Problem: Find the area of intersection between a circle $c$ and a polygon $g$. The center coordinate of the circle is ($0, 0$).

The polygon $g$ is represented by a sequence of points $p_1$, $p_2$,..., $p_n$ where line segments connecting $p_i$ and $p_{i+1}$ ($1 \leq i \leq n−1$) are sides of the polygon. The line segment connecting $p_n$ and $p_1$ is also a side of the polygon.

Note that the polygon is not necessarily convex.

Constraints

* $3 \leq n \leq 100$
* $1 \leq r \leq 100$
* $-100 \leq x_i, y_i \leq 100$

Input

The input is given in the following format.

$n$ $r$
$x_1$ $y_1$
$x_2$ $y_2$
:
$x_n$ $y_n$


In the first line, an integer n representing the number of points in the polygon is given. The coordinate of a point $p_i$ is given by two integers $x_i$ and $y_i$. The coordinates of the points are given in the order of counter-clockwise visit of them. All input values are given in integers.

Output

Print the area of intersection in a line. The output values should be in a decimal fraction with an error less than 0.00001.

Examples

Input

3 5
1 1
4 1
5 5


Output

4.639858417607


Input

4 5
0 0
-3 -6
1 -3
5 -4


Output

11.787686807576

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
using namespace std;

//////////////// 幾何 /////////////////
#pragma region geometry
using Real = long double;  //時間がかかる割にdoubleと大差ない
using Point = complex<Real>;
const Real EPS = 1e-10, /*1e-8はたまに落ちる*/ PI = acos(-1);

inline bool eq(Real a, Real b) { return fabs(b - a) < EPS; }  //誤差無視の等式

//実スカラー倍
Point operator*(const Point &p, const Real &d) { return Point(real(p) * d, imag(p) * d); }

// point入力受け取り
istream &operator>>(istream &is, Point &p) {
	Real a, b;
	is >> a >> b;
	p = Point(a, b);
	return is;
}
// point出力
ostream &operator<<(ostream &os, Point &p) { return os << fixed << setprecision(10) << p.real() << " " << p.imag(); }

// 点 p を反時計回りに theta 回転
Point rotate(Real theta, const Point &p) {
	return Point(cos(theta) * p.real() - sin(theta) * p.imag(), sin(theta) * p.real() + cos(theta) * p.imag());
}

Real radian_to_degree(Real r) { return (r * 180.0 / PI); }

Real degree_to_radian(Real d) { return (d * PI / 180.0); }

// a-b-c の角度のうち小さい方を返す
Real get_angle(const Point &a, const Point &b, const Point &c) {
	const Point v(a - b), /* もとはv(b-a)になってたので修正 */ w(c - b);
	Real alpha = atan2(v.imag(), v.real()), beta = atan2(w.imag(), w.real());
	if(alpha > beta) swap(alpha, beta);
	Real theta = (beta - alpha);
	return min(theta, 2 * acos(-1) - theta);
}

namespace std {
bool operator<(const Point &a, const Point &b) { return a.real() != b.real() ? a.real() < b.real() : a.imag() < b.imag(); }
}  // namespace std

struct Line {
	Point a, b;

	Line() = default;

	Line(Point a, Point b) : a(a), b(b) {}

	Line(Real A, Real B, Real C)  // Ax + By = C
	{
		if(eq(A, 0))
			a = Point(0, C / B), b = Point(1, C / B);
		else if(eq(B, 0))
			b = Point(C / A, 0), b = Point(C / A, 1);
		else
			a = Point(0, C / B), b = Point(C / A, 0);
	}

	friend ostream &operator<<(ostream &os, Line &p) { return os << p.a << " to " << p.b; }

	friend istream &operator>>(istream &is, Line &a) { return is >> a.a >> a.b; }
};

struct Segment : Line {
	Segment() = default;

	Segment(Point a, Point b) : Line(a, b) {}
};

struct Circle {
	Point p;
	Real r;

	Circle() = default;

	Circle(Point p, Real r) : p(p), r(r) {}
};

using Points = vector<Point>;
using Polygon = vector<Point>;
using Segments = vector<Segment>;
using Lines = vector<Line>;
using Circles = vector<Circle>;

