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>
#define int long long
#define endl '\n'
#define FOR(i, a, n) for (int i = (a); i < (n); ++i)
#define REP(i, n) FOR(i, 0, n)
using namespace std;

using T = long double;
const T EPS = 1e-10;
const T PI = acos(-1.0);
T torad(int deg) {return (T)(deg) * PI / 180;}
T todeg(T ang) {return ang * 180 / PI;}

/* Point */
using P = complex<T>;
#define x real()
#define y imag()
template <typename T> inline bool eq(T p, T q) { return abs(p - q) < EPS; }
inline int sgn(T x) { return (0.0 < x) - (x < 0.0); }
inline T dot(P v, P w) { return (conj(v) * w).x; }
inline T cross(P v, P w) { return (conj(v) * w).y; }
inline bool isOrth(P v, P w) { return dot(v, w) == 0; }
inline bool isPara(P v, P w) { return cross(v, w) == 0; }
inline P unit(P v) { return v / abs(v); }
inline P translate(P p, P v) { return p + v; }
inline P scale(P p, P c, T k) { return c + (p - c) * k; }
inline P rot(P p, T a) { return p * polar(T(1), a); }
inline P rot90(P p) { return {-p.y, p.x}; }
inline T orient(P a, P b, P c) { return cross(b - a, c - a); }
istream& operator>>(istream& is, P& p) { T xx, yy; is >> xx >> yy; p = P(xx, yy); return is; }
bool cmpX(const P& a, const P& b) { return a.x != b.x ? a.x < b.x : a.y < b.y; }
bool cmpY(const P& a, const P& b) { return a.y != b.y ? a.y < b.y : a.x < b.x; }
namespace std {
    bool operator < (const P& a, const P& b) { return cmpX(a, b); }
}

P linearTransform(P p, P q, P fp, P fq, P r) {
    return fp + (r - p) * (fq - fp) / (q - p);
}

int ccw(P a, P b, P c) {
    b -= a; c -= a;
    if (cross(b, c) > EPS) return +1;      // counter clockwise
    if (cross(b, c) < -EPS) return -1;      // clockwise
    if (dot(b, c) < -EPS) return +2;        // c--a--b on line
    if (norm(b) < norm(c) - EPS) return -2;    // a--b--c on line or a==b
    return 0;                            // a--c--b on line or a==c or b==c
}

T angle(P v, P w) {
    T cosTheta = dot(v, w) / abs(v) / abs(w);
    return acos(max(T(-1.0), min(T(1.0), cosTheta)));
}
T orientedAngle(P a, P b, P c) {
    if (orient(a, b, c) >= 0) return angle(b - a, c - a);
    return 2 * PI - angle(b - a, c - a);
}
bool inAngle(P a, P b, P c, P p) {
    assert(abs(orient(a, b, c)) != 0);
    if (orient(a, b, c) < 0) swap(b, c);
    return orient(a, b, p) >= 0 && orient(a, c, p) <= 0;
}

// polar sort
bool half(P p) {
    assert(p.x != 0 || p.y != 0);
    return p.y > 0 || (p.y == 0 && p.x < 0);
}
void polarSort(vector<P>& v, P o = P(0.0, 0.0)) {
    sort(v.begin(), v.end(), [&](P v, P w) {
        return make_tuple(half(v - o), 0, norm(v - o)) <
            make_tuple(half(w - o), cross(v - o, w - o), norm(w - o));
    });
}

/* Line */
struct L : array<P, 2> {
    P v; T c;
    L(P a = P(0.0, 0.0), P b = P(0.0, 0.0)) : array<P, 2>({a, b}), v(b - a), c(cross(v, a)) {}
    T side(P p) { return cross(v, p) - c; }
    P proj(P p) { return p - rot90(v) * side(p) / norm(v); }
    P refl(P p) { return p - rot90(v) * T(2) * side(p) / norm(v); }
    friend istream& operator>>(istream& is, L& l) { P p, q; is >> p >> q; l = L(p, q); return is; }
    friend ostream& operator<<(ostream& os, const L& l) { return os << '{' << l[0] << ", " << l[1] << '}'; }
};

/* Circle */
struct C : P {
    T r;
    C(P p = P(0.0, 0.0), T r = 0.0) : P(p), r(r) {}
    friend istream& operator>>(istream& is, C& c) { P p; T r; is >> p >> r; c = C(p, r); return is; }
    friend ostream& operator<<(ostream& os, const C& c) { return os << '(' << c.x << ", " << c.y << ", " << c.r << ')'; }
};

