test/aoj/CGL_7_H.test.cpp

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Last update: 2024-09-28 19:54:09+09:00
Problem: https://onlinejudge.u-aizu.ac.jp/courses/library/4/CGL/7/CGL_7_H

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// competitive-verifier: PROBLEM https://onlinejudge.u-aizu.ac.jp/courses/library/4/CGL/7/CGL_7_H
// competitive-verifier: ERROR 0.00000001
// competitive-verifier: TLE 0.5
// competitive-verifier: MLE 64
#include <iostream>
#include <iomanip>
#include "src/Geometry/intersection_area.hpp"
using namespace std;
signed main() {
 cin.tie(0);
 ios::sync_with_stdio(false);
 using namespace geo;
 Circle<long double> c;
 int n;
 cin >> n >> c.r;
 vector<Point<long double>> ps(n);
 for (int i= 0; i < n; ++i) cin >> ps[i];
 cout << fixed << setprecision(12) << intersection_area(c, Polygon(ps)) << '\n';
 return 0;
}

#line 1 "test/aoj/CGL_7_H.test.cpp"
// competitive-verifier: PROBLEM https://onlinejudge.u-aizu.ac.jp/courses/library/4/CGL/7/CGL_7_H
// competitive-verifier: ERROR 0.00000001
// competitive-verifier: TLE 0.5
// competitive-verifier: MLE 64
#include <iostream>
#include <iomanip>
#line 2 "src/Geometry/angle.hpp"
#include <vector>
#line 3 "src/Geometry/Point.hpp"
#include <fstream>
#line 5 "src/Geometry/Point.hpp"
#include <cmath>
#include <cassert>
#line 2 "src/Internal/long_traits.hpp"
// clang-format off
template<class T>struct make_long{using type= T;};
template<>struct make_long<char>{using type= short;};
template<>struct make_long<unsigned char>{using type= unsigned short;};
template<>struct make_long<short>{using type= int;};
template<>struct make_long<unsigned short>{using type= unsigned;};
template<>struct make_long<int>{using type= long long;};
template<>struct make_long<unsigned>{using type= unsigned long long;};
template<>struct make_long<long long>{using type= __int128_t;};
template<>struct make_long<unsigned long long>{using type= __uint128_t;};
template<>struct make_long<float>{using type= double;};
template<>struct make_long<double>{using type= long double;};
template<class T> using make_long_t= typename make_long<T>::type;
// clang-format on
#line 8 "src/Geometry/Point.hpp"
namespace geo {
using namespace std;
struct Visualizer {
 ofstream ofs;
 Visualizer(string s= "visualize.txt"): ofs(s) { ofs << fixed << setprecision(10); }
 friend Visualizer &operator<<(Visualizer &vis, const string &s) { return vis.ofs << s, vis; }
};
template <class K> int sgn(K x) {
 if constexpr (is_floating_point_v<K>) {
  static constexpr K EPS= 1e-9;
  return x < -EPS ? -1 : x > EPS;
 } else return x < 0 ? -1 : x > 0;
}
template <class K> K err_floor(K x) {
 K y= floor(x);
 if constexpr (is_floating_point_v<K>)
  if (K z= y + 1, w= x - z; 0 <= sgn(w) && sgn(w - 1) < 0) return z;
 return y;
}
template <class K> K err_ceil(K x) {
 K y= ceil(x);
 if constexpr (is_floating_point_v<K>)
  if (K z= y - 1, w= x - z; 0 < sgn(w + 1) && sgn(w) <= 0) return z;
 return y;
}
template <class K> struct Point {
 K x, y;
 Point(K x= K(), K y= K()): x(x), y(y) {}
 Point &operator+=(const Point &p) { return x+= p.x, y+= p.y, *this; }
 Point &operator-=(const Point &p) { return x-= p.x, y-= p.y, *this; }
 Point &operator*=(K a) { return x*= a, y*= a, *this; }
 Point &operator/=(K a) { return x/= a, y/= a, *this; }
 Point operator+(const Point &p) const { return {x + p.x, y + p.y}; }
