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:heavy_check_mark: verify/geometry/voronoi_diagram.test.cpp

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Code

#define PROBLEM "https://onlinejudge.u-aizu.ac.jp/problems/2160"
#define ERROR "1e-4"

#include "../../geometry/voronoi_diagram.hpp"
#include "../../geometry/half_plane_intersection.hpp"

#include <algorithm>
#include <array>
#include <cassert>
#include <cmath>
#include <cstdint>
#include <iomanip>
#include <iostream>
#include <limits>
#include <set>
#include <utility>
#include <vector>

namespace {

using m1une::geometry::Point;
using m1une::geometry::VoronoiDiagram;
using m1une::geometry::VoronoiEdge;
using m1une::geometry::VoronoiEdgeKind;
using Site = Point<long long>;
using RealPoint = Point<long double>;

long double squared_distance(const RealPoint& first, const RealPoint& second) {
    long double x = first.x - second.x;
    long double y = first.y - second.y;
    return x * x + y * y;
}

bool close(long double first, long double second, long double eps = 1e-8L) {
    return std::fabs(first - second) <=
           eps * std::max(
               1.0L,
               std::max(std::fabs(first), std::fabs(second))
           );
}

void check_boundary_point(
    const std::vector<Site>& sites,
    const VoronoiEdge& edge,
    const RealPoint& point
) {
    RealPoint first(sites[edge.first_site]);
    RealPoint second(sites[edge.second_site]);
    long double first_distance = squared_distance(point, first);
    long double second_distance = squared_distance(point, second);
    assert(close(first_distance, second_distance, 1e-7L));
    for (const Site& site : sites) {
        long double candidate = squared_distance(point, RealPoint(site));
        long double tolerance =
            1e-7L * std::max(1.0L, std::max(first_distance, candidate));
        assert(first_distance <= candidate + tolerance);
    }
}

bool naive_has_voronoi_edge(
    const std::vector<Site>& sites,
    int first_site,
    int second_site
) {
    RealPoint first(sites[first_site]);
    RealPoint second(sites[second_site]);
    RealPoint midpoint = (first + second) / 2.0L;
    RealPoint difference = second - first;
    RealPoint direction(difference.y, -difference.x);
    long double lower = -std::numeric_limits<long double>::infinity();
    long double upper = std::numeric_limits<long double>::infinity();

    for (const Site& integer_site : sites) {
        RealPoint site(integer_site);
        long double constant =
            squared_distance(midpoint, first) - squared_distance(midpoint, site);
        RealPoint shifted = midpoint + direction;
        long double coefficient =
            squared_distance(shifted, first) - squared_distance(shifted, site) -
            constant;
        if (std::fabs(coefficient) <= 1e-14L) {
            if (constant > 1e-12L) return false;
        } else {
            long double bound = -constant / coefficient;
            if (coefficient > 0) {
                upper = std::min(upper, bound);
            } else {
                lower = std::max(lower, bound);
            }
        }
    }
    return lower + 1e-10L < upper;
}

void check_diagram(const std::vector<Site>& sites) {
    VoronoiDiagram diagram = m1une::geometry::voronoi_diagram(sites);
    assert(diagram.cell_edges.size() == sites.size());

    std::set<std::pair<int, int>> actual_pairs;
    std::vector<int> cell_occurrences(diagram.edges.size(), 0);
    for (int site = 0; site < int(sites.size()); ++site) {
        for (int edge_index : diagram.cell_edges[site]) {
            assert(0 <= edge_index && edge_index < int(diagram.edges.size()));
            const VoronoiEdge& edge = diagram.edges[edge_index];
            assert(edge.first_site == site || edge.second_site == site);
            ++cell_occurrences[edge_index];
        }
    }

    for (int edge_index = 0; edge_index < int(diagram.edges.size()); ++edge_index) {
        const VoronoiEdge& edge = diagram.edges[edge_index];
        assert(cell_occurrences[edge_index] == 2);
        assert(0 <= edge.first_site && edge.first_site < int(sites.size()));
        assert(0 <= edge.second_site && edge.second_site < int(sites.size()));
        assert(edge.first_site < edge.second_site);
        assert(actual_pairs.emplace(edge.first_site, edge.second_site).second);