P circumCenter(P a, P b, P c) {
    b -= a, c -= a;
    assert(cross(b, c) != 0);
    return a + rot90(b * norm(c) - c * norm(b)) / cross(b, c) / T(2);
}

vector<P> tangentPoints(C c, P p) {
    if (eq(abs(p - c), c.r)) return vector<P>{p};
    vector<P> res;
    T t = acos(c.r / abs(p - c));
    P q = (p - c) / abs(p - c) * c.r;
    res.push_back(c + q * polar(T(1), t));
    res.push_back(c + q * polar(T(1), -t));
    sort(res.begin(), res.end(), cmpX);
    return res;
}

vector<L> tangentLines(C c1, C c2, bool inner) {
    if (inner) c2.r = -c2.r;
    vector<L> res;
    P d = c2 - c1;
    T dr = c1.r - c2.r, d2 = norm(d), h2 = d2 - dr * dr;
    if (d2 == 0 || h2 < 0) { assert(h2 != 0); return res; }
    for (T sign : {-1, 1}) {
        P v = (d * dr + rot90(d) * sqrt(h2) * sign) / d2;
        res.emplace_back(c1 + v * c1.r, c2 + v * c2.r);
    }
    return res;
}

// intersect
bool isInterPL(P p, L l) { return (ccw(l[0], l[1], p) != 1); }
bool isInterPS(P p, L s) { return (ccw(s[0], s[1], p) == 0); }
bool isInterLL(L l, L m) { return (!isPara(l.v, m.v)); }
bool isInterLS(L l, L s) {
    if (eq(s[0], s[1])) return isInterPL(s[0], l);
    return (ccw(l[0], l[1], s[0]) * ccw(l[0], l[1], s[1]) <= 0);
}
bool isInterSS(L s, L t) {
    if (eq(s[0], s[1])) return isInterPL(s[0], t);
    if (eq(t[0], t[1])) return isInterPL(t[0], s);
    return (ccw(s[0], s[1], t[0]) * ccw(s[0], s[1], t[1]) <= 0 &&
            ccw(t[0], t[1], s[0]) * ccw(t[0], t[1], s[1]) <= 0);
}
int isInterCC(C a, C b) {
    T d = abs(b - a);
    if (a == b && a.r == b.r) return 0;         // ∞ cp (A = B)
    if (a.r + d - b.r < -EPS) return +1;        // 0 cp (A in B)
    if (b.r + d - a.r < -EPS) return -1;        // 0 cp (B in A)
    if (abs(a.r + d - b.r) < EPS) return +2;    // 1 cp (A in B)
    if (abs(b.r + d - a.r) < EPS) return -2;    // 1 cp (B in A)
    if (abs(a.r + b.r - d) < EPS) return 3;     // 1 cp (outside)
    if (a.r + b.r - d < -EPS) return 4;         // 0 cp (outside)
    return 5;                                   // 2 cp
}
 
// distance
T distancePL(P p, L l) { return (abs(p - l.proj(p))); }
T distancePS(P p, L s) {
    P h = s.proj(p);
    if (isInterPS(h, s)) return abs(p - h);
    return min(abs(p - s[0]), abs(p - s[1]));
}
T distanceLL(L l, L m) {
    if (isInterLL(l, m)) return 0.0;
    return distancePL(m[0], l);
}
T distanceLS(L l, L s) {
    if (isInterLS(l, s)) return 0;
    return min(distancePL(s[0], l), distancePL(s[1], l));
}
T distanceSS(L s, L t) {
    if (isInterSS(s, t)) return 0;
    return min({distancePS(s[0], t), distancePS(s[1], t), distancePS(t[0], s), distancePS(t[1], s)});
}

// crosspoint
vector<P> crosspointLL(L l, L m) {
    if (!isInterLL(l, m)) return vector<P>();
    vector<P> res;
    T d = cross(m[1] - m[0], l[1] - l[0]);
    res.push_back(l[0] + (l[1] - l[0]) * cross(m[1] - m[0], m[1] - l[0]) / d);
    return res;
}
vector<P> crosspointLS(L l, L s) {
    if (!isInterLS(l, s)) return vector<P>();
    vector<P> res;
    T d = cross(s[1] - s[0], l[1] - l[0]);
    res.push_back(l[0] + (l[1] - l[0]) * cross(s[1] - s[0], s[1] - l[0]) / d);
    return res;
}
vector<P> crosspointLC(L l, C c, bool dupl = false) {
    if (abs(c - l.proj(c)) > c.r) return vector<P>();
    vector<P> res;
    T d = distancePL(c, l), h2 = c.r * c.r - d * d;
    assert(h2 >= 0);
    P p = l.proj(c), h = l.v * sqrt(h2) / abs(l.v);
    res.push_back(p - h);
    if (dupl || abs(h) > 0) res.push_back(p + h);
    return res;
}
vector<P> crosspointSS(L s, L t) {
    if (!isInterSS(s, t)) return vector<P>();
    vector<P> res;
    if (unit(s.v) != unit(t.v)) {
        T d = cross(t[1] - t[0], s[1] - s[0]);
        res.push_back(s[0] + (s[1] - s[0]) * cross(t[1] - t[0], t[1] - s[0]) / d);
    } else {
        if (ccw(s[0], s[1], t[0]) == 0) res.push_back(t[0]);
        if (ccw(s[0], s[1], t[1]) == 0) res.push_back(t[1]);
        if (ccw(t[0], t[1], s[0]) == 0) res.push_back(s[0]);
        if (ccw(t[0], t[1], s[1]) == 0) res.push_back(s[1]);
    }
    return res;
}
vector<P> crosspointSC(L s, C c, bool dupl = false) {
    if (abs(c - s.proj(c)) > c.r) return vector<P>();
    vector<P> res;
    T d = distancePL(c, s), h2 = c.r * c.r - d * d;
    assert(h2 >= 0);
    P p = s.proj(c), h = s.v * sqrt(h2) / abs(s.v);
    if (isInterPS(p - h, s)) res.push_back(p - h);
    if (dupl || (abs(h) > 0 && isInterPS(p + h, s))) res.push_back(p + h);
    return res;
}
vector<P> crosspointCC(C c1, C c2, bool dupl = false) {
    P d = c2 - c1; T d2 = norm(d);
    if (d2 == 0) { assert(c1.r != c2.r); return vector<P>(); }
    vector<P> res;
    T pd = (d2 + c1.r * c1.r - c2.r * c2.r) / 2;
    T h2 = c1.r * c1.r - pd * pd / d2;
    if (h2 >= 0) {
        P p = c1 + d * pd / d2, h = rot90(d) * sqrt(h2 / d2);
        res.push_back(p - h);
        if (dupl || abs(h) > 0) res.push_back(p + h);
    }
    return res;
}