 Point operator-(const Point &p) const { return {x - p.x, y - p.y}; }
 Point operator*(K a) const { return {x * a, y * a}; }
 Point operator/(K a) const { return {x / a, y / a}; }
 friend Point operator*(K a, const Point &p) { return {a * p.x, a * p.y}; }
 Point operator-() const { return {-x, -y}; }
 bool operator<(const Point &p) const {
  int s= sgn(x - p.x);
  return s ? s < 0 : sgn(y - p.y) < 0;
 }
 bool operator>(const Point &p) const { return p < *this; }
 bool operator<=(const Point &p) const { return !(p < *this); }
 bool operator>=(const Point &p) const { return !(*this < p); }
 bool operator==(const Point &p) const { return !sgn(x - p.x) && !sgn(y - p.y); }
 bool operator!=(const Point &p) const { return sgn(x - p.x) || sgn(y - p.y); }
 Point operator!() const { return {-y, x}; }  // rotate 90 degree
 friend istream &operator>>(istream &is, Point &p) { return is >> p.x >> p.y; }
 friend ostream &operator<<(ostream &os, const Point &p) { return os << "(" << p.x << ", " << p.y << ")"; }
 friend Visualizer &operator<<(Visualizer &vis, const Point &p) { return vis.ofs << p.x << " " << p.y << "\n", vis; }
};
template <class K> make_long_t<K> dot(const Point<K> &p, const Point<K> &q) { return make_long_t<K>(p.x) * q.x + make_long_t<K>(p.y) * q.y; }
// left turn: > 0, right turn: < 0
template <class K> make_long_t<K> cross(const Point<K> &p, const Point<K> &q) { return make_long_t<K>(p.x) * q.y - make_long_t<K>(p.y) * q.x; }
template <class K> make_long_t<K> norm2(const Point<K> &p) { return dot(p, p); }
template <class K> long double norm(const Point<K> &p) { return sqrt(norm2(p)); }
template <class K> make_long_t<K> dist2(const Point<K> &p, const Point<K> &q) { return norm2(p - q); }
template <class T, class U> long double dist(const T &a, const U &b) { return sqrt(dist2(a, b)); }
enum CCW { COUNTER_CLOCKWISE, CLOCKWISE, ONLINE_BACK, ONLINE_FRONT, ON_SEGMENT };
ostream &operator<<(ostream &os, CCW c) { return os << (c == COUNTER_CLOCKWISE ? "COUNTER_CLOCKWISE" : c == CLOCKWISE ? "CLOCKWISE" : c == ONLINE_BACK ? "ONLINE_BACK" : c == ONLINE_FRONT ? "ONLINE_FRONT" : "ON_SEGMENT"); }
template <class K> CCW ccw(const Point<K> &p0, const Point<K> &p1, const Point<K> &p2) {
 Point a= p1 - p0, b= p2 - p0;
 int s;
 if constexpr (is_floating_point_v<K>) s= sgn(sgn(cross(a, b) / sqrt(norm2(a) * norm2(b))));
 else s= sgn(cross(a, b));
 if (s) return s > 0 ? COUNTER_CLOCKWISE : CLOCKWISE;
 if (K d= dot(a, b); sgn(d) < 0) return ONLINE_BACK;
 else return sgn(d - norm2(a)) > 0 ? ONLINE_FRONT : ON_SEGMENT;
}
template <class K> struct Line;
template <class K> struct Segment;
template <class K> class Polygon;
template <class K> struct Convex;
template <class K> struct Affine {
 K a00= 1, a01= 0, a10= 0, a11= 1;
 Point<K> b;
 Point<K> operator()(const Point<K> &p) const { return {a00 * p.x + a01 * p.y + b.x, a10 * p.x + a11 * p.y + b.y}; }
 Line<K> operator()(const Line<K> &l);
 Segment<K> operator()(const Segment<K> &s);
 Polygon<K> operator()(const Polygon<K> &p);
 Convex<K> operator()(const Convex<K> &c);
 Affine operator*(const Affine &r) const { return {a00 * r.a00 + a01 * r.a10, a00 * r.a01 + a01 * r.a11, a10 * r.a00 + a11 * r.a10, a10 * r.a01 + a11 * r.a11, (*this)(r)}; }
 Affine &operator*=(const Affine &r) { return *this= *this * r; }
};
template <class K> Affine<K> translate(const Point<K> &p) { return {1, 0, 0, 1, p}; }
}
#line 4 "src/Geometry/angle.hpp"