//外積
Real cross(const Point &a, const Point &b) { return real(a) * imag(b) - imag(a) * real(b); }
//内積
Real dot(const Point &a, const Point &b) { return real(a) * real(b) + imag(a) * imag(b); }

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_C
// 点の回転方向
int ccw(const Point &a, Point b, Point c) {
	b = b - a, c = c - a;
	if(cross(b, c) > EPS) return +1;   // "COUNTER_CLOCKWISE"
	if(cross(b, c) < -EPS) return -1;  // "CLOCKWISE"
	if(dot(b, c) < 0) return +2;	   // "ONLINE_BACK"
	if(norm(b) < norm(c)) return -2;   // "ONLINE_FRONT"
	return 0;						   // "ON_SEGMENT"
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_A
// 平行判定
bool parallel(const Line &a, const Line &b) { return eq(cross(a.b - a.a, b.b - b.a), 0.0); }

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_A
// 垂直判定
bool orthogonal(const Line &a, const Line &b) { return eq(dot(a.a - a.b, b.a - b.b), 0.0); }

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_A
// 射影
// 直線 l に p から垂線を引いた交点を求める
Point projection(const Line &l, const Point &p) {
	double t = dot(p - l.a, l.a - l.b) / norm(l.a - l.b);
	return l.a + (l.a - l.b) * t;
}

Point projection(const Segment &l, const Point &p) {
	double t = dot(p - l.a, l.a - l.b) / norm(l.a - l.b);
	return l.a + (l.a - l.b) * t;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_B
// 反射
// 直線 l を対称軸として点 p  と線対称にある点を求める
Point reflection(const Line &l, const Point &p) { return p + (projection(l, p) - p) * 2.0; }

bool intersect(const Line &l, const Point &p) { return abs(ccw(l.a, l.b, p)) != 1; }

bool intersect(const Line &l, const Line &m) { return abs(cross(l.b - l.a, m.b - m.a)) > EPS || abs(cross(l.b - l.a, m.b - l.a)) < EPS; }

bool intersect(const Segment &s, const Point &p) { return ccw(s.a, s.b, p) == 0; }

bool intersect(const Line &l, const Segment &s) { return cross(l.b - l.a, s.a - l.a) * cross(l.b - l.a, s.b - l.a) < EPS; }

Real distance(const Line &l, const Point &p);

bool intersect(const Circle &c, const Line &l) { return distance(l, c.p) <= c.r + EPS; }

bool intersect(const Circle &c, const Point &p) { return abs(abs(p - c.p) - c.r) < EPS; }

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_B
// 線分同士の交差判定　(端点が他方の線分上にあるならば交差とみなす)
bool intersect(const Segment &s, const Segment &t) {
	return ccw(s.a, s.b, t.a) * ccw(s.a, s.b, t.b) <= 0 && ccw(t.a, t.b, s.a) * ccw(t.a, t.b, s.b) <= 0;
}

int intersect(const Circle &c, const Segment &l) {
	if(norm(projection(l, c.p) - c.p) - c.r * c.r > EPS) return 0;
	auto d1 = abs(c.p - l.a), d2 = abs(c.p - l.b);
	if(d1 < c.r + EPS && d2 < c.r + EPS) return 0;
	if(d1 < c.r - EPS && d2 > c.r + EPS || d1 > c.r + EPS && d2 < c.r - EPS) return 1;
	const Point h = projection(l, c.p);
	if(dot(l.a - h, l.b - h) < 0) return 2;
	return 0;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_A&lang=jp
int intersect(Circle c1, Circle c2) {
	if(c1.r < c2.r) swap(c1, c2);
	Real d = abs(c1.p - c2.p);
	if(c1.r + c2.r < d) return 4;	  // 外に離れている
	if(eq(c1.r + c2.r, d)) return 3;  // 外接
	if(c1.r - c2.r < d) return 2;	  // 交差
	if(eq(c1.r - c2.r, d)) return 1;  // 内接
	return 0;						  // 内包
}

