geo/convex_polygon_edge_voronoi.hpp

• View this file on GitHub
• Last update: 2026-08-19 20:31:51+09:00
• Include: #include "geo/convex_polygon_edge_voronoi.hpp"

Depends on

• geo/base.hpp

Code

#include "geo/base.hpp"

// 20260812, written by GPT-5.6 Sol
//
// A: CCW convex polygon, consecutive edges should not be collinear.
// Voronoi diagram of the edges of a convex polygon, restricted to its interior.
// Each Voronoi edge is represented by
//   center(t) = p1 + t (p2 - p1)
//   radius(t) = r1 + t (r2 - r1), 0 <= t <= 1.
// O(N log N) time.
// If a circle is tangent to 4 or more edges simultaneously,
// the combinatorial structure may be tie-broken arbitrarily by floating errors.
template <typename P, typename Re>
struct Convex_Polygon_Edge_Voronoi {
  using RP = Point<Re>;
  struct Edge {
    int i, j;
    RP p1, p2;
    Re r1, r2;
  };

  int N;
  vc<P> A;

  // polygon edge i:
  //   U[i].dot(x) <= H[i]
  // where U[i] is the unit outward normal.
  vc<RP> U;
  vc<Re> H;

  vc<Edge> edges;
  vc<vc<int>> incident;

  Convex_Polygon_Edge_Voronoi(vc<P> A) : N(len(A)), A(A) { build(); }

  RP point(const Edge& e, Re t) const { return e.p1 + (e.p2 - e.p1) * t; }
  Re radius(const Edge& e, Re t) const { return e.r1 + (e.r2 - e.r1) * t; }
  pair<RP, RP> segment(const Edge& e) const { return {e.p1, e.p2}; }

  // Voronoi cell of polygon edge i.
  // Collinear points on the boundary may be removed.
  vc<RP> cell(int i) const {
    vc<RP> X;
    X.eb(RP(A[i]));
    X.eb(RP(A[(i + 1) % N]));

    for (int k : incident[i]) {
      X.eb(edges[k].p1);
      X.eb(edges[k].p2);
    }

    sort(all(X), [&](const RP& a, const RP& b) -> bool {
      if (a.x != b.x) return a.x < b.x;
      return a.y < b.y;
    });
    X.erase(unique(all(X),
                   [&](const RP& a, const RP& b) -> bool {
                     return a.x == b.x && a.y == b.y;
                   }),
            X.end());
    if (len(X) <= 2) return X;
    vc<RP> H;
    auto push = [&](RP p) -> void {
      while (len(H) >= 2) {
        RP a = H[len(H) - 2];
        RP b = H[len(H) - 1];
        if ((b - a).det(p - b) > 0) break;
        H.pop_back();
      }
      H.eb(p);
    };
    for (auto& p : X) push(p);
    int k = len(H);
    for (int i = len(X) - 2; i >= 0; --i) push(X[i]);
    H.pop_back();
    return H;
  }

 private:
  void add_edge(int i, int j, RP p1, Re r1, RP p2, Re r2) {
    // Except for tiny errors around simultaneous events,
    // radius is nondecreasing along a medial-axis branch.
    if (r1 > r2) {
      swap(p1, p2);
      swap(r1, r2);
    }

    int k = len(edges);
    edges.eb(Edge{i, j, p1, p2, r1, r2});
    incident[i].eb(k);
    incident[j].eb(k);
  }

  // For three consecutive active polygon edges a,b,c,
  // returns the center and radius where b disappears.
  pair<RP, Re> event(int a, int b, int c) const {
    RP p = U[a] - U[b];
    RP q = U[b] - U[c];
    Re s = H[a] - H[b];
    Re t = H[b] - H[c];
    Re det = p.det(q);
    assert(det != Re(0));
    RP x((s * q.y - p.y * t) / det, (p.x * t - s * q.x) / det);
    Re r = H[b] - U[b].dot(x);
    return {x, r};
  }

  void build() {
    assert(N >= 3);

    U.resize(N);
    H.resize(N);
    incident.resize(N);