namespace geo {
long double radian_to_degree(long double r) { return r * 180.0 / M_PI; }
long double degree_to_radian(long double d) { return d * M_PI / 180.0; }
long double normalize_radian(long double r) { return r= fmod(r + M_PI, 2 * M_PI), r > 0 ? r - M_PI : r + M_PI; }
template <class K> long double angle(const Point<K> &p) { return atan2(p.y, p.x); }
template <class K> long double angle(const Point<K> &p, const Point<K> &q) { return atan2(cross(p, q), dot(p, q)); }
template <class K> Affine<K> rotate(long double theta) {
 K c= cos(theta), s= sin(theta);
 return {c, -s, s, c, Point<K>{0, 0}};
}
template <class K> Affine<K> rotate(const Point<K> &p, long double theta) {
 K c= cos(theta), s= sin(theta);
 return {c, -s, s, c, Point<K>{p.x - c * p.x + s * p.y, p.y - s * p.x - c * p.y}};
}
template <class K> Affine<K> rotate90(const Point<K> &p) { return {0, -1, 1, 0, p - !p}; }
// (-PI,PI], counter-clockwise
template <class K> class AngleComp {
 using P= Point<K>;
 static int quad(const P &p) {
  if (int s= sgn(p.y); s) return s;
  return sgn(p.x) < 0 ? 2 : 0;
 }
public:
 bool operator()(const P &p, const P &q) const {
  if (int a= quad(p), b= quad(q); a != b) return a < b;
  return cross(p, q) > 0;
 }
};
}
#line 2 "src/Geometry/Segment.hpp"
#include <algorithm>
#line 4 "src/Geometry/Line.hpp"
namespace geo {
template <class K> struct Line {
 using P= Point<K>;
 P p, d;  // p+td
 Line() {}
 // p + td
 Line(const P &p, const P &d): p(p), d(d) { assert(sgn(norm2(d))); }
 // ax+by+c=0 ................. ax+by+c>0: left, ax+by+c=0: on, ax+by+c<0: right
 Line(K a, K b, K c) {
  int sa= sgn(a), sb= sgn(b);
  assert(sa || sb);
  d= P{b, -a}, p= sb ? P{0, -c / b} : P{-c / a, 0};
 }
 bool operator==(const Line &l) const { return !sgn(cross(d, l.d)) && !where(l.p); }
 bool operator!=(const Line &l) const { return sgn(cross(d, l.d)) || where(l.p); }
 // +1: left, 0: on, -1: right
 int where(const P &q) const { return sgn(cross(d, q - p)); }
 P project(const P &q) const { return p + dot(q - p, d) / norm2(d) * d; }
 // return  a,b,c of ax+by+c=0
 tuple<K, K, K> coef() const { return make_tuple(-d.y, d.x, cross(p, d)); }
 friend ostream &operator<<(ostream &os, const Line &l) { return os << l.p << " + t" << l.d; }
 friend Visualizer &operator<<(Visualizer &vis, const Line &l) {
  auto [a, b, c]= l.coef();
  return vis.ofs << "Line " << a << " " << b << " " << c << "\n", vis;
 }
};
// p + t(q-p)
template <class K> Line<K> line_through(const Point<K> &p, const Point<K> &q) { return Line(p, q - p); }
template <class K> bool is_parallel(const Line<K> &l, const Line<K> &m) { return !sgn(cross(l.d, m.d)); }
template <class K> bool is_orthogonal(const Line<K> &l, const Line<K> &m) { return !sgn(dot(l.d, m.d)); }
// 1 : properly crossing, 0 : disjoint parallel, 2 : same line
template <class K> vector<Point<K>> cross_points(const Line<K> &l, const Line<K> &m) {
 K a= cross(m.d, l.d), b= cross(l.p - m.p, l.d);
 if (sgn(a)) return {m.p + b / a * m.d};  // properly crossing
 if (sgn(b)) return {};                   // disjoint parallel
 return {m.p, m.p + m.d};                 // same line
}
// perpendicular bisector ............ p on leftside
template <class K> Line<K> bisector(const Point<K> &p, const Point<K> &q) { return Line((p + q) / 2, !(q - p)); }
// angle bisector ........... parallel -> 1 line, non-parallel -> 2 lines
