HDU

题目链接：http://acm.hdu.edu.cn/showproblem.php?pid=5130

题目大意：求多边形内的某个点P，使dis(P,B)<=dis(P,A)，问P所构成的面积

解题思路：把上面的方程化开，可以得到一个圆的不等式，题目于是转化成既在圆内，又在多边形内的点所构成的面积，即两者相交的面积(为了使表达式简单点，我把每个坐标都进行了平移，使A的坐标为(0,0))，对于几何模板有不懂得可以参考博客：http://blog.csdn.net/Nightmare_ak/article/details/77944442

AC代码：

#include<cstdio>#include<cmath>#include<iostream>#include<algorithm>#include<iomanip>#include<vector>using namespace std;const int MAXN = 200000 + 5;const double PI = 3.1415926;const double EPS = 1e-10;double R;//反演圆半径int judgeZero(double a)//a>0为1,<0为-1，=0为0{    return (a > EPS) - (a < -EPS);}struct Point{    double _x, _y;    Point(double x = 0.0, double y = 0.0) :_x(x), _y(y) {}    Point(const Point& p) { _x = p._x, _y = p._y; }    bool operator<(const Point& p)const    {        if (judgeZero(_x - p._x) != 0) return _x < p._x;        return _y < p._y;    }    bool operator==(const Point& p)const    {        return judgeZero(_x - p._x) == 0 && judgeZero(_y - p._y) == 0;    }    void toMove(Point a, double rad, double d)    {        _x = a._x + cos(rad)*d;        _y = a._y + sin(rad)*d;    }    Point operator+(Point a) { return Point(_x + a._x, _y + a._y); }    Point operator-(Point a) { return Point(_x - a._x, _y - a._y); }    friend Point operator*(double a, Point p) { return Point(a*p._x, a*p._y); }    friend istream& operator >> (istream& in, Point& point)    {        in >> point._x >> point._y;        return in;    }    friend ostream& operator<<(ostream& out, const Point& point)    {        out << fixed << setprecision(8) << point._x << ' ' << point._y;        return out;    }}crossp[MAXN];double getDis(Point a, Point b){    return sqrt((a._x - b._x)*(a._x - b._x) + (a._y - b._y)*(a._y - b._y));}typedef vector<Point> Polygon;struct Circle{    Point _o;    double _r;    Circle(double x = 0.0, double y = 0.0, double r = 0.0) :_o(x, y), _r(r) {}    Circle(const Point& o, double r) :_o(o), _r(r) {}    Circle getAnti(const Point& point)    {        Circle antic;        double dis = getDis(point, _o);        double tmp = R*R / (dis*dis - _r*_r);        antic._r = tmp*_r;        antic._o._x = point._x + tmp*(_o._x - point._x);        antic._o._y = point._y + tmp*(_o._y - point._y);        return antic;    }    friend istream& operator >> (istream& in, Circle& circle)    {        in >> circle._o >> circle._r;        return in;    }    friend ostream& operator<<(ostream& out, const Circle& circle)    {        out << circle._o << ' ' << circle._r;        return out;    }};double getCross(Point p1, Point p2, Point p){    return (p1._x - p._x)*(p2._y - p._y) - (p2._x - p._x)*(p1._y - p._y);}bool onSegment(Point p, Point a, Point b)//点p与线段ab{    if (judgeZero(getCross(a, b, p)) != 0)        return false;    if (p._x<min(a._x, b._x) || p._x>max(a._x, b._x))//trick：垂直或平行坐标轴;        return false;    if (p._y<min(a._y, b._y) || p._y>max(a._y, b._y))        return false;    return true;}bool segmentIntersect(Point a, Point b, Point c, Point d)//线段ab与线段cd{    if (onSegment(c, a, b) || onSegment(d, a, b) || onSegment(a, c, d) || onSegment(b, c, d))        return true;    if (judgeZero(getCross(a, b, c)*getCross(a, b, d)) < 0 && judgeZero(getCross(c, d, a)*getCross(c, d, b)) < 0)//==0的特殊情况已经在前面排除        return true;    return false;}bool onLine(Point p, Point a, Point b)//点p与直线ab{    return judgeZero(getCross(a, b, p)) == 0;}bool lineIntersect(Point a, Point b, Point c, Point d)//直线ab与直线cd{    if (judgeZero((a._x - b._x)*(c._y - d._y) - (c._x - d._x)*(a._y - b._y)) != 0)        return true;    if (onLine(a, c, d))//两直线重合        return true;    return false;}bool segment_intersectLine(Point a, Point b, Point c, Point d)//线段ab与直线cd{    if (judgeZero(getCross(c, d, a)*getCross(c, d, b)) <= 0)        return true;    return false;}Point point_line_intersectLine(Point p1, Point p2, Point p3, Point