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#include "geo/polygon.hpp"
#include "geo/base.hpp" #include "geo/cross_point.hpp" template <typename T> struct Polygon { vc<Point<T>> point; T a; Polygon(vc<Point<T>> point) : point(point) { build(); } int size() { return len(point); } template <typename REAL> REAL area() { return a * 0.5; } T area_2() { return a; } bool is_convex() { FOR(j, len(point)) { int i = (j == 0 ? len(point) - 1 : j - 1); int k = (j == len(point) - 1 ? 0 : j + 1); if ((point[j] - point[i]).det(point[k] - point[j]) < 0) return false; } return true; } // 中:1, 境界:0, 外:-1. int side(Point<T> p) { int n = len(point); FOR(i, n) if (point[i] == p) return 0; FOR(i, n) { Point<T> A = point[i], B = point[(i + 1) % n]; if ((p - A).det(B - A) != 0) continue; if ((p - A).dot(B - A) >= 0 && (p - B).dot(A - B) >= 0) return 0; } // p から x 方向に (+1, +eps) 方向にのびる半直線との交差を考える int cnt = 0; FOR(i, n) { Point<T> A = point[i], B = point[(i + 1) % n]; FOR(2) { swap(A, B); if (A.y > p.y) continue; if (B.y <= p.y) continue; if ((A - p).det(B - p) > 0) ++cnt; } } return (cnt % 2 == 0 ? -1 : 1); } // point[i] の近傍だけで見た side // https://github.com/maspypy/library/blob/main/geo/polygon_side.png int side_at(int i, Point<T> p) { int n = len(point); p -= point[i]; if (p.x == T(0) && p.y == T(0)) return 0; Point<T> L = point[(i + 1) % n] - point[i]; Point<T> R = point[(i + n - 1) % n] - point[i]; auto sgn = [&](T x) -> int { if (x == T(0)) return 0; return (x > T(0) ? 1 : -1); }; int x = sgn(L.det(p)) + sgn(p.det(R)) + sgn(R.det(L)); if (x == 0) return x; return (x > 0 ? 1 : -1); } // 線分が内部・外部それぞれを通るか // https://atcoder.jp/contests/jag2016-domestic/tasks/jag2016secretspring_e // https://atcoder.jp/contests/JAG2014Spring/tasks/icpc2014spring_f pair<bool, bool> side_segment(Point<T> L, Point<T> R) { Segment<T> S(L, R); // まず線分と非自明に交わるパターン int n = len(point); FOR(i, n) { Segment<T> S2(point[i], point[(i + 1) % n]); if (count_cross(S, S2, false) == 1) { return {1, 1}; } } bool in = 0, out = 0; if (side(L) == 1 || side(R) == 1) in = 1; if (side(L) == -1 || side(R) == -1) out = 1; FOR(i, n) { if (!S.contain(point[i])) continue; for (auto& p: {L, R}) { int k = side_at(i, p); if (k == 1) in = 1; if (k == -1) out = 1; } } return {in, out}; } private: void build() { a = 0; FOR(i, len(point)) { int j = (i + 1 == len(point) ? 0 : i + 1); a += point[i].det(point[j]); } assert(a > 0); } };
#line 2 "geo/base.hpp" template <typename T> struct Point { T x, y; Point() : x(0), y(0) {} template <typename A, typename B> Point(A x, B y) : x(x), y(y) {} template <typename A, typename B> Point(pair<A, B> p) : x(p.fi), y(p.se) {} Point operator+=(const Point p) { x += p.x, y += p.y; return *this; } Point operator-=(const Point p) { x -= p.x, y -= p.y; return *this; } Point operator+(Point p) const { return {x + p.x, y + p.y}; } Point operator-(Point p) const { return {x - p.x, y - p.y}; } bool operator==(Point p) const { return x == p.x && y == p.y; } bool operator!=(Point p) const { return x != p.x || y != p.y; } Point operator-() const { return {-x, -y}; } Point operator*(T t) const { return {x * t, y * t}; } Point operator/(T t) const { return {x / t, y / t}; } bool operator<(Point p) const { if (x != p.x) return x < p.x; return y < p.y; } T dot(const Point& other) const { return x * other.x + y * other.y; } T det(const Point& other) const { return x * other.y - y * other.x; } double norm() { return sqrtl(x * x + y * y); } double angle() { return atan2(y, x); } Point rotate(double theta) { static_assert(!is_integral<T>::value); double c = cos(theta), s = sin(theta); return Point{c * x - s * y, s * x + c * y}; } Point rot90(bool ccw) { return (ccw ? Point{-y, x} : Point{y, -x}); } }; #ifdef FASTIO template <typename T> void rd(Point<T>& p) { fastio::rd(p.x), fastio::rd(p.y); } template <typename T> void wt(Point<T>& p) { fastio::wt(p.x); fastio::wt(' '); fastio::wt(p.y); } #endif // A -> B -> C と進むときに、左に曲がるならば +1、右に曲がるならば -1 template <typename T> int ccw(Point<T> A, Point<T> B, Point<T> C) { T x = (B - A).det(C - A); if (x > 0) return 1; if (x < 0) return -1; return 0; } template <typename REAL, typename T, typename U> REAL dist(Point<T> A, Point<U> B) { REAL dx = REAL(A.x) - REAL(B.x); REAL dy = REAL(A.y) - REAL(B.y); return sqrt(dx * dx + dy * dy); } // ax+by+c template <typename T> struct Line { T a, b, c; Line(T a, T b, T c) : a(a), b(b), c(c) {} Line(Point<T> A, Point<T> B) { a = A.y - B.y, b = B.x - A.x, c = A.x * B.y - A.y * B.x; } Line(T x1, T y1, T x2, T y2) : Line(Point<T>(x1, y1), Point<T>(x2, y2)) {} template <typename U> U eval(Point<U> P) { return a * P.x + b * P.y + c; } template <typename U> T eval(U x, U y) { return a * x + b * y + c; } // 同じ直線が同じ a,b,c で表現されるようにする void normalize() { static_assert(is_same_v<T, int> || is_same_v<T, long long>); T g = gcd(gcd(abs(a), abs(b)), abs(c)); a /= g, b /= g, c /= g; if (b < 0) { a = -a, b = -b, c = -c; } if (b == 0 && a < 0) { a = -a, b = -b, c = -c; } } bool is_parallel(Line other) { return a * other.b - b * other.a == 0; } bool is_orthogonal(Line other) { return a * other.a + b * other.b == 0; } }; template <typename T> struct Segment { Point<T> A, B; Segment(Point<T> A, Point<T> B) : A(A), B(B) {} Segment(T x1, T y1, T x2, T y2) : Segment(Point<T>(x1, y1), Point<T>(x2, y2)) {} bool contain(Point<T> C) { T det = (C - A).det(B - A); if (det != 0) return 0; return (C - A).dot(B - A) >= 0 && (C - B).dot(A - B) >= 0; } Line<T> to_Line() { return Line(A, B); } }; template <typename REAL> struct Circle { Point<REAL> O; REAL r; Circle(Point<REAL> O, REAL r) : O(O), r(r) {} Circle(REAL x, REAL y, REAL r) : O(x, y), r(r) {} template <typename T> bool contain(Point<T> p) { REAL dx = p.x - O.x, dy = p.y - O.y; return dx * dx + dy * dy <= r * r; } }; #line 2 "geo/cross_point.hpp" #line 4 "geo/cross_point.hpp" // 平行でないことを仮定 template <typename REAL, typename T> Point<REAL> cross_point(const Line<T> L1, const Line<T> L2) { T det = L1.a * L2.b - L1.b * L2.a; assert(det != 0); REAL x = -REAL(L1.c) * L2.b + REAL(L1.b) * L2.c; REAL y = -REAL(L1.a) * L2.c + REAL(L1.c) * L2.a; return Point<REAL>(x / det, y / det); } // 浮動小数点数はエラー // 0: 交点なし // 1: 一意な交点 // 2:2 つ以上の交点(整数型を利用して厳密にやる) template <typename T> int count_cross(Segment<T> S1, Segment<T> S2, bool include_ends) { static_assert(!std::is_floating_point<T>::value); Line<T> L1 = S1.to_Line(); Line<T> L2 = S2.to_Line(); if (L1.is_parallel(L2)) { if (L1.eval(S2.A) != 0) return 0; // 4 点とも同一直線上にある T a1 = S1.A.x, b1 = S1.B.x; T a2 = S2.A.x, b2 = S2.B.x; if (a1 == b1) { a1 = S1.A.y, b1 = S1.B.y; a2 = S2.A.y, b2 = S2.B.y; } if (a1 > b1) swap(a1, b1); if (a2 > b2) swap(a2, b2); T a = max(a1, a2); T b = min(b1, b2); if (a < b) return 2; if (a > b) return 0; return (include_ends ? 1 : 0); } // 平行でない場合 T a1 = L2.eval(S1.A), b1 = L2.eval(S1.B); T a2 = L1.eval(S2.A), b2 = L1.eval(S2.B); if (a1 > b1) swap(a1, b1); if (a2 > b2) swap(a2, b2); bool ok1 = 0, ok2 = 0; if (include_ends) { ok1 = (a1 <= T(0)) && (T(0) <= b1); ok2 = (a2 <= T(0)) && (T(0) <= b2); } else { ok1 = (a1 < T(0)) && (T(0) < b1); ok2 = (a2 < T(0)) && (T(0) < b2); } return (ok1 && ok2 ? 