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:heavy_check_mark: geo/convex_polygon.hpp

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Code

#include "geo/base.hpp"
#include "geo/convex_hull.hpp"

// n=2 は現状サポートしていない
template <typename T>
struct ConvexPolygon {
  using P = Point<T>;
  int n;
  vc<P> point;

  ConvexPolygon(vc<P> point_) : n(len(point_)), point(point_) {
    assert(n >= 3);
    FOR(i, n) {
      int j = nxt_idx(i), k = nxt_idx(j);
      assert((point[j] - point[i]).det(point[k] - point[i]) > 0);
    }
  }

  // 比較関数 comp(i,j)
  template <typename F>
  int periodic_min_comp(F comp) {
    int L = 0, M = n, R = n + n;
    while (1) {
      if (R - L == 2) break;
      int L1 = (L + M) / 2, R1 = (M + R + 1) / 2;
      if (comp(L1 % n, M % n)) { R = M, M = L1; }
      elif (comp(R1 % n, M % n)) { L = M, M = R1; }
      else {
        L = L1, R = R1;
      }
    }
    return M % n;
  }

  int nxt_idx(int i) { return (i + 1 == n ? 0 : i + 1); }
  int prev_idx(int i) { return (i == 0 ? n - 1 : i - 1); }

  // 中:1, 境界:0, 外:-1. test した.
  int side(P p) {
    int L = 1, R = n - 1;
    T a = (point[L] - point[0]).det(p - point[0]);
    T b = (point[R] - point[0]).det(p - point[0]);
    if (a < 0 || b > 0) return -1;
    // p は 0 から見て [L,R] 方向
    while (R - L >= 2) {
      int M = (L + R) / 2;
      T c = (point[M] - point[0]).det(p - point[0]);
      if (c < 0)
        R = M, b = c;
      else
        L = M, a = c;
    }
    T c = (point[R] - point[L]).det(p - point[L]);
    T x = min({a, -b, c});
    if (x < 0) return -1;
    if (x > 0) return 1;
    // on triangle p[0]p[L]p[R]
    if (p == point[0]) return 0;
    if (c != 0 && a == 0 && L != 1) return 1;
    if (c != 0 && b == 0 && R != n - 1) return 1;
    return 0;
  }

  // return {min, idx}. test した.
  pair<T, int> min_dot(P p) {
    int idx = periodic_min_comp([&](int i, int j) -> bool {
      return point[i].dot(p) < point[j].dot(p);
    });
    return {point[idx].dot(p), idx};
  }

  // return {max, idx}. test した.
  pair<T, int> max_dot(P p) {
    int idx = periodic_min_comp([&](int i, int j) -> bool {
      return point[i].dot(p) > point[j].dot(p);
    });
    return {point[idx].dot(p), idx};
  }

  // p から見える範囲. p 辺に沿って見えるところも見えるとする. test した.
  // 多角形からの反時計順は [l,r] だが p から見た偏角順は [r,l] なので注意
  pair<int, int> visible_range(P p) {
    int a = periodic_min_comp([&](int i, int j) -> bool {
      return ((point[i] - p).det(point[j] - p) < 0);
    });
    int b = periodic_min_comp([&](int i, int j) -> bool {
      return ((point[i] - p).det(point[j] - p) > 0);
    });
    if ((p - point[a]).det(p - point[prev_idx(a)]) == T(0)) a = prev_idx(a);
    if ((p - point[b]).det(p - point[nxt_idx(b)]) == T(0)) b = nxt_idx(b);
    return {a, b};
  }

  // 線分が「内部と」交わるか
  // https://codeforces.com/contest/1906/problem/D
  bool check_cross(P A, P B) {
    FOR(2) {
      swap(A, B);
      auto [a, b] = visible_range(A);
      if ((point[a] - A).det(B - A) >= 0) return 0;
      if ((point[b] - A).det(B - A) <= 0) return 0;
    }
    return 1;
  }
};
#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+(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(Point other) { return x * other.x + y * other.y; }
  T det(Point other) { 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};
  }
};