template <class K> vector<Line<K>> bisector(const Line<K> &l, const Line<K> &m) {
 auto cp= cross_points(l, m);
 if (cp.size() != 1) return {Line((l.p + m.p) / 2, l.d)};
 auto d= l.d / norm(l.d) + m.d / norm(m.d);
 return {Line(cp[0], d), Line(cp[0], !d)};
}
template <class K> make_long_t<K> dist2(const Line<K> &l, const Point<K> &p) {
 make_long_t<K> a= cross(l.d, p - l.p);
 return a * a / norm2(l.d);
}
template <class K> make_long_t<K> dist2(const Point<K> &p, const Line<K> &l) { return dist2(l, p); }
template <class K> make_long_t<K> dist2(const Line<K> &l, const Line<K> &m) { return is_parallel(l, m) ? dist2(l, m.p) : 0; }
template <class K> Affine<K> reflect(const Line<K> &l) {
 K a= l.d.x * l.d.x, b= l.d.x * l.d.y * 2, c= l.d.y * l.d.y, d= a + c;
 a/= d, b/= d, c/= d, d= a - c;
 return {d, b, b, -d, Point<K>{c * 2 * l.p.x - b * l.p.y, a * 2 * l.p.y - b * l.p.x}};
}
template <class K> Line<K> Affine<K>::operator()(const Line<K> &l) { return line_through((*this)(l.p), (*this)(l.p + l.d)); }
}
#line 4 "src/Geometry/Segment.hpp"
namespace geo {
template <class K> struct Segment {
 using P= Point<K>;
 P p, q;
 Segment() {}
 Segment(const P &p, const P &q): p(p), q(q) {}
 // do not consider the direction
 bool operator==(const Segment &s) const { return (p == s.p && q == s.q) || (p == s.q && q == s.p); }
 bool operator!=(const Segment &s) const { return !(*this == s); }
 bool on(const P &r) const { return ccw(p, q, r) == ON_SEGMENT; }
 P &operator[](int i) { return i ? q : p; }
 const P &operator[](int i) const { return i ? q : p; }
 long double length() const { return dist(p, q); }
 P closest_point(const P &r) const {
  P d= q - p;
  K a= dot(r - p, d), b;
  return sgn(a) > 0 ? sgn(a - (b= norm2(d))) < 0 ? p + a / b * d : q : p;
 }
 friend ostream &operator<<(ostream &os, const Segment &s) { return os << s.p << "---" << s.q; }
 friend Visualizer &operator<<(Visualizer &vis, const Segment &s) { return vis.ofs << "Segment " << s.p.x << " " << s.p.y << " " << s.q.x << " " << s.q.y << "\n", vis; }
};
// 1: properly crossing, 0: no intersect, 2: same line
template <class K> vector<Point<K>> cross_points(const Segment<K> &s, const Line<K> &l) {
 Point d= s.q - s.p;
 K a= cross(d, l.d), b= cross(l.p - s.p, l.d);
 if (sgn(a)) {
  if (b/= a; sgn(b) < 0 || sgn(b - 1) > 0) return {};  // no intersect
  else return {s.p + b * d};                           // properly crossing}
 }
 if (sgn(b)) return {};  // disjoint parallel
 return {s.p, s.q};      // same line
}
template <class K> vector<Point<K>> cross_points(const Line<K> &l, const Segment<K> &s) { return cross_points(s, l); }
// 2: same line, 0: no intersect, 1: ...
template <class K> vector<Point<K>> cross_points(const Segment<K> &s, const Segment<K> &t) {
 Point d= s.q - s.p, e= t.q - t.p;
 K a= cross(d, e), b= cross(t.p - s.p, e);
 if (sgn(a)) {
  if (b/= a; sgn(b) < 0 || sgn(b - 1) > 0) return {};                       // no intersect
  if (b= cross(d, s.p - t.p) / a; sgn(b) < 0 || sgn(b - 1) > 0) return {};  // no intersect
  return {t.p + b * e};                                                     // properly crossing
 }
 if (sgn(b)) return {};  // disjoint parallel
 vector<Point<K>> ps;    // same line
 auto insert_if_possible= [&](const Point<K> &p) {
  for (auto q: ps)
   if (p == q) return;
  ps.emplace_back(p);
 };
 if (sgn(dot(t.p - s.p, t.q - s.p)) <= 0) insert_if_possible(s.p);
