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:warning: geo/polygon.hpp

Depends on

Code

#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 U(a) * P.x + U(b) * P.y + U(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() {}
  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);
}

// 4 次式まで登場している、オーバーフロー注意！
// 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;
  T abs_a = (a < 0 ? -a : a);
  T abs_b = (b < 0 ? -b : b);
  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);
  }
};
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