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#include "geometry/common_area_cp.hpp"

#include "base.hpp" #include "point.hpp" #include "polygon.hpp" #include "distance_sp.hpp" #include "cross_point_cl.hpp" #include "is_intersect_cs.hpp" namespace geometry { // http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_H Real ca_cp_impl(const Circle &c, const Point &a, const Point &b) { auto va = c.p - a, vb = c.p - b; Real f = cross(va, vb), ret = 0; if(sign(f) == 0) return ret; if(sign(max(abs(va), abs(vb)) - c.r) <= 0) return f; if(sign(distance_sp(Segment(a, b), c.p) - c.r) >= 0) return norm(c.r) * arg(vb * conj(va)); auto tot = cross_point_cl(c, Line(a, b)); if(is_intersect_cs(c, Segment(a, b)) != 2 and dot(a - tot[0], b - tot[0]) < 0) { swap(tot[0], tot[1]); } tot.emplace(begin(tot), a); tot.emplace_back(b); for(int i = 1; i < (int) tot.size(); i++) { ret += ca_cp_impl(c, tot[i - 1], tot[i]); } return ret; } Real common_area_cp(const Circle &c, const Polygon &p) { if(p.size() < 3) return 0; Real A = 0; for(int i = 0; i < p.size(); i++) { A += ca_cp_impl(c, p[i], p[(i + 1) % p.size()]); } return A * 0.5; } }

#line 2 "geometry/base.hpp" namespace geometry { using Real = double; const Real EPS = 1e-8; const Real PI = acos(static_cast< Real >(-1)); enum { OUT, ON, IN }; inline int sign(const Real &r) { return r <= -EPS ? -1 : r >= EPS ? 1 : 0; } inline bool equals(const Real &a, const Real &b) { return sign(a - b) == 0; } } #line 3 "geometry/point.hpp" namespace geometry { using Point = complex< Real >; istream &operator>>(istream &is, Point &p) { Real a, b; is >> a >> b; p = Point(a, b); return is; } ostream &operator<<(ostream &os, const Point &p) { return os << real(p) << " " << imag(p); } Point operator*(const Point &p, const Real &d) { return Point(real(p) * d, imag(p) * d); } // rotate point p counterclockwise by theta rad Point rotate(Real theta, const Point &p) { return Point(cos(theta) * real(p) - sin(theta) * imag(p), sin(theta) * real(p) + cos(theta) * imag(p)); } Real cross(const Point &a, const Point &b) { return real(a) * imag(b) - imag(a) * real(b); } Real dot(const Point &a, const Point &b) { return real(a) * real(b) + imag(a) * imag(b); } bool compare_x(const Point &a, const Point &b) { return equals(real(a), real(b)) ? imag(a) < imag(b) : real(a) < real(b); } bool compare_y(const Point &a, const Point &b) { return equals(imag(a), imag(b)) ? real(a) < real(b) : imag(a) < imag(b); } using Points = vector< Point >; } #line 2 "geometry/polygon.hpp" #line 4 "geometry/polygon.hpp" namespace geometry { using Polygon = vector< Point >; using Polygons = vector< Polygon >; } #line 3 "geometry/line.hpp" namespace geometry { struct Line { Point a, b; Line() = default; Line(const Point &a, const Point &b) : a(a), b(b) {} Line(const Real &A, const Real &B, const Real &C) { // Ax+By=C if(equals(A, 0)) { assert(!equals(B, 0)); a = Point(0, C / B); b = Point(1, C / B); } else if(equals(B, 0)) { a = 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 &l) { return os << l.a << " to " << l.b; } friend istream &operator>>(istream &is, Line &l) { return is >> l.a >> l.b; } }; using Lines = vector< Line >; } #line 3 "geometry/segment.hpp" namespace geometry { struct Segment : Line { Segment() = default; using