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area_common_circle_polygon.cpp
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area_common_circle_polygon.cpp
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//
// 円と多角形の共通部分の面積
//
// verified:
// AOJ Course CGL_7_H: Intersection of a Circle and a Polygon
// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_H&lang=ja
//
#include <iostream>
#include <vector>
#include <cmath>
#include <iomanip>
using namespace std;
//------------------------------//
// 基本要素 (点, 線分, 円)
//------------------------------//
using DD = long double;
const DD INF = 1LL<<60; // to be set appropriately
const DD EPS = 1e-6; // to be set appropriately
const DD PI = acosl(-1.0);
DD torad(int deg) {return (DD)(deg) * PI / 180;}
DD todeg(DD ang) {return ang * 180 / PI;}
/* Point */
struct Point {
DD x, y;
Point(DD x = 0.0, DD y = 0.0) : x(x), y(y) {}
friend ostream& operator << (ostream &s, const Point &p) {return s << '(' << p.x << ", " << p.y << ')';}
};
inline Point operator + (const Point &p, const Point &q) {return Point(p.x + q.x, p.y + q.y);}
inline Point operator - (const Point &p, const Point &q) {return Point(p.x - q.x, p.y - q.y);}
inline Point operator * (const Point &p, DD a) {return Point(p.x * a, p.y * a);}
inline Point operator * (DD a, const Point &p) {return Point(a * p.x, a * p.y);}
inline Point operator * (const Point &p, const Point &q) {return Point(p.x * q.x - p.y * q.y, p.x * q.y + p.y * q.x);}
inline Point operator / (const Point &p, DD a) {return Point(p.x / a, p.y / a);}
inline Point conj(const Point &p) {return Point(p.x, -p.y);}
inline Point rot(const Point &p, DD ang) {return Point(cos(ang) * p.x - sin(ang) * p.y, sin(ang) * p.x + cos(ang) * p.y);}
inline Point rot90(const Point &p) {return Point(-p.y, p.x);}
inline DD cross(const Point &p, const Point &q) {return p.x * q.y - p.y * q.x;}
inline DD dot(const Point &p, const Point &q) {return p.x * q.x + p.y * q.y;}
inline DD norm(const Point &p) {return dot(p, p);}
inline DD abs(const Point &p) {return sqrt(dot(p, p));}
inline DD amp(const Point &p) {DD res = atan2(p.y, p.x); if (res < 0) res += PI*2; return res;}
inline bool eq(const Point &p, const Point &q) {return abs(p - q) < EPS;}
inline bool operator < (const Point &p, const Point &q) {return (abs(p.x - q.x) > EPS ? p.x < q.x : p.y < q.y);}
inline bool operator > (const Point &p, const Point &q) {return (abs(p.x - q.x) > EPS ? p.x > q.x : p.y > q.y);}
inline Point operator / (const Point &p, const Point &q) {return p * conj(q) / norm(q);}
/* Line */
struct Line : vector<Point> {
Line(Point a = Point(0.0, 0.0), Point b = Point(0.0, 0.0)) {
this->push_back(a);
this->push_back(b);
}
friend ostream& operator << (ostream &s, const Line &l) {return s << '{' << l[0] << ", " << l[1] << '}';}
};
/* Circle */
struct Circle : Point {
DD r;
Circle(Point p = Point(0.0, 0.0), DD r = 0.0) : Point(p), r(r) {}
friend ostream& operator << (ostream &s, const Circle &c) {return s << '(' << c.x << ", " << c.y << ", " << c.r << ')';}
};
//------------------------------//
// 円と線分の交点
//------------------------------//
int ccw_for_crosspoint_cs(const Point &a, const Point &b, const Point &c) {
if (cross(b-a, c-a) > EPS) return 1;
if (cross(b-a, c-a) < -EPS) return -1;
if (dot(b-a, c-a) < -EPS) return 2;
if (norm(b-a) < norm(c-a) - EPS) return -2;
return 0;
}
bool isinterPS_crosspoint_cs(const Point &p, const Line &s) {
return (ccw_for_crosspoint_cs(s[0], s[1], p) == 0);
}
Point proj_for_crosspoint(const Point &p, const Line &l) {
DD t = dot(p - l[0], l[1] - l[0]) / norm(l[1] - l[0]);
return l[0] + (l[1] - l[0]) * t;
}
vector<Point> crosspoint_CS(const Circle &e, const Line &s) {
vector<Point> res;
Point p = proj_for_crosspoint(e, s);
DD rcos = abs(e - p), rsin;
if (rcos > e.r + EPS) return vector<Point>();
else if (e.r - rcos < EPS) rsin = 0;
else rsin = sqrt(e.r * e.r - rcos * rcos);
Point dir = (s[1] - s[0]) / abs(s[1] - s[0]);
Point p1 = p - dir * rsin;
Point p2 = p + dir * rsin;
if (isinterPS_crosspoint_cs(p1, s)) res.push_back(p1);
if (isinterPS_crosspoint_cs(p2, s) && !eq(p1, p2)) res.push_back(p2);
return res;
}
//------------------------------//
// 円と多角形の共通部分の面積
//------------------------------//
// 原点, 点 x, 点 y とで囲まれる領域の面積 (三角形 ver と扇型 ver)
DD calc_element(const Point &x, const Point &y, DD r, bool triangle) {
if (triangle) return cross(x, y) / 2;
else {
Point tmp = y * Point(x.x, -x.y);
DD ang = atan2(tmp.y, tmp.x);
return r * r * ang / 2;
}
}
// 円 C と、三角形 ((0, 0), ia, ib) との共通部分の面積
DD calc_common_area(const Circle &c, const Point &ia, const Point &ib) {
Point a = ia - c, b = ib - c;
if (abs(a - b) < EPS) return 0;
bool isin_a = (abs(a) < c.r + EPS);
bool isin_b = (abs(b) < c.r + EPS);
if (isin_a && isin_b) return calc_element(a, b, c.r, true);
Circle oc(Point(0, 0), c.r);
Line seg(a, b);
auto cr = crosspoint_CS(oc, seg);
if (cr.empty()) return calc_element(a, b, c.r, false);
auto s = cr[0], t = cr.back();
return calc_element(s, t, c.r, true)
+ calc_element(a, s, c.r, isin_a) + calc_element(t, b, c.r, isin_b);
}
// 円 c と多角形 pol の共通部分の面積
DD calc_common_area(const Circle &c, const vector<Point> &pol) {
DD res = 0;
int N = pol.size();
for (int i = 0; i < N; ++i) {
res += calc_common_area(c, pol[i], pol[(i+1)%N]);
}
return res;
}
int main() {
int N; DD r; cin >> N >> r;
Circle c(Point(0, 0), r);
vector<Point> pol(N);
for (int i = 0; i < N; ++i) cin >> pol[i].x >> pol[i].y;
cout << fixed << setprecision(10) << calc_common_area(c, pol) << endl;
}