 if (sgn(dot(t.p - s.q, t.q - s.q)) <= 0) insert_if_possible(s.q);
 if (sgn(dot(s.p - t.p, s.q - t.p)) <= 0) insert_if_possible(t.p);
 if (sgn(dot(s.p - t.q, s.q - t.q)) <= 0) insert_if_possible(t.q);
 return ps;
}
enum INTERSECTION { CROSSING, TOUCHING, DISJOINT, OVERLAP };
ostream &operator<<(ostream &os, INTERSECTION i) { return os << (i == CROSSING ? "CROSSING" : i == TOUCHING ? "TOUCHING" : i == DISJOINT ? "DISJOINT" : "OVERLAP"); }
template <class K> INTERSECTION intersection(const Segment<K> &s, const Segment<K> &t) {
 auto cp= cross_points(s, t);
 return cp.size() == 0 ? DISJOINT : cp.size() == 2 ? OVERLAP : cp[0] == s.p || cp[0] == s.q || cp[0] == t.p || cp[0] == t.q ? TOUCHING : CROSSING;
}
template <class K> make_long_t<K> dist2(const Segment<K> &s, const Point<K> &p) { return dist2(p, s.closest_point(p)); }
template <class K> make_long_t<K> dist2(const Point<K> &p, const Segment<K> &s) { return dist2(s, p); }
template <class K> make_long_t<K> dist2(const Segment<K> &s, const Line<K> &l) { return cross_points(s, l).size() ? 0 : min(dist2(s.p, l), dist2(s.q, l)); }
template <class K> make_long_t<K> dist2(const Line<K> &l, const Segment<K> &s) { return dist2(s, l); }
template <class K> make_long_t<K> dist2(const Segment<K> &s, const Segment<K> &t) { return cross_points(s, t).size() ? 0 : min({dist2(s, t.p), dist2(s, t.q), dist2(t, s.p), dist2(t, s.q)}); }
template <class K> Segment<K> Affine<K>::operator()(const Segment<K> &s) { return {(*this)(s.p), (*this)(s.q)}; }
}
#line 3 "src/Geometry/Polygon.hpp"
namespace geo {
// build counterclockwise
template <class K> class Polygon {
 using P= Point<K>;
 make_long_t<K> a2= 0;
protected:
 vector<P> dat;
 void build() {
  if (dat.empty()) return;
  a2= cross(dat.back(), dat[0]);
  for (int i= this->size(); --i;) a2+= cross(dat[i - 1], dat[i]);
  if (a2 < 0) reverse(dat.begin(), dat.end()), a2= -a2;
 }
 template <int opp> inline bool contain(const Segment<K> &s) const {
  assert(dat.size());
  if (where(s.p) == opp || where(s.q) == opp) return false;
  vector<P> ps, qs;
  for (const auto &e: edges())
   if (auto cp= cross_points(s, e); cp.size()) ps.insert(ps.end(), cp.begin(), cp.end());
  if (ps.empty()) return true;
  sort(ps.begin(), ps.end()), ps.erase(unique(ps.begin(), ps.end()), ps.end());
  for (int i= ps.size(); --i;) qs.emplace_back((ps[i] + ps[i - 1]) / 2);
  for (const auto &q: qs)
   if (where(q) == opp) return false;
  return true;
 }
public:
 Polygon() {}
 Polygon(const vector<P> &ps): dat(ps) { build(); }
 inline int prev(int i) const { return i ? i - 1 : (int)this->size() - 1; }
 inline int next(int i) const { return i + 1 >= (int)this->size() ? 0 : i + 1; }
 const P &operator[](int i) const { return dat[i]; }
 auto begin() const { return dat.begin(); }
 auto end() const { return dat.end(); }
 size_t size() const { return dat.size(); }
 vector<Segment<K>> edges() const {
  vector<Segment<K>> es;
  for (int i= 0, e= dat.size(); i < e; ++i) es.emplace_back(dat[i], dat[next(i)]);
  return es;
 }
 // assuming no self-intersections
 bool is_convex() const {
  assert(dat.size());
  for (int i= dat.size(); i--;)
   if (P a= dat[i], b= a - dat[prev(i)], c= dat[next(i)] - a; sgn(cross(b, c)) < 0) return false;
  return true;
 }
 make_long_t<K> area() const { return a2 / 2; }
 // for integer
 make_long_t<K> area2() const { return a2; }