p4)//求出交点,直线(线段)p1p2与直线(线段)p3p4{//t=lamta/(lamta+1),必须用t取代lamta,不然算lamta可能分母为0    double x1 = p1._x, y1 = p1._y;    double x2 = p2._x, y2 = p2._y;    double x3 = p3._x, y3 = p3._y;    double x4 = p4._x, y4 = p4._y;    double t = ((x2 - x1)*(y3 - y1) - (x3 - x1)*(y2 - y1)) / ((x2 - x1)*(y3 - y4) - (x3 - x4)*(y2 - y1));    return Point(x3 + t*(x4 - x3), y3 + t*(y4 - y3));}bool inPolygon(Point a, Polygon polygon)//点是否含于多边形{    Point b(-1e15 + a._x, a._y);//向左无穷远的线段    int count = 0;    for (int i = 0;i < polygon.size();++i)    {        Point c = polygon[i], d = polygon[(i + 1) % polygon.size()];        if (onSegment(a, c, d)) //如果点在线段上，那么一定在多边形内            return true;        if (judgeZero((a._x - b._x)*(c._y - d._y) - (c._x - d._x)*(a._y - b._y)) == 0)            continue;//如果边与射线平行,trick1        if (!segmentIntersect(a, b, c, d))            continue;//如果边与射线没有交点        Point lower;        if (c._y < d._y) lower = c;        else lower = d;        if (onSegment(lower, a, b)) //如果纵坐标小的点与射线有交点,trick2            continue;        count++;    }    return count % 2 == 1;}bool segment_inPolygon(Point a, Point b, Polygon polygon)//线段是否含于多边形{    if (!inPolygon(a, polygon) || !inPolygon(b, polygon))//两个端点都不在多边形内        return false;    int tot = 0;    for (int i = 0;i < polygon.size();++i)    {        Point c = polygon[i], d = polygon[(i + 1) % polygon.size()];        if (onSegment(a, c, d))//以下只用记录一个交点，多的要么重复，要么一定在边上            crossp[tot++] = a;        else if (onSegment(b, c, d))            crossp[tot++] = b;        else if (onSegment(c, a, b))            crossp[tot++] = c;        else if (onSegment(d, a, b))            crossp[tot++] = d;        else if (segmentIntersect(a, b, c, d))//端点没有在线段上且相交            return false;    }    sort(crossp, crossp + tot);//按x，y排序    tot = unique(crossp, crossp + tot) - crossp;    for (int i = 0;i < tot - 1;++i)    {        Point tmp = 0.5*(crossp[i] + crossp[i + 1]);        if (!inPolygon(tmp, polygon))//中点不在多边形内            return false;    }    return true;}double lineDis_inPolygon(Point a, Point b, Polygon polygon)//求出直线在多边形内的长度{    int tot = 0;    for (int i = 0;i < polygon.size();++i)    {        Point c = polygon[i], d = polygon[(i + 1) % polygon.size()];        if (onLine(c, a, b) && onLine(d, a, b))//如果边与直线重合，记录两个端点        {            crossp[tot++] = c;            crossp[tot++] = d;        }        else if (segment_intersectLine(c, d, a, b))            crossp[tot++] = point_line_intersectLine(a, b, c, d);    }    sort(crossp, crossp + tot);    tot = unique(crossp, crossp + tot) - crossp;    double ans = 0.0;    for (int i = 0;i < tot - 1;++i)    {        Point tmp = 0.5*(crossp[i] + crossp[i + 1]);        if (inPolygon(tmp, polygon))            ans += getDis(crossp[i], crossp[i + 1]);    }    return ans;}bool inCircle(Point p, Circle a)//点是否在圆内{    return judgeZero(getDis(p, a._o) - a._r) <= 0;}double dis_toSegment(Point c, Point a, Point b)//点到线段的最短距离{    Point d = a - b;    double tmp = d._y;//得到与ab垂直的向量    d._y = d._x;    d._x = -tmp;    d.toMove(c, atan2(d._y, d._x), 10.0);    Point intsect = point_line_intersectLine(a, b, c, d);    if (onSegment(intsect, a, b))        return getDis(c, intsect);    return min(getDis(c, a), getDis(c, b));}Polygon dividePolygon(Polygon p, Point a, Point b)//直线ab划分多边形{    int n = p.size();    Polygon newp;    for (int i = 0;i < n;i++)    {        Point c = p[i], d = p[(i + 1) % n];        if (judgeZero(getCross(b, c, a)) > 0)//只记录向量ab左侧的多边形            newp.push_back(c);        if (onLine(c, a, b))//防止重复记录点            newp.push_back(c);        else if (onLine(d, a, b))            continue;        else if (lineIntersect(a, b, c, d))        {            Point tmp = point_line_intersectLine(a, b, c, d);            if (onSegment(tmp, c, d))                newp.push_back(tmp);        }    }    return newp;//要判断newp.size()>=3}double polygonArea(Polygon