1 : 0); } // https://codeforces.com/contest/607/problem/E template <typename REAL, typename T> vc<Point<REAL>> cross_point(const Circle<T> C, const Line<T> L) { T a = L.a, b = L.b, c = L.a * (C.O.x) + L.b * (C.O.y) + L.c; T r = C.r; bool SW = 0; if (abs(a) < abs(b)) { swap(a, b); SW = 1; } // ax+by+c=0, x^2+y^2=r^2 T D = 4 * c * c * b * b - 4 * (a * a + b * b) * (c * c - a * a * r * r); if (D < 0) return {}; REAL sqD = sqrtl(D); REAL y1 = (-2 * b * c + sqD) / (2 * (a * a + b * b)); REAL y2 = (-2 * b * c - sqD) / (2 * (a * a + b * b)); REAL x1 = (-b * y1 - c) / a; REAL x2 = (-b * y2 - c) / a; if (SW) swap(x1, y1), swap(x2, y2); x1 += C.O.x, x2 += C.O.x; y1 += C.O.y, y2 += C.O.y; if (D == 0) return {Point<REAL>(x1, y1)}; return {Point<REAL>(x1, y1), Point<REAL>(x2, y2)}; } // https://codeforces.com/contest/2/problem/C template <typename REAL, typename T> tuple<bool, Point<T>, Point<T>> cross_point_circle(Circle<T> C1, Circle<T> C2) { using P = Point<T>; P O{0, 0}; P A = C1.O, B = C2.O; if (A == B) return {false, O, O}; T d = (B - A).norm(); REAL cos_val = (C1.r * C1.r + d * d - C2.r * C2.r) / (2 * C1.r * d); if (cos_val < -1 || 1 < cos_val) return {false, O, O}; REAL t = acos(cos_val); REAL u = (B - A).angle(); P X = A + P{C1.r * cos(u + t), C1.r * sin(u + t)}; P Y = A + P{C1.r * cos(u - t), C1.r * sin(u - t)}; return {true, X, Y}; } #line 3 "geo/polygon.hpp" template <typename T> struct Polygon { vc<Point<T>> point; T a; Polygon(vc<Point<T>> point) : point(point) { build(); } int size() { return len(point); } template <typename REAL> REAL area() { return a * 0.5; } T area_2() { return a; } bool is_convex() { FOR(j, len(point)) { int i = (j == 0 ? len(point) - 1 : j - 1); int k = (j == len(point) - 1 ? 0 : j + 1); if ((point[j] - point[i]).det(point[k] - point[j]) < 0) return false; } return true; } // 中:1, 境界:0, 外:-1. int side(Point<T> p) { int n = len(point); FOR(i, n) if (point[i] == p) return 0; FOR(i, n) { Point<T> A = point[i], B = point[(i + 1) % n]; if ((p - A).det(B - A) != 0) continue; if ((p - A).dot(B - A) >= 0 && (p - B).dot(A - B) >= 0) return 0; } // p から x 方向に (+1, +eps) 方向にのびる半直線との交差を考える int cnt = 0; FOR(i, n) { Point<T> A = point[i], B = point[(i + 1) % n]; FOR(2) { swap(A, B); if (A.y > p.y) continue; if (B.y <= p.y) continue; if ((A - p).det(B - p) > 0) ++cnt; } } return (cnt % 2 == 0 ? -1 : 1); } // point[i] の近傍だけで見た side // https://github.com/maspypy/library/blob/main/geo/polygon_side.png int side_at(int i, Point<T> p) { int n = len(point); p -= point[i]; if (p.x == T(0) && p.y == T(0)) return 0; Point<T> L = point[(i + 1) % n] - point[i]; Point<T> R = point[(i + n - 1) % n] - point[i]; auto sgn = [&](T x) -> int { if (x == T(0)) return 0; return (x > T(0) ? 1 : -1); }; int x = sgn(L.det(p)) + sgn(p.det(R)) + sgn(R.det(L)); if (x == 0) return x; return (x > 0 ? 1 : -1); } // 線分が内部・外部それぞれを通るか // https://atcoder.jp/contests/jag2016-domestic/tasks/jag2016secretspring_e // https://atcoder.jp/contests/JAG2014Spring/tasks/icpc2014spring_f pair<bool, bool> side_segment(Point<T> L, Point<T> R) { Segment<T> S(L, R); // まず線分と非自明に交わるパターン int n = len(point); FOR(i, n) { Segment<T> S2(point[i], point[(i + 1) % n]); if (count_cross(S, S2, false) == 1) { return {1, 1}; } } bool in = 0, out = 0; if (side(L) == 1 || side(R) == 1) in = 1; if (side(L) == -1 || side(R) == -1) out = 1; FOR(i, n) { if (!S.contain(point[i])) continue; for (auto& p: {L, R}) { int k = side_at(i, p); if (k == 1) in = 1; if (k == -1) out = 1; } } return {in, out}; } private: void build() { a = 0; FOR(i, len(point)) { int j = (i + 1 == len(point) ? 0 : i + 1); a += point[i].det(point[j]); } assert(a > 0); } };