#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) {
    static_assert(is_integral<T>::value);
    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;
  }
};

template <typename T>
struct Polygon {
  vc<Point<T>> points;
  T a;

  template <typename A, typename B>
  Polygon(vc<pair<A, B>> pairs) {
    for (auto&& [a, b]: pairs) points.eb(Point<T>(a, b));
    build();
  }
  Polygon(vc<Point<T>> points) : points(points) { build(); }

  int size() { return len(points); }

  template <typename REAL>
  REAL area() {
    return a * 0.5;
  }

  template <enable_if_t<is_integral<T>::value, int> = 0>
  T area_2() {
    return a;
  }

  bool is_convex() {
    FOR(j, len(points)) {
      int i = (j == 0 ? len(points) - 1 : j - 1);
      int k = (j == len(points) - 1 ? 0 : j + 1);
      if ((points[j] - points[i]).det(points[k] - points[j]) < 0) return false;
    }
    return true;
  }

private:
  void build() {
    a = 0;
    FOR(i, len(points)) {
      int j = (i + 1 == len(points) ? 0 : i + 1);
      a += points[i].det(points[j]);
    }
    if (a < 0) {
      a = -a;
      reverse(all(points));
    }
  }
};
#line 2 "geo/convex_hull.hpp"

#line 4 "geo/convex_hull.hpp"

template <typename T>
vector<int> ConvexHull(vector<pair<T, T>>& XY, string mode = "full",
                       bool inclusive = false, bool sorted = false) {
  assert(mode == "full" || mode == "lower" || mode == "upper");
  ll N = XY.size();
  if (N == 1) return {0};
  if (N == 2) {
    if (XY[0] < XY[1]) return {0, 1};
    if (XY[1] < XY[0]) return {1, 0};
    if (inclusive) return {0, 1};
    return {0};
  }
  vc<int> I(N);
  if (sorted) {
    FOR(i, N) I[i] = i;
  } else {
    I = argsort(XY);
  }

  auto check = [&](ll i, ll j, ll k) -> bool {
    auto xi = XY[i].fi, yi = XY[i].se;
    auto xj = XY[j].fi, yj = XY[j].se;
    auto xk = XY[k].fi, yk = XY[k].se;
    auto dx1 = xj - xi, dy1 = yj - yi;
    auto dx2 = xk - xj, dy2 = yk - yj;
    T det = dx1 * dy2 - dy1 * dx2;
    return (inclusive ? det >= 0 : det > 0);
  };

  auto calc = [&]() {
    vector<int> P;
    for (auto&& k: I) {
      while (P.size() > 1) {
        auto i = P[P.size() - 2];
        auto j = P[P.size() - 1];
        if (check(i, j, k)) break;
        P.pop_back();
      }
      P.eb(k);
    }
    return P;
  };

  vc<int> P;
  if (mode == "full" || mode == "lower") {
    vc<int> Q = calc();
    P.insert(P.end(), all(Q));
  }
  if (mode == "full" || mode == "upper") {
    if (!P.empty()) P.pop_back();
    reverse(all(I));
    vc<int> Q = calc();
    P.insert(P.end(), all(Q));
  }
  if (mode == "upper") reverse(all(P));
  while (len(P) >= 2 && XY[P[0]] == XY[P.back()]) P.pop_back();
  return P;
}

template <typename T>
vector<int> ConvexHull(vector<Point<T>>& XY, string mode = "full",
                       bool inclusive = false, bool sorted = false) {
  assert(mode == "full" || mode == "lower" || mode == "upper");
  ll N = XY.size();
  if (N == 1) return {0};
  if (N == 2) {
    if (XY[0] < XY[1]) return {0, 1};
    if (XY[1] < XY[0]) return {1, 0};
    if (inclusive) return {0, 1};
    return {0};
  }
  vc<int> I(N);
  if (sorted) {
    FOR(i, N) I[i] = i;
  } else {
    I = argsort(XY);
  }

  auto check = [&](ll i, ll j, ll k) -> bool {
    auto xi = XY[i].x, yi = XY[i].y;
    auto xj = XY[j].x, yj = XY[j].y;
    auto xk = XY[k].x, yk = XY[k].y;
    auto dx1 = xj - xi, dy1 = yj - yi;
    auto dx2 = xk - xj, dy2 = yk - yj;
    T det = dx1 * dy2 - dy1 * dx2;
    return (inclusive ? det >= 0 : det > 0);
  };

  auto calc = [&]() {
    vector<int> P;
    for (auto&& k: I) {
      while (P.size() > 1) {
        auto i = P[P.size() - 2];
        auto j = P[P.size() - 1];
        if (check(i, j, k)) break;
        P.pop_back();
      }
      P.eb(k);
    }
    return P;
  };

  vc<int> P;
  if (mode == "full" || mode == "lower") {
    vc<int> Q = calc();
    P.insert(P.end(), all(Q));
  }
  if (mode == "full" || mode == "upper") {
    if (!P.empty()) P.pop_back();
    reverse(all(I));
    vc<int> Q = calc();
    P.insert(P.end(), all(Q));
  }
  if (mode == "upper") reverse(all(P));
  while (len(P) >= 2 && XY[P[0]] == XY[P.back()]) P.pop_back();
  return P;
}
#line 3 "geo/convex_polygon.hpp"