Line::Line; }; using Segments = vector< Segment >; } #line 2 "geometry/projection.hpp" #line 5 "geometry/projection.hpp" namespace geometry { // http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_A Point projection(const Line &l, const Point &p) { auto t = dot(p - l.a, l.a - l.b) / norm(l.a - l.b); return l.a + (l.a - l.b) * t; } } #line 3 "geometry/ccw.hpp" namespace geometry { // http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_C constexpr int COUNTER_CLOCKWISE = +1; constexpr int CLOCKWISE = -1; constexpr int ONLINE_BACK = +2; // c-a-b constexpr int ONLINE_FRONT = -2; // a-b-c constexpr int ON_SEGMENT = 0; // a-c-b int ccw(const Point &a, Point b, Point c) { b = b - a, c = c - a; if(sign(cross(b, c)) == +1) return COUNTER_CLOCKWISE; if(sign(cross(b, c)) == -1) return CLOCKWISE; if(sign(dot(b, c)) == -1) return ONLINE_BACK; if(norm(b) < norm(c)) return ONLINE_FRONT; return ON_SEGMENT; } } #line 4 "geometry/is_intersect_sp.hpp" namespace geometry { bool is_intersect_sp(const Segment &s, const Point &p) { return ccw(s.a, s.b, p) == ON_SEGMENT; } } #line 5 "geometry/distance_sp.hpp" namespace geometry { Real distance_sp(const Segment &s, const Point &p) { Point r = projection(s, p); if(is_intersect_sp(s, r)) return abs(r - p); return min(abs(s.a - p), abs(s.b - p)); } } #line 3 "geometry/circle.hpp" namespace geometry { struct Circle { Point p; Real r{}; Circle() = default; Circle(const Point &p, const Real &r) : p(p), r(r) {} }; using Circles = vector< Circle >; } #line 6 "geometry/cross_point_cl.hpp" namespace geometry { // http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_D Points cross_point_cl(const Circle &c, const Line &l) { Point pr = projection(l, c.p); if(equals(abs(pr - c.p), c.r)) return {pr}; Point e = (l.b - l.a) / abs(l.b - l.a); auto k = sqrt(norm(c.r) - norm(pr - c.p)); return {pr - e * k, pr + e * k}; } } #line 6 "geometry/is_intersect_cs.hpp" namespace geometry { int is_intersect_cs(const Circle &c, const Segment &l) { Point h = projection(l, c.p); if(sign(norm(h - c.p) - norm(c.r)) > 0) return 0; auto d1 = abs(c.p - l.a), d2 = abs(c.p - l.b); if(sign(c.r - d1) >= 0 && sign(c.r - d2) >= 0) return 0; if(sign(c.r - d1) < 0 && sign(d2 - c.r) > 0 || sign(d1 - c.r) > 0 && sign(c.r - d2) < 0) return 1; if(sign(dot(l.a - h, l.b - h)) < 0) return 2; return 0; } } #line 7 "geometry/common_area_cp.hpp" namespace geometry { // http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_H Real ca_cp_impl(const Circle &c, const Point &a, const Point &b) { auto va = c.p - a, vb = c.p - b; Real f = cross(va, vb), ret = 0; if(sign(f) == 0) return ret; if(sign(max(abs(va), abs(vb)) - c.r) <= 0) return f; if(sign(distance_sp(Segment(a, b), c.p) - c.r) >= 0) return norm(c.r) * arg(vb * conj(va)); auto tot = cross_point_cl(c, Line(a, b)); if(is_intersect_cs(c, Segment(a, b)) != 2 and dot(a - tot[0], b - tot[0]) < 0) { swap(tot[0], tot[1]); } tot.emplace(begin(tot), a); tot.emplace_back(b); for(int i = 1; i < (int) tot.size(); i++) { ret += ca_cp_impl(c, tot[i - 1], tot[i]); } return ret; } Real common_area_cp(const Circle &c, const Polygon &p) { if(p.size() < 3) return 0; Real A = 0; for(int i = 0; i < p.size(); i++) { A += ca_cp_impl(c, p[i], p[(i + 1) % p.size()]); } return A * 0.5; } }