    FOR(i, N) {
      int j = (i + 1) % N;
      Re dx = Re(A[j].x) - Re(A[i].x);
      Re dy = Re(A[j].y) - Re(A[i].y);
      Re d = sqrt(dx * dx + dy * dy);
      // A is CCW, so the right normal is outward.
      U[i] = RP(dy / d, -dx / d);
      H[i] = U[i].dot(RP(A[i]));
    }

    vc<int> nxt(N), pre(N);
    FOR(i, N) {
      nxt[i] = (i + 1) % N;
      pre[nxt[i]] = i;
    }

    vc<bool> alive(N, true);

    vc<RP> born_point(N);
    vc<Re> born_radius(N, Re(0));
    FOR(i, N) { born_point[i] = RP(A[(i + 1) % N]); }

    vc<Re> rm_time(N);
    vc<int> version(N, 0);

    pq_min<tuple<Re, int, int>> que;

    auto upd = [&](int b) -> void {
      if (!alive[b]) return;
      int a = pre[b];
      int c = nxt[b];
      auto [x, t] = event(a, b, c);
      rm_time[b] = t;
      ++version[b];
      que.emplace(t, b, version[b]);
    };

    FOR(i, N) upd(i);
    int n_alive = N;
    while (n_alive > 3) {
      Re t;
      int b, ver;
      while (1) {
        tie(t, b, ver) = que.top();
        que.pop();
        if (!alive[b]) continue;
        if (ver != version[b]) continue;
        break;
      }

      int a = pre[b], c = nxt[b];
      auto [x, r] = event(a, b, c);

      add_edge(a, b, born_point[a], born_radius[a], x, r);
      add_edge(b, c, born_point[b], born_radius[b], x, r);

      alive[b] = false;
      --n_alive;
      nxt[a] = c, pre[c] = a;

      born_point[a] = x;
      born_radius[a] = r;

      upd(a);
      upd(c);
    }

    // three active edges remain
    int a = -1;
    FOR(i, N) if (alive[i]) {
      a = i;
      break;
    }
    int b = nxt[a], c = nxt[b];
    assert(nxt[c] == a);

    auto [x, r] = event(a, b, c);
    add_edge(a, b, born_point[a], born_radius[a], x, r);
    add_edge(b, c, born_point[b], born_radius[b], x, r);
    add_edge(c, a, born_point[c], born_radius[c], x, r);
  }
};

#line 1 "geo/convex_polygon_edge_voronoi.hpp"

#line 1 "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) {}

  template <typename U>
  Point(Point<U> p) : x(p.x), y(p.y) {
    static_assert(!is_integral_v<T> || is_integral_v<U>);
  }

  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; }
  bool is_same(Line other) {
    if (a * other.b != b * other.a) return 0;
    if (a * other.c != c * other.a) return 0;
    if (b * other.c != c * other.b) return 0;
    return 1;
  }
};

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 3 "geo/convex_polygon_edge_voronoi.hpp"

// 20260812, written by GPT-5.6 Sol
//
// A: CCW convex polygon, consecutive edges should not be collinear.
// Voronoi diagram of the edges of a convex polygon, restricted to its interior.
// Each Voronoi edge is represented by
//   center(t) = p1 + t (p2 - p1)
//   radius(t) = r1 + t (r2 - r1), 0 <= t <= 1.
// O(N log N) time.
// If a circle is tangent to 4 or more edges simultaneously,
// the combinatorial structure may be tie-broken arbitrarily by floating errors.
template <typename P, typename Re>
struct Convex_Polygon_Edge_Voronoi {
  using RP = Point<Re>;
  struct Edge {
    int i, j;
    RP p1, p2;
    Re r1, r2;
  };

  int N;
  vc<P> A;

  // polygon edge i:
  //   U[i].dot(x) <= H[i]
  // where U[i] is the unit outward normal.
  vc<RP> U;
  vc<Re> H;

  vc<Edge> edges;
  vc<vc<int>> incident;

  Convex_Polygon_Edge_Voronoi(vc<P> A) : N(len(A)), A(A) { build(); }

  RP point(const Edge& e, Re t) const { return e.p1 + (e.p2 - e.p1) * t; }
  Re radius(const Edge& e, Re t) const { return e.r1 + (e.r2 - e.r1) * t; }
  pair<RP, RP> segment(const Edge& e) const { return {e.p1, e.p2}; }