 // 1: in, 0: on, -1: out
 int where(const P &p) const {
  assert(dat.size());
  bool in= false;
  for (int i= dat.size(); i--;) {
   Point a= dat[i] - p, b= dat[next(i)] - p;
   if (a.y > b.y) swap(a, b);
   int s= sgn(cross(a, b));
   if (!s && sgn(dot(a, b)) <= 0) return 0;
   if (s < 0 && sgn(a.y) <= 0 && 0 < sgn(b.y)) in= !in;
  }
  return in ? 1 : -1;
 }
 bool in(const Segment<K> &s) const { return contain<-1>(s); }
 bool out(const Segment<K> &s) const { return contain<+1>(s); }
 friend ostream &operator<<(ostream &os, const Polygon &g) {
  for (int i= 0, e= g.size(); i < e; ++i) os << "--" << g[i] << "-";
  return os;
 }
 friend Visualizer &operator<<(Visualizer &vis, const Polygon &g) {
  vis.ofs << "Polygon" << '\n';
  for (const auto &p: g) vis << p;
  return vis.ofs << "..." << '\n', vis;
 }
};
template <class K> make_long_t<K> dist2(const Polygon<K> &g, const Point<K> &p) {
 if (g.where(p) != -1) return 0;
 make_long_t<K> ret= numeric_limits<make_long_t<K>>::max();
 for (const auto &e: g.edges()) ret= min(ret, dist2(e, p));
 return ret;
}
template <class K> make_long_t<K> dist2(const Point<K> &p, const Polygon<K> &g) { return dist2(g, p); }
template <class K> make_long_t<K> dist2(const Polygon<K> &g, const Line<K> &l) {
 make_long_t<K> ret= numeric_limits<make_long_t<K>>::max();
 for (const auto &e: g.edges()) ret= min(ret, dist2(e, l));
 return ret;
}
template <class K> make_long_t<K> dist2(const Line<K> &l, const Polygon<K> &g) { return dist2(g, l); }
template <class K> make_long_t<K> dist2(const Polygon<K> &g, const Segment<K> &s) {
 if (g.where(s.p) != -1 || g.where(s.q) != -1) return 0;
 make_long_t<K> ret= numeric_limits<make_long_t<K>>::max();
 for (const auto &e: g.edges()) ret= min(ret, dist2(e, s));
 return ret;
}
template <class K> make_long_t<K> dist2(const Segment<K> &s, const Polygon<K> &g) { return dist2(g, s); }
template <class K> make_long_t<K> dist2(const Polygon<K> &g, const Polygon<K> &h) {
 make_long_t<K> ret= numeric_limits<make_long_t<K>>::max();
 for (const auto &e: g.edges()) ret= min(ret, dist2(h, e));
 return ret;
}
template <class K> Polygon<K> Affine<K>::operator()(const Polygon<K> &g) {
 vector<Point<K>> ps;
 for (const auto &p: g) ps.emplace_back((*this)(p));
 return Polygon(ps);
}
}
#line 3 "src/Geometry/Circle.hpp"
namespace geo {
template <class R> struct Circle {
 using P= Point<R>;
 P o;
 R r;
 Circle() {}
 Circle(const P &o, R r): o(o), r(r) {}
 long double area() const { return r * r * M_PI; }
 // +1: in, 0: on, -1: out
 int where(const P &p) const { return sgn(r * r - dist2(p, o)); }
 // +1: intersect, 0: contact, -1: disjoint
 int where(const Line<R> &l) const { return sgn(r * r - dist2(l, o)); }
 // true: c in *this
 bool in(const Circle &c) const {
  R a= c.r - r;
  return sgn(a) <= 0 && sgn(dist2(o, c.o) - a * a) <= 0;
 }
 vector<Line<R>> tangent(const P &p) const {
  P d= p - o, e= !d;
  R b= norm2(d), a= b - r * r;
  if (int s= sgn(a); s < 0) return {};
  else if (s == 0) return {{p, e}};
  d*= r, e*= sqrt(a);
  return {Line(p, !(d + e)), Line(p, !(d - e))};
 }
 friend ostream &operator<<(ostream &os, const Circle &c) { return os << c.o << " " << c.r; }
 friend Visualizer &operator<<(Visualizer &vis, const Circle &c) { return vis.ofs << "Circle " << c.o.x << " " << c.o.y << " " << c.r << '\n', vis; }
};
// 2: properly intersect, 1: contact, 0: disjoint, 3: same