Real distance(const Point &a, const Point &b) { return abs(a - b); }

Real distance(const Line &l, const Point &p) { return abs(p - projection(l, p)); }

Real distance(const Line &l, const Line &m) { return intersect(l, m) ? 0 : distance(l, m.a); }

Real distance(const Segment &s, const Point &p) {
	Point r = projection(s, p);
	if(intersect(s, r)) return abs(r - p);
	return min(abs(s.a - p), abs(s.b - p));
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_D
Real distance(const Segment &a, const Segment &b) {
	if(intersect(a, b)) return 0;
	return min({distance(a, b.a), distance(a, b.b), distance(b, a.a), distance(b, a.b)});
}

Real distance(const Line &l, const Segment &s) {
	if(intersect(l, s)) return 0;
	return min(distance(l, s.a), distance(l, s.b));
}

Point crosspoint(const Line &l, const Line &m) {
	Real A = cross(l.b - l.a, m.b - m.a);
	Real B = cross(l.b - l.a, l.b - m.a);
	if(eq(abs(A), 0.0) && eq(abs(B), 0.0)) return m.a;
	return m.a + (m.b - m.a) * B / A;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_C
Point crosspoint(const Segment &l, const Segment &m) { return crosspoint(Line(l), Line(m)); }

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_D
pair<Point, Point> crosspoint(const Circle &c, const Line l) {
	Point pr = projection(l, c.p);
	Point e = (l.b - l.a) / abs(l.b - l.a);
	if(eq(distance(l, c.p), c.r)) return {pr, pr};
	double base = sqrt(c.r * c.r - norm(pr - c.p));
	return {pr - e * base, pr + e * base};
}

pair<Point, Point> crosspoint(const Circle &c, const Segment &l) {
	Line aa = Line(l.a, l.b);
	if(intersect(c, l) == 2) return crosspoint(c, aa);
	auto ret = crosspoint(c, aa);
	if(dot(l.a - ret.first, l.b - ret.first) < 0)
		ret.second = ret.first;
	else
		ret.first = ret.second;
	return ret;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_E
pair<Point, Point> crosspoint(const Circle &c1, const Circle &c2) {
	Real d = abs(c1.p - c2.p);
	Real a = acos((c1.r * c1.r + d * d - c2.r * c2.r) / (2 * c1.r * d));  // 余弦定理
	Real t = atan2(c2.p.imag() - c1.p.imag(), c2.p.real() - c1.p.real());
	Point p1 = c1.p + Point(cos(t + a) * c1.r, sin(t + a) * c1.r);
	Point p2 = c1.p + Point(cos(t - a) * c1.r, sin(t - a) * c1.r);
	return {p1, p2};
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_F
// "点 p を通る円 c の接線"の接点2つ
pair<Point, Point> tangent(const Circle &c1, const Point &p2) { return crosspoint(c1, Circle(p2, sqrt(norm(c1.p - p2) - c1.r * c1.r))); }

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_G
// 円 c1, c2 の共通接線
Lines tangent(Circle c1, Circle c2) {
	Lines ret;
	if(c1.r < c2.r) swap(c1, c2);
	Real g = norm(c1.p - c2.p);
	if(eq(g, 0)) return ret;
	Point u = (c2.p - c1.p) / sqrt(g);
	Point v = rotate(PI * 0.5, u);
	for(int s : {-1, 1}) {
		Real h = (c1.r + s * c2.r) / sqrt(g);
		if(eq(1 - h * h, 0)) {
			ret.emplace_back(c1.p + u * c1.r, c1.p + (u + v) * c1.r);
		} else if(1 - h * h > 0) {
			Point uu = u * h, vv = v * sqrt(1 - h * h);
			ret.emplace_back(c1.p + (uu + vv) * c1.r, c2.p - (uu + vv) * c2.r * s);
			ret.emplace_back(c1.p + (uu - vv) * c1.r, c2.p - (uu - vv) * c2.r * s);
		}
	}
	return ret;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_3_B
// 凸性判定 (反時計回り)
bool is_convex(const Polygon &p) {
	int n = (int)p.size();
	for(int i = 0; i < n; i++) {
		if(ccw(p[(i + n - 1) % n], p[i], p[(i + 1) % n]) == -1) return false;
	}
	return true;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_4_A
// 凸包
Polygon convex_hull(Polygon &p) {
	int n = (int)p.size(), k = 0;
	if(n <= 2) return p;
	sort(p.begin(), p.end());  // x -> yの順にソート
	vector<Point> ch(2 * n);
	for(int i = 0; i < n; ch[k++] = p[i++]) {  //下半分を構成
		while(k >= 2 && cross(ch[k - 1] - ch[k - 2], p[i] - ch[k - 1]) < EPS) --k;
	}
	for(int i = n - 2, t = k + 1; i >= 0; ch[k++] = p[i--]) {  //上半分を構成
		while(k >= t && cross(ch[k - 1] - ch[k - 2], p[i] - ch[k - 1]) < EPS) --k;
	}
	ch.resize(k - 1);
	return ch;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_3_C
// 多角形と点の包含判定
enum { OUT, ON, IN };

