SPOJ 149 FSHEEP Fencing in the Sheep （ 计算几何 + 二分 ）

以下摘自SPOJ泛做表格：

题意：给定一个星形多边形，而且给出了一个可以看到形内所有点的位置（我们称这个点为观察点），让你判断有多少个点位于多边形内。

时间复杂度：O(mlogn)

将多边形上的点按极角排序，观察点与多边形上任意相邻两点形成若干个三角形，再按照极角查找给定点可能位于哪个三角形，最后用叉积判断它是否真的在那个三角形内。

注意细节，给几组数据：

input

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output

#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <cmath>
#include <algorithm>

using namespace std;

const double eps = 1e-;
const double PI = acos( -1.0 );
const int MAXN = ;

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

struct Point
{
    double x, y;
    double ang;      //极角
    int id;
    Point( double x = , double y =  ):x(x), y(y) { }
    void readPoint()
    {
        scanf("%lf%lf", &x, &y );
        return;
    }
    void GetAngle()
    {
        ang = atan2( y, x );
        if ( dcmp( ang ) <  ) ang = PI + PI+ ang;
        return;
    }
    void showP()
    {
        printf( "(%.6f, %.6f): ang=%.6f id=%d\n", x, y, ang, id );
        return;
    }
};

typedef Point Vector;

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 )   //两点比较小于
{
    return dcmp( A.x - B.x) <  || ( dcmp(A.x - B.x ) ==  && dcmp( A.y - B.y ) <  );
}

bool operator>( const Point& A, const Point& B )   //两点比较大于
{
    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;
}

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 ) ) < ;
}

int N, M;
Point Poly[MAXN];
int Dcnt;

//找到第一个大于的
void BiSearch( double tar, int l, int r, int& u, int& v )
{
    //printf("%.6f %.6f %.6f\n", tar, Poly[1].ang, Poly[N].ang );
    if ( dcmp( tar - Poly[].ang ) <=  || dcmp( tar - Poly[N].ang ) >=  )
    {
        u = N;
        v = ;
        return;
    }
    int mid;
    int ans = N;
    while ( l <= r )
    {
        mid = ( l + r ) >> ;
        if ( dcmp( Poly[mid].ang - tar ) >  )
        {
            ans = mid;
            r = mid - ;
        }
        else l = mid + ;
    }
    v = ans;
    u = v - ;
    return;
}

bool cmp( Point a, Point b )
{
    if ( dcmp( a.ang - b.ang ) !=  )
        return dcmp( a.ang - b.ang ) < ;
    return a.id < b.id;
}

void init()
{
    scanf( "%d%d", &N, &M );
    Dcnt = ;
    for ( int i = ; i <= N; ++i )
    {
        Poly[i].readPoint();
        Poly[i].GetAngle();
        Poly[i].id = i;
    }
    sort( Poly + , Poly +  + N, cmp );
    Poly[] = Poly[N];
    Poly[].ang -= *PI;
    Poly[N+] = Poly[];
    Poly[N+].ang += *PI;
    return;
}

bool check( Point a, Point *p )
{
    if ( a == p[] || a == p[] || a == p[] ) return true;
    if ( OnSegment( a, p[], p[] ) ) return true;
    if ( OnSegment( a, p[], p[] ) ) return true;
    if ( OnSegment( a, p[], p[] ) ) return true;
    int pre = dcmp( Cross( p[] - p[], a - p[] ) );
    for ( int i = ; i < ; ++i )
    {
        int tmp = dcmp( Cross( p[i + ] - p[i], a - p[i] ) );
        if ( tmp != pre ) return false;
    }
    return true;
}

int main()
{
    //freopen("in.txt", "r", stdin);
    //freopen("out.txt", "w", stdout);
    int T;
    scanf( "%d", &T );
    while ( T-- )
    {
        init();

        int ans = ;
        Point sanjiao[];
        sanjiao[] = Point(, );
        for ( int i = ; i < M; ++i )
        {
            Point p;
            p.readPoint();
            p.GetAngle();
            int u, v;
            BiSearch( p.ang, , N, u, v );
            sanjiao[] = Poly[u];
            sanjiao[] = Poly[v];
            //printf( "u=%d v=%d\n", u, v );
            if ( check( p, sanjiao ) )
            {
                ++ans;
                continue;
            }
            while ( dcmp( Poly[u-].ang - p.ang ) ==  && u != v )
            {
                --u;
                if ( u ==  ) u = N;
                //printf( "**u=%d v=%d\n", u, v );
                sanjiao[] = Poly[u];
                sanjiao[] = Poly[v];
                if ( check( p, sanjiao ) )
                {
                    ++ans;
                    break;
                }
            }
        }
        printf( "%d\n", ans );
    }
    return ;
}