  // Voronoi cell of polygon edge i.
  // Collinear points on the boundary may be removed.
  vc<RP> cell(int i) const {
    vc<RP> X;
    X.eb(RP(A[i]));
    X.eb(RP(A[(i + 1) % N]));

    for (int k : incident[i]) {
      X.eb(edges[k].p1);
      X.eb(edges[k].p2);
    }

    sort(all(X), [&](const RP& a, const RP& b) -> bool {
      if (a.x != b.x) return a.x < b.x;
      return a.y < b.y;
    });
    X.erase(unique(all(X),
                   [&](const RP& a, const RP& b) -> bool {
                     return a.x == b.x && a.y == b.y;
                   }),
            X.end());
    if (len(X) <= 2) return X;
    vc<RP> H;
    auto push = [&](RP p) -> void {
      while (len(H) >= 2) {
        RP a = H[len(H) - 2];
        RP b = H[len(H) - 1];
        if ((b - a).det(p - b) > 0) break;
        H.pop_back();
      }
      H.eb(p);
    };
    for (auto& p : X) push(p);
    int k = len(H);
    for (int i = len(X) - 2; i >= 0; --i) push(X[i]);
    H.pop_back();
    return H;
  }

 private:
  void add_edge(int i, int j, RP p1, Re r1, RP p2, Re r2) {
    // Except for tiny errors around simultaneous events,
    // radius is nondecreasing along a medial-axis branch.
    if (r1 > r2) {
      swap(p1, p2);
      swap(r1, r2);
    }

    int k = len(edges);
    edges.eb(Edge{i, j, p1, p2, r1, r2});
    incident[i].eb(k);
    incident[j].eb(k);
  }

  // For three consecutive active polygon edges a,b,c,
  // returns the center and radius where b disappears.
  pair<RP, Re> event(int a, int b, int c) const {
    RP p = U[a] - U[b];
    RP q = U[b] - U[c];
    Re s = H[a] - H[b];
    Re t = H[b] - H[c];
    Re det = p.det(q);
    assert(det != Re(0));
    RP x((s * q.y - p.y * t) / det, (p.x * t - s * q.x) / det);
    Re r = H[b] - U[b].dot(x);
    return {x, r};
  }

  void build() {
    assert(N >= 3);

    U.resize(N);
    H.resize(N);
    incident.resize(N);

    FOR(i, N) {
      int j = (i + 1) % N;
      Re dx = Re(A[j].x) - Re(A[i].x);
      Re dy = Re(A[j].y) - Re(A[i].y);
      Re d = sqrt(dx * dx + dy * dy);
      // A is CCW, so the right normal is outward.
      U[i] = RP(dy / d, -dx / d);
      H[i] = U[i].dot(RP(A[i]));
    }

    vc<int> nxt(N), pre(N);
    FOR(i, N) {
      nxt[i] = (i + 1) % N;
      pre[nxt[i]] = i;
    }

    vc<bool> alive(N, true);

    vc<RP> born_point(N);
    vc<Re> born_radius(N, Re(0));
    FOR(i, N) { born_point[i] = RP(A[(i + 1) % N]); }

    vc<Re> rm_time(N);
    vc<int> version(N, 0);

    pq_min<tuple<Re, int, int>> que;

    auto upd = [&](int b) -> void {
      if (!alive[b]) return;
      int a = pre[b];
      int c = nxt[b];
      auto [x, t] = event(a, b, c);
      rm_time[b] = t;
      ++version[b];
      que.emplace(t, b, version[b]);
    };

    FOR(i, N) upd(i);
    int n_alive = N;
    while (n_alive > 3) {
      Re t;
      int b, ver;
      while (1) {
        tie(t, b, ver) = que.top();
        que.pop();
        if (!alive[b]) continue;
        if (ver != version[b]) continue;
        break;
      }

      int a = pre[b], c = nxt[b];
      auto [x, r] = event(a, b, c);

      add_edge(a, b, born_point[a], born_radius[a], x, r);
      add_edge(b, c, born_point[b], born_radius[b], x, r);

      alive[b] = false;
      --n_alive;
      nxt[a] = c, pre[c] = a;

      born_point[a] = x;
      born_radius[a] = r;

      upd(a);
      upd(c);
    }

    // three active edges remain
    int a = -1;
    FOR(i, N) if (alive[i]) {
      a = i;
      break;
    }
    int b = nxt[a], c = nxt[b];
    assert(nxt[c] == a);

    auto [x, r] = event(a, b, c);
    add_edge(a, b, born_point[a], born_radius[a], x, r);
    add_edge(b, c, born_point[b], born_radius[b], x, r);
    add_edge(c, a, born_point[c], born_radius[c], x, r);
  }
};

Back to top page