int contains(const Polygon &Q, const Point &p) {
	bool in = false;
	for(int i = 0; i < Q.size(); i++) {
		Point a = Q[i] - p, b = Q[(i + 1) % Q.size()] - p;
		if(a.imag() > b.imag()) swap(a, b);
		if(a.imag() <= 0 && 0 < b.imag() && cross(a, b) < 0) in = !in;
		if(cross(a, b) == 0 && dot(a, b) <= 0) return ON;
	}
	return in ? IN : OUT;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=1033
// 線分の重複除去
void merge_segments(vector<Segment> &segs) {
	auto merge_if_able = [](Segment &s1, const Segment &s2) {
		if(abs(cross(s1.b - s1.a, s2.b - s2.a)) > EPS) return false;
		if(ccw(s1.a, s2.a, s1.b) == 1 || ccw(s1.a, s2.a, s1.b) == -1) return false;
		if(ccw(s1.a, s1.b, s2.a) == -2 || ccw(s2.a, s2.b, s1.a) == -2) return false;
		s1 = Segment(min(s1.a, s2.a), max(s1.b, s2.b));
		return true;
	};

	for(int i = 0; i < segs.size(); i++) {
		if(segs[i].b < segs[i].a) swap(segs[i].a, segs[i].b);
	}
	for(int i = 0; i < segs.size(); i++) {
		for(int j = i + 1; j < segs.size(); j++) {
			if(merge_if_able(segs[i], segs[j])) {
				segs[j--] = segs.back(), segs.pop_back();
			}
		}
	}
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=1033
// 線分アレンジメント
// 任意の2線分の交点を頂点としたグラフを構築する
vector<vector<int>> segment_arrangement(vector<Segment> &segs, vector<Point> &ps) {
	vector<vector<int>> g;
	int N = (int)segs.size();
	for(int i = 0; i < N; i++) {
		ps.emplace_back(segs[i].a);
		ps.emplace_back(segs[i].b);
		for(int j = i + 1; j < N; j++) {
			const Point p1 = segs[i].b - segs[i].a;
			const Point p2 = segs[j].b - segs[j].a;
			if(cross(p1, p2) == 0) continue;
			if(intersect(segs[i], segs[j])) {
				ps.emplace_back(crosspoint(segs[i], segs[j]));
			}
		}
	}
	sort(begin(ps), end(ps));
	ps.erase(unique(begin(ps), end(ps)), end(ps));

