简单几何(圆与多边形公共面积) UVALive 7072 Signal Interference (14广州D)

 

题目传送门

题意：一个多边形，A点和B点，满足PB <= k * PA的P的范围与多边形的公共面积。

分析：这是个阿波罗尼斯圆。既然是圆，那么设圆的一般方程：(x + D/2) ^ 2 + (y + E/2) ^ 2 = (D ^ 2 + E ^ 2 - 4 * F)  / 4，通过PB == PA * k解方程来求解圆心以及半径。然后就是套模板啦，上海交大的红书。

 

#include <bits/stdc++.h>
using namespace std;
 
#define lson l, mid, rt << 1
#define rson mid + 1, r, rt << 1 | 1
typedef long long ll;
const int N = 5e2 + 10;
const int INF = 0x3f3f3f3f;
const int MOD = 1e9 + 7;
const double EPS = 1e-10;
const double PI = acos (-1.0);

int dcmp(double x)  {
    if (fabs (x) < EPS)   return 0;
    else    return x < 0 ? -1 : 1;
}
struct Point    {
    double x, y;
    Point ()    {}
    Point (double x, double y) : x (x), y (y) {}
    Point operator + (const Point &r) const {       //向量加法
        return Point (x + r.x, y + r.y);
    }
    Point operator - (const Point &r) const {
        return Point (x - r.x, y - r.y);
    }
    Point operator * (double p) const {       //向量乘以标量
        return Point (x * p, y * p);
    }
    Point operator / (double p) const {       //向量除以标量
        return Point (x / p, y / p);
    }
    bool operator < (const Point &r) const  {
        return x < r.x || (x == r.x && y < r.y);
    }
    bool operator == (const Point &r) const {
        return dcmp (x - r.x) == 0 && dcmp (y - r.y) == 0;
    }
};
typedef Point Vector;
Point read_point(void)  {
    double x, y;    scanf ("%lf%lf", &x, &y);
    return Point (x, y);
}
double polar_angle(Vector V)    {
    return atan2 (V.y, V.x);
}
double dot(Point a, Point b)    {
    return a.x * b.x + a.y * b.y;
}
double cross(Point a, Point b)  {
    return a.x * b.y - a.y * b.x;
}
double length(Vector V) {
    return sqrt (dot (V, V));
}
double my_sqrt(double x)    {
    return sqrt (max (0.0, x));
}
 
Point ps[N];
double r, k;
double x1, _y1, x2, _y2;
int n;
 
double sqr(double x)    {
    return x * x;
}
 
struct  Line    {
    Point p;
    Vector v;
    double r;
    Line () {}
    Line (const Point &p, const Vector &v) : p (p), v (v) {
        r = polar_angle (v);
    }
    Point point(double a)   {
        return p + v * a;
    }
};
 
struct Circle   {
    Point c;
    double r;
    Circle () {}
    Circle (Point c, double r) : c (c), r (r) {}
    Point point(double a)   {
        return Point (c.x + cos (a) * r, c.y + sin (a) * r);
    }
};
 
int line_cir_inter(Line L, Circle C, double &t1, double &t2, vector<Point> &P)    {
    double a = L.v.x, b = L.p.x - C.c.x, c = L.v.y, d = L.p.y - C.c.y;
    double e = a * a + c * c, f = 2 * (a * b + c * d), g = b * b + d * d - C.r * C.r;
    double delta = f * f - 4 * e * g;
    if (dcmp (delta) < 0)   return 0;
    if (dcmp (delta) == 0)  {
        t1 = t2 = -f / (2 * e); P.push_back (L.point (t1));
        return 1;
    }
    t1 = (-f - sqrt (delta)) / (2 * e);
    t2 = (-f + sqrt (delta)) / (2 * e);
    if (t1 > t2)    swap (t1, t2);
    if (dcmp (t1) > 0 && dcmp (t1 - 1) < 0) P.push_back (L.point (t1));
    if (dcmp (t2) > 0 && dcmp (t2 - 1) < 0) P.push_back (L.point (t2));
    return (int) P.size ();
}
 
double sector_area(Point a, Point b)    {
    double theta = polar_angle (a) - polar_angle (b);
    while (dcmp (theta) <= 0)   theta += 2 * PI;
    while (theta > 2 * PI)  theta -= 2 * PI;
    theta = min (theta, 2 * PI - theta);
    return r * r * theta / 2;
}
 
double cal(Point a, Point b)    {
    double t1, t2;
    bool ina = dcmp (length (a) - r) < 0;
    bool inb = dcmp (length (b) - r) < 0;
    if (ina && inb) return fabs (cross (a, b)) / 2.0;
    vector<Point> p;
    int num = line_cir_inter (Line (a, b - a), Circle (Point (0, 0), r), t1, t2, p);
    if (ina)    return sector_area (b, p[0]) + fabs (cross (a, p[0])) / 2.0;
    if (inb)    return sector_area (p[0], a) + fabs (cross (p[0], b)) / 2.0;
    if (num == 2)   return sector_area (a, p[0]) + sector_area (p[1], b) + fabs (cross (p[0], p[1])) / 2.0;
    return sector_area (a, b);
}
 
double cir_poly_area()  {
    double ret = 0;
    for (int i=0; i<n; ++i) {
        int sgn = dcmp (cross (ps[i], ps[i+1]));
        if (sgn != 0)   {
            ret += sgn * cal (ps[i], ps[i+1]);
        }
    }
    return ret;
}
 
void init(void) {
    double D = (2 * x2 - 2 * sqr (k) * x1) / (1 - sqr (k));
    double E = (2 * _y2 - 2 * sqr (k) * _y1) / (1 - sqr (k));
    double F = (sqr (k*x1)+sqr (k*_y1)-sqr (x2)-sqr(_y2)) / (1 - sqr (k));
    double x0 = D / 2, y0 = E / 2;
    r = sqrt (F + sqr (D)/4 + sqr (E)/4);
    for (int i=0; i<n; ++i) {
        ps[i].x -= x0;  ps[i].y -= y0;
    }
    ps[n] = ps[0];
}
 
int main(void)    {
    int cas = 0;
    while (scanf ("%d%lf", &n, &k) == 2)    {
        for (int i=0; i<n; ++i) {
            ps[i] = read_point ();
        }
        scanf ("%lf%lf", &x1, &_y1);
        scanf ("%lf%lf", &x2, &_y2);
        init ();
        printf ("Case %d: %.10f\n", ++cas, fabs (cir_poly_area ()));
    }
 
   //cout << "Time elapsed: " << 1.0 * clock() / CLOCKS_PER_SEC << " s.\n";
 
    return 0;
}