dcmp( dis - limit ) <  ) return false;
    }
    return true;
}

int main()
{
    //freopen( "in.txt", "r", stdin );
    //freopen( "s.txt", "w", stdout );
    while ( ~scanf( "%d%d", &N, &M ) )
    {
        for ( int i = ; i <= N; ++i )
            scanf( "%lf%lf", &city[i].x, &city[i].y );

        for ( int i = ; i <= M; ++i )
            scanf( "%lf%lf", &GSM[i].x, &GSM[i].y );

        init();   //初始化所有中垂线
        int Q;
        scanf( "%d", &Q );
        while ( Q-- )
        {
            int a, b;
            scanf( "%d%d", &a, &b );
            if ( a > b ) swap( a, b );
            Line train = Line( city[a], city[b] - city[a] );  //火车行进路线
            int huan = ;         //换基站次数
            for ( int i = ; i <= M; ++i )
                for ( int j = i + ; j <= M; ++j )
                {
                    if ( dcmp( Cross( train.v, L[i][j].v ) ) ==  ) //如果中垂线与火车行进路线平行
                        continue;
                    Point tmp = GetLineIntersection( train.s, train.v, L[i][j].s, L[i][j].v );  //求交点//交点到形成中垂线的线段的其中一个端点的距离
                    double limit = Length( GSM[i] - tmp );
                    Point st = city[a], ed = city[b];
                    if ( ed < st ) swap( st, ed );

                    if ( check( st, ed, tmp ) )  //如果在线段上
                    {
                        if ( check2( limit, tmp ) ) //如果确实在这点交换基站
                            ++huan;
                    }
                }
            printf( "%d\n", huan );
        }
    }
    return ;
}