// n=2 は現状サポートしていない
template <typename T>
struct ConvexPolygon {
  using P = Point<T>;
  int n;
  vc<P> point;

  ConvexPolygon(vc<P> point_) : n(len(point_)), point(point_) {
    assert(n >= 3);
    FOR(i, n) {
      int j = nxt_idx(i), k = nxt_idx(j);
      assert((point[j] - point[i]).det(point[k] - point[i]) > 0);
    }
  }

  // 比較関数 comp(i,j)
  template <typename F>
  int periodic_min_comp(F comp) {
    int L = 0, M = n, R = n + n;
    while (1) {
      if (R - L == 2) break;
      int L1 = (L + M) / 2, R1 = (M + R + 1) / 2;
      if (comp(L1 % n, M % n)) { R = M, M = L1; }
      elif (comp(R1 % n, M % n)) { L = M, M = R1; }
      else {
        L = L1, R = R1;
      }
    }
    return M % n;
  }

  int nxt_idx(int i) { return (i + 1 == n ? 0 : i + 1); }
  int prev_idx(int i) { return (i == 0 ? n - 1 : i - 1); }

  // 中:1, 境界:0, 外:-1. test した.
  int side(P p) {
    int L = 1, R = n - 1;
    T a = (point[L] - point[0]).det(p - point[0]);
    T b = (point[R] - point[0]).det(p - point[0]);
    if (a < 0 || b > 0) return -1;
    // p は 0 から見て [L,R] 方向
    while (R - L >= 2) {
      int M = (L + R) / 2;
      T c = (point[M] - point[0]).det(p - point[0]);
      if (c < 0)
        R = M, b = c;
      else
        L = M, a = c;
    }
    T c = (point[R] - point[L]).det(p - point[L]);
    T x = min({a, -b, c});
    if (x < 0) return -1;
    if (x > 0) return 1;
    // on triangle p[0]p[L]p[R]
    if (p == point[0]) return 0;
    if (c != 0 && a == 0 && L != 1) return 1;
    if (c != 0 && b == 0 && R != n - 1) return 1;
    return 0;
  }

  // return {min, idx}. test した.
  pair<T, int> min_dot(P p) {
    int idx = periodic_min_comp([&](int i, int j) -> bool {
      return point[i].dot(p) < point[j].dot(p);
    });
    return {point[idx].dot(p), idx};
  }

  // return {max, idx}. test した.
  pair<T, int> max_dot(P p) {
    int idx = periodic_min_comp([&](int i, int j) -> bool {
      return point[i].dot(p) > point[j].dot(p);
    });
    return {point[idx].dot(p), idx};
  }

  // p から見える範囲. p 辺に沿って見えるところも見えるとする. test した.
  // 多角形からの反時計順は [l,r] だが p から見た偏角順は [r,l] なので注意
  pair<int, int> visible_range(P p) {
    int a = periodic_min_comp([&](int i, int j) -> bool {
      return ((point[i] - p).det(point[j] - p) < 0);
    });
    int b = periodic_min_comp([&](int i, int j) -> bool {
      return ((point[i] - p).det(point[j] - p) > 0);
    });
    if ((p - point[a]).det(p - point[prev_idx(a)]) == T(0)) a = prev_idx(a);
    if ((p - point[b]).det(p - point[nxt_idx(b)]) == T(0)) b = nxt_idx(b);
    return {a, b};
  }

  // 線分が「内部と」交わるか
  // https://codeforces.com/contest/1906/problem/D
  bool check_cross(P A, P B) {
    FOR(2) {
      swap(A, B);
      auto [a, b] = visible_range(A);
      if ((point[a] - A).det(B - A) >= 0) return 0;
      if ((point[b] - A).det(B - A) <= 0) return 0;
    }
    return 1;
  }
};
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