// counter-clockwise of c and clockwise of d
template <class R> vector<Point<R>> cross_points(const Circle<R> &c, const Circle<R> &d) {
 Point v= d.o - c.o;
 R g= norm2(v), a= c.r - d.r, b= c.r + d.r;
 if (!sgn(g)) {
  if (sgn(a)) return {};
  return {{c.o.x + c.r, c.o.y}, {c.o.x - c.r, c.o.y}, {c.o.x, c.o.y + c.r}};
 }
 int in= sgn(g - a * a), out= sgn(g - b * b);
 if (in < 0 || out > 0) return {};
 if (!in) return {(c.r * d.o - d.r * c.o) / a};
 if (!out) return {(c.r * d.o + d.r * c.o) / b};
 R e= (a * b + g) / (g * 2);
 Point q= c.o + e * v, n= !v * sqrt(c.r * c.r / g - e * e);
 return {q - n, q + n};
}
// 2: properly intersect, 1: contact, 0: disjoint
// direction of l
template <class R> vector<Point<R>> cross_points(const Circle<R> &c, const Line<R> &l) {
 Point<R> v= l.p - c.o;
 R a= norm2(l.d), b= dot(l.d, v) / a, d= b * b - (norm2(v) - c.r * c.r) / a;
 int s= sgn(d);
 if (s < 0) return {};
 if (!s) return {l.p - b * l.d};
 d= sqrt(d);
 return {l.p - (b + d) * l.d, l.p - (b - d) * l.d};
}
template <class R> vector<Point<R>> cross_points(const Line<R> &l, const Circle<R> &c) { return cross_points(c, l); }
template <class R> vector<Point<R>> cross_points(const Circle<R> &c, const Segment<R> &s) {
 Point<R> u= s.q - s.p, v= s.p - c.o;
 R a= norm2(u), b= dot(u, v) / a, d= b * b - (norm2(v) - c.r * c.r) / a;
 int t= sgn(d);
 if (t < 0) return {};
 if (!t && sgn(b) <= 0 && sgn(1 + b) >= 0) return {s.p - b * u};
 d= sqrt(d), a= -b - d, b= -b + d;
 vector<Point<R>> ps;
 if (0 <= sgn(a) && sgn(a - 1) <= 0) ps.emplace_back(s.p + a * u);
 if (0 <= sgn(b) && sgn(b - 1) <= 0) ps.emplace_back(s.p + b * u);
 return ps;
}
template <class R> vector<Point<R>> cross_points(const Segment<R> &s, const Circle<R> &c) { return cross_points(c, s); }
template <class R> Circle<R> circumscribed_circle(const Point<R> &A, const Point<R> &B, const Point<R> &C) {
 Point u= !(B - A), v= C - A, o= (A + B + dot(C - B, v) / dot(u, v) * u) / 2;
 return {o, dist(A, o)};
}
template <class R> Circle<R> inscribed_circle(const Point<R> &A, const Point<R> &B, const Point<R> &C) {
 R a= dist(B, C), b= dist(C, A), c= dist(A, B), s= (a + b + c) / 2;
 return {(a * A + b * B + c * C) / (s * 2), sqrt((s - a) * (s - b) * (s - c) / s)};
}
template <class R> vector<Line<R>> common_tangent(const Circle<R> &c, const Circle<R> &d) {
 Point u= d.o - c.o, v= !u;
 R g= norm2(u), b;
 if (!sgn(g)) return {};  // same origin
 vector<Line<R>> ls;
 for (R a: {c.r - d.r, c.r + d.r}) {
  if (int s= sgn(b= g - a * a); !s) ls.emplace_back(Line(c.o + c.r * a / g * u, v));
  else if (s > 0) {
   Point x= a / g * u, y= sqrt(b) / g * v, e= x + y, f= x - y;
   ls.emplace_back(Line(c.o + c.r * e, !e)), ls.emplace_back(Line(c.o + c.r * f, !f));
  }
 }
 return ls;
}
}
#line 5 "src/Geometry/intersection_area.hpp"
namespace geo {
template <class R> R intersection_area(const Circle<R> &c, const Circle<R> &d) {
 R l= dist2(d.o, c.o), g= c.r + d.r;
 if (sgn(l - g * g) >= 0) return 0;
 R h= c.r - d.r, a= c.r * c.r, b= d.r * d.r;
 if (sgn(l - h * h) <= 0) return min(a, b) * M_PI;
 R m= sqrt(l), z= (a - b + l) / (m * 2), e= z / c.r, f= (m - z) / d.r;
 return a * acos(e) + b * acos(f) - m * sqrt(a - z * z);
}
template <class R> R intersection_area(const Circle<R> &c, const Polygon<R> &g) {
 using P= Point<R>;
 auto sub= [&](const P &p, const P &q) {
  P u= q - p;
  R a= norm2(u), b= dot(u, p) / a, e= c.r * c.r, d= b * b - (norm2(p) - e) / a;