polygon)//多边形面积{    double ans = 0.0;    for (int i = 1;i < polygon.size() - 1;i++)        ans += getCross(polygon[i], polygon[i + 1], polygon[0]);    return fabs(ans) / 2.0;}bool circle_inPolygon(Circle c, Polygon p)//圆含于多边形{    if (!inPolygon(c._o, p))        return false;    for (int i = 0;i < p.size();i++)    {        Point a = p[i], b = p[(i + 1) % p.size()];        if (judgeZero(dis_toSegment(c._o, a, b) - c._r) < 0)            return false;    }    return true;}bool polygon_inCircle(Circle c, Polygon p)//多边形含于圆{    for (int i = 0;i < p.size();i++)        if (!inCircle(p[i], c))            return false;    return true;}bool polygon_intersectCircle(Circle c, Polygon p)//圆含于多边形，多边形含于圆，圆和多边形相交，相切没算进去{    if (polygon_inCircle(c, p) || circle_inPolygon(c, p))        return true;    for (int i = 0;i < p.size();i++)    {        Point a = p[i], b = p[(i + 1) % p.size()];        if (judgeZero(dis_toSegment(c._o, a, b) - c._r) < 0)//算相切要<=0            return true;    }    return false;}int point_line_intersectCircle(Circle c, Point a, Point b, Point p[])//线段ab与圆c的交点{    double A = (b._x - a._x)*(b._x - a._x) + (b._y - a._y)*(b._y - a._y);    double B = 2.0 * ((b._x - a._x)*(a._x - c._o._x) + (b._y - a._y)*(a._y - c._o._y));    double C = (a._x - c._o._x)*(a._x - c._o._x) + (a._y - c._o._y)*(a._y - c._o._y) - c._r*c._r;    double deta = B*B - 4.0 * A*C;    if (judgeZero(deta) < 0) return 0;    double t1 = (-B - sqrt(deta)) / (2.0 * A);    double t2 = (-B + sqrt(deta)) / (2.0 * A);    int tot = 0;    if (judgeZero(t1) >= 0 && judgeZero(1 - t1) >= 0)        p[tot++] = Point(a._x + t1*(b._x - a._x), a._y + t1*(b._y - a._y));    if (judgeZero(t2) >= 0 && judgeZero(1 - t2) >= 0)        p[tot++] = Point(a._x + t2*(b._x - a._x), a._y + t2*(b._y - a._y));    if (tot == 2 && p[0] == p[1]) return 1;    return tot;}double sectorArea(Circle c, Point p1, Point p2)//扇形面积op1p2{    Point op1 = p1 - c._o, op2 = p2 - c._o;    double sita = atan2(op2._y, op2._x) - atan2(op1._y, op1._x);    while (judgeZero(sita) <= 0) sita += 2 * PI;    while (judgeZero(sita - 2 * PI) > 0) sita -= 2 * PI;    sita = min(sita, 2 * PI - sita);    return c._r*c._r*sita / 2.0;}double area_triangle_fromCircle(Circle c, Point a, Point b)//圆点，a，b构成的三角形与圆相交的面积{    bool flag1 = inCircle(a, c);    bool flag2 = inCircle(b, c);    if (flag1 && flag2)//两点在圆内        return fabs(getCross(a, b, c._o) / 2.0);    Point p[2];    int tot = point_line_intersectCircle(c, a, b, p);    if (flag1) return fabs(getCross(a, p[0], c._o) / 2.0) + sectorArea(c, p[0], b);    if (flag2) return fabs(getCross(b, p[0], c._o) / 2.0) + sectorArea(c, p[0], a);    if (tot == 2) return sectorArea(c, p[0], a) + sectorArea(c, p[1], b) + fabs(getCross(p[0], p[1], c._o)) / 2;    return sectorArea(c, a, b);}double area_polygon_intersectCircle(Circle c, Polygon p)//多边形和圆的相交面积{    double ans = 0.0;    for (int i = 0;i < p.size();i++)//trick：只需每相邻的点与圆心相连求相交面积，有向面积可以相互抵消    {        Point a, b;        a = p[i], b = p[(i + 1) % p.size()];        int flag = judgeZero(getCross(a, b, c._o));        ans += flag*area_triangle_fromCircle(c, a, b);    }    return fabs(ans);}int main(){    int n, cas = 1;    double k;    while (scanf("%d%lf", &n, &k) == 2)    {        Polygon p;        for (int i = 0;i < n;++i)        {            double x1, y1;            scanf("%lf%lf", &x1, &y1);            p.push_back(Point(x1, y1));        }        Point A, B;        cin >> A >> B;        for (int i = 0;i < n;++i)            p[i]._x -= A._x, p[i]._y -= A._y;        B._x -= A._x, B._y -= A._y;        A._x = 0, A._y = 0;        double a = B._x / (1 - k*k);        double b = B._y / (1 - k*k);        double c = (-B._x*B._x - B._y*B._y) / (1 - k*k);        Circle C(Point(a, b), sqrt(c + a*a + b*b));        printf("Case %d: %.10f\n", cas++, area_polygon_intersectCircle(C, p));    }    return 0;}