/* Polygon */
// p[0] has minimum y and p is counterclockwise
vector<P> makeStandard(vector<P> p) {
    int n = p.size(), j = -1;
    for (int i = 0; i < n; ++i) {
        if (j == -1 || p[i].y < p[j].y) j = i;
    }
    vector<P> res;
    if (p[(j - 1 + n) % n].x <= p[j].x && p[j].x <= p[(j + 1) % n].x) {
        for (int i = 0; i < n; ++i) {
            res.push_back(p[(i + j) % n]);
        }
    } else {
        for (int i = 0; i < n; ++i) {
            res.push_back(p[(i - j + n) % n]);
        }
    }
    return res;
}
bool isConvex(vector<P> p) {
    p = makeStandard(p);
    for (int i = 0, n = p.size(); i < n; ++i) {
        P a = p[i], b = p[(i + 1) % n], c = p[(i + 2) % n];
        if (orient(a, b, c) < 0) return false;
        if (b.y != p[0].y && b.y < min(a.y, c.y)) return false;
        if (b.y != p[0].y && a.y == b.y && b.y == c.y && a.x <= b.x && b.x <= c.x) return false;
    }
    return true;
}
T areaTriangle(P a, P b, P c) {
    return abs(cross(b - a, c - a)) / 2.0;
}
T areaPolygon(vector<P> p) {
    T res = 0;
    for (int i = 0, n = p.size(); i < n; ++i) {
        res += cross(p[i], p[(i + 1) % n]);
    }
    return abs(res) / 2.0;
}
T areaTwoCircle(C p, C q) {
    int in = isInterCC(p, q);
    if (abs(in) <= 2) return pow(min(p.r, q.r), 2) * PI;
    if (in <= 4) return 0.0;
    T d = abs(p - q), res = 0;
    T pt = acos((p.r * p.r + d * d - q.r * q.r) / (2 * p.r * d));
    res += p.r * p.r * pt - p.r * p.r * sin(pt) * cos(pt);
    T qt = acos((q.r * q.r + d * d - p.r * p.r) / (2 * q.r * d));
    res += q.r * q.r * qt - q.r * q.r * sin(qt) * cos(qt);
    return res;
}
T calcElem(P a, P b, T r, bool triangle) {
    if (triangle) return cross(a, b) / 2;
    P tmp = b * P(a.x, -a.y);
    return r * r * atan2(tmp.y, tmp.x) / 2;
}
T areaCircleTriange(C c, P a, P b) {
    a -= c, b -= c;
    if (eq(a, b)) return 0;
    bool inA = (abs(a) < c.r + EPS);
    bool inB = (abs(b) < c.r + EPS);
    if (inA && inB) return calcElem(a, b, c.r, true);
    L s(a, b);
    auto cr = crosspointSC(s, C(P(0, 0), c.r));
    if (cr.empty()) return calcElem(a, b, c.r, false);
    auto p = cr[0], q = cr.back();
    return calcElem(p, q, c.r, true) + calcElem(a, p, c.r, inA) + calcElem(q, b, c.r, inB);
}
T areaCirclePolygon(C c, vector<P> p) {
    T res = 0;
    for (int i = 0, n = p.size(); i < n; ++i) {
        res += areaCircleTriange(c, p[i], p[(i + 1) % n]);
    }
    return res;
}

signed main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);
    cout << fixed << setprecision(10);

    int n, r;
    cin >> n >> r;
    vector<P> p(n);
    REP (i, n) cin >> p[i];
    C c(P(0, 0), r);
    cout << areaCirclePolygon(c, p) << endl;
}