  if (sgn(d) <= 0) return e * angle(p, q);
  d= sqrt(d);
  P x= p + clamp(-b - d, R(0), R(1)) * u, y= p + clamp(-b + d, R(0), R(1)) * u;
  return (cross(x, y) + e * (angle(p, x) + angle(y, q)));
 };
 vector<P> ps;
 for (const auto &p: g) ps.emplace_back(p - c.o);
 R ret= 0;
 for (int i= ps.size(); i--;) ret+= sub(ps[i], ps[g.next(i)]);
 return ret / 2;
}
template <class R> R intersection_area(const Polygon<R> &g, const Circle<R> &c) { return intersection_area(c, g); }
}
#line 8 "test/aoj/CGL_7_H.test.cpp"
using namespace std;
signed main() {
 cin.tie(0);
 ios::sync_with_stdio(false);
 using namespace geo;
 Circle<long double> c;
 int n;
 cin >> n >> c.r;
 vector<Point<long double>> ps(n);
 for (int i= 0; i < n; ++i) cin >> ps[i];
 cout << fixed << setprecision(12) << intersection_area(c, Polygon(ps)) << '\n';
 return 0;
}

Test cases

Env	Name	Status	Elapsed	Memory
g++-13	00_sample_00.in	AC	5 ms	4 MB
g++-13	00_sample_01.in	AC	4 ms	4 MB
g++-13	01_triangle_00.in	AC	4 ms	4 MB
g++-13	01_triangle_01.in	AC	4 ms	4 MB
g++-13	01_triangle_02.in	AC	4 ms	4 MB
g++-13	01_triangle_03.in	AC	4 ms	4 MB
g++-13	01_triangle_04.in	AC	4 ms	4 MB
g++-13	01_triangle_05.in	AC	4 ms	4 MB
g++-13	01_triangle_06.in	AC	4 ms	4 MB
g++-13	01_triangle_07.in	AC	4 ms	4 MB
g++-13	02_small_00.in	AC	4 ms	4 MB
g++-13	02_small_01.in	AC	4 ms	4 MB
g++-13	02_small_02.in	AC	4 ms	4 MB
g++-13	03_medium_00.in	AC	4 ms	4 MB
g++-13	03_medium_01.in	AC	4 ms	4 MB
g++-13	04_corner_00.in	AC	4 ms	4 MB
g++-13	04_corner_01.in	AC	4 ms	4 MB
g++-13	04_corner_02.in	AC	4 ms	4 MB
g++-13	04_corner_03.in	AC	4 ms	4 MB
g++-13	04_corner_04.in	AC	4 ms	4 MB
g++-13	05_corner_00.in	AC	4 ms	4 MB
g++-13	05_corner_01.in	AC	4 ms	4 MB
g++-13	05_corner_02.in	AC	4 ms	4 MB
g++-13	05_corner_03.in	AC	4 ms	4 MB
g++-13	05_corner_04.in	AC	4 ms	4 MB
g++-13	05_corner_05.in	AC	4 ms	4 MB
g++-13	05_corner_06.in	AC	4 ms	4 MB
g++-13	05_corner_07.in	AC	4 ms	4 MB
g++-13	06_maximum_00.in	AC	4 ms	4 MB
g++-13	06_maximum_01.in	AC	4 ms	4 MB
g++-13	06_maximum_02.in	AC	4 ms	4 MB
g++-13	06_maximum_03.in	AC	4 ms	4 MB
clang++-18	00_sample_00.in	AC	5 ms	4 MB
clang++-18	00_sample_01.in	AC	4 ms	4 MB
clang++-18	01_triangle_00.in	AC	4 ms	4 MB
clang++-18	01_triangle_01.in	AC	4 ms	4 MB
clang++-18	01_triangle_02.in	AC	4 ms	4 MB
clang++-18	01_triangle_03.in	AC	4 ms	4 MB
clang++-18	01_triangle_04.in	AC	4 ms	4 MB
clang++-18	01_triangle_05.in	AC	4 ms	4 MB
clang++-18	01_triangle_06.in	AC	4 ms	4 MB
clang++-18	01_triangle_07.in	AC	4 ms	4 MB
clang++-18	02_small_00.in	AC	4 ms	4 MB
clang++-18	02_small_01.in	AC	4 ms	4 MB
clang++-18	02_small_02.in	AC	4 ms	4 MB
clang++-18	03_medium_00.in	AC	4 ms	4 MB
clang++-18	03_medium_01.in	AC	4 ms	4 MB
clang++-18	04_corner_00.in	AC	4 ms	4 MB
clang++-18	04_corner_01.in	AC	4 ms	4 MB
clang++-18	04_corner_02.in	AC	4 ms	4 MB
clang++-18	04_corner_03.in	AC	4 ms	4 MB
clang++-18	04_corner_04.in	AC	4 ms	4 MB
clang++-18	05_corner_00.in	AC	4 ms	4 MB
clang++-18	05_corner_01.in	AC	4 ms	4 MB
clang++-18	05_corner_02.in	AC	4 ms	4 MB
clang++-18	05_corner_03.in	AC	4 ms	4 MB
clang++-18	05_corner_04.in	AC	4 ms	4 MB
clang++-18	05_corner_05.in	AC	4 ms	4 MB
clang++-18	05_corner_06.in	AC	4 ms	4 MB
clang++-18	05_corner_07.in	AC	4 ms	4 MB
clang++-18	06_maximum_00.in	AC	4 ms	4 MB
clang++-18	06_maximum_01.in	AC	4 ms	4 MB
clang++-18	06_maximum_02.in	AC	4 ms	4 MB
clang++-18	06_maximum_03.in	AC	4 ms	4 MB