	int M = (int)ps.size();
	g.resize(M);
	for(int i = 0; i < N; i++) {
		vector<int> vec;
		for(int j = 0; j < M; j++) {
			if(intersect(segs[i], ps[j])) {
				vec.emplace_back(j);
			}
		}
		for(int j = 1; j < vec.size(); j++) {
			g[vec[j - 1]].push_back(vec[j]);
			g[vec[j]].push_back(vec[j - 1]);
		}
	}
	return (g);
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_4_C
// 凸多角形の切断
// 直線 l.a-l.b で切断しその左側にできる凸多角形を返す
Polygon convex_cut(const Polygon &U, Line l) {
	Polygon ret;
	for(int i = 0; i < U.size(); i++) {
		Point now = U[i], nxt = U[(i + 1) % U.size()];
		if(ccw(l.a, l.b, now) != -1) ret.push_back(now);
		if(ccw(l.a, l.b, now) * ccw(l.a, l.b, nxt) < 0) {
			ret.push_back(crosspoint(Line(now, nxt), l));
		}
	}
	return (ret);
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_3_A
// 多角形の面積
Real area(const Polygon &p) {
	Real A = 0;
	for(int i = 0; i < p.size(); ++i) {
		A += cross(p[i], p[(i + 1) % p.size()]);
	}
	return A * 0.5;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_H
// 円と多角形の共通部分の面積
Real area(const Polygon &p, const Circle &c) {
	if(p.size() < 3) return 0.0;
	function<Real(Circle, Point, Point)> cross_area = [&](const Circle &c, const Point &a, const Point &b) {
		Point va = c.p - a, vb = c.p - b;
		Real f = cross(va, vb), ret = 0.0;
		if(eq(f, 0.0)) return ret;
		if(max(abs(va), abs(vb)) < c.r + EPS) return f;
		if(distance(Segment(a, b), c.p) > c.r - EPS) return c.r * c.r * arg(vb * conj(va));
		auto u = crosspoint(c, Segment(a, b));
		vector<Point> tot{a, u.first, u.second, b};
		for(int i = 0; i + 1 < tot.size(); i++) {
			ret += cross_area(c, tot[i], tot[i + 1]);
		}
		return ret;
	};
	Real A = 0;
	for(int i = 0; i < p.size(); i++) {
		A += cross_area(c, p[i], p[(i + 1) % p.size()]);
	}
	return A * 0.5;	 // 0.5倍が抜けていたので修正
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_4_B
// 凸多角形の直径(最遠頂点対間距離)
Real convex_diameter(const Polygon &p) {
	int N = (int)p.size();
	int is = 0, js = 0;
	for(int i = 1; i < N; i++) {
		if(p[i].imag() > p[is].imag()) is = i;
		if(p[i].imag() < p[js].imag()) js = i;
	}
	Real maxdis = norm(p[is] - p[js]);

	int maxi, maxj, i, j;
	i = maxi = is;
	j = maxj = js;
	do {
		if(cross(p[(i + 1) % N] - p[i], p[(j + 1) % N] - p[j]) >= 0) {
			j = (j + 1) % N;
		} else {
			i = (i + 1) % N;
		}
		if(norm(p[i] - p[j]) > maxdis) {
			maxdis = norm(p[i] - p[j]);
			maxi = i;
			maxj = j;
		}
	} while(i != is || j != js);
	return sqrt(maxdis);
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_5_A
// 最近点対
Real closest_pair(Points ps) {
	if(ps.size() <= 1) throw(0);
	sort(begin(ps), end(ps));

	auto compare_y = [&](const Point &a, const Point &b) { return imag(a) < imag(b); };
	vector<Point> beet(ps.size());
	const Real INF = 1e18;

	function<Real(int, int)> rec = [&](int left, int right) {
		if(right - left <= 1) return INF;
		int mid = (left + right) >> 1;
		auto x = real(ps[mid]);
		auto ret = min(rec(left, mid), rec(mid, right));
		inplace_merge(begin(ps) + left, begin(ps) + mid, begin(ps) + right, compare_y);
		int ptr = 0;
		for(int i = left; i < right; i++) {
			if(abs(real(ps[i]) - x) >= ret) continue;
			for(int j = 0; j < ptr; j++) {
				auto luz = ps[i] - beet[ptr - j - 1];
				if(imag(luz) >= ret) break;
				ret = min(ret, abs(luz));
			}
			beet[ptr++] = ps[i];
		}
		return ret;
	};
	return rec(0, (int)ps.size());
}

#pragma endregion

int main() {
	cout << fixed << setprecision(15);
	int n;
	Real r;
	cin >> n >> r;
	Polygon p(n);
	for(int i = 0; i < n; i++) {
		cin >> p[i];
	}
	Circle c(Point(0, 0), r);
	Real ans = area(p, c);
	cout << ans << endl;
}