        if (edge.kind == VoronoiEdgeKind::Segment) {
            assert(0 <= edge.first_vertex);
            assert(edge.first_vertex < int(diagram.vertices.size()));
            assert(0 <= edge.second_vertex);
            assert(edge.second_vertex < int(diagram.vertices.size()));
            assert(edge.first_vertex < edge.second_vertex);
            assert(close(edge.point.x, diagram.vertices[edge.first_vertex].x));
            assert(close(edge.point.y, diagram.vertices[edge.first_vertex].y));
            RealPoint expected_direction =
                diagram.vertices[edge.second_vertex] -
                diagram.vertices[edge.first_vertex];
            assert(close(edge.direction.x, expected_direction.x));
            assert(close(edge.direction.y, expected_direction.y));
            for (long double parameter : std::array<long double, 3>{0, 0.5L, 1}) {
                check_boundary_point(
                    sites,
                    edge,
                    edge.point + edge.direction * parameter
                );
            }
        } else if (edge.kind == VoronoiEdgeKind::Ray) {
            assert(0 <= edge.first_vertex);
            assert(edge.first_vertex < int(diagram.vertices.size()));
            assert(edge.second_vertex == -1);
            assert(close(edge.point.x, diagram.vertices[edge.first_vertex].x));
            assert(close(edge.point.y, diagram.vertices[edge.first_vertex].y));
            assert(close(m1une::geometry::norm(edge.direction), 1));
            for (long double parameter : std::array<long double, 3>{0, 1, 100}) {
                check_boundary_point(
                    sites,
                    edge,
                    edge.point + edge.direction * parameter
                );
            }
        } else {
            assert(edge.kind == VoronoiEdgeKind::Line);
            assert(edge.first_vertex == -1);
            assert(edge.second_vertex == -1);
            assert(close(m1une::geometry::norm(edge.direction), 1));
            for (long double parameter : std::array<long double, 3>{-100, 0, 100}) {
                check_boundary_point(
                    sites,
                    edge,
                    edge.point + edge.direction * parameter
                );
            }
        }
    }

    for (int first = 0; first < int(sites.size()); ++first) {
        for (int second = first + 1; second < int(sites.size()); ++second) {
            bool actual = actual_pairs.contains(std::pair(first, second));
            bool expected = naive_has_voronoi_edge(sites, first, second);
            assert(actual == expected);
        }
    }
}

int count_kind(const VoronoiDiagram& diagram, VoronoiEdgeKind kind) {
    return int(std::count_if(
        diagram.edges.begin(),
        diagram.edges.end(),
        [&](const VoronoiEdge& edge) { return edge.kind == kind; }
    ));
}

void test_fixed() {
    check_diagram({});
    check_diagram(std::vector<Site>{Site(4, -2)});

    std::vector<Site> two_sites{Site(0, 0), Site(4, 0)};
    check_diagram(two_sites);
    VoronoiDiagram two = m1une::geometry::voronoi_diagram(two_sites);
    assert(two.vertices.empty());
    assert(two.edges.size() == 1);
    assert(two.edges[0].kind == VoronoiEdgeKind::Line);
    assert(close(two.edges[0].point.x, 2));
    assert(close(two.edges[0].point.y, 0));

    std::vector<Site> triangle{
        Site(0, 0),
        Site(6, 0),
        Site(0, 8),
    };
    check_diagram(triangle);
    VoronoiDiagram three = m1une::geometry::voronoi_diagram(triangle);
    assert(three.vertices.size() == 1);
    assert(three.edges.size() == 3);
    assert(count_kind(three, VoronoiEdgeKind::Ray) == 3);
    assert(close(three.vertices[0].x, 3));
    assert(close(three.vertices[0].y, 4));

    std::vector<Site> square{
        Site(0, 0),
        Site(2, 0),
        Site(2, 2),
        Site(0, 2),
    };
    check_diagram(square);
    VoronoiDiagram four = m1une::geometry::voronoi_diagram(square);
    assert(four.vertices.size() == 1);
    assert(four.edges.size() == 4);
    assert(count_kind(four, VoronoiEdgeKind::Ray) == 4);

    std::vector<Site> cocircular{
        Site(5, 0),
        Site(3, 4),
        Site(0, 5),
        Site(-3, 4),
        Site(-5, 0),
        Site(-3, -4),
        Site(0, -5),
        Site(3, -4),
    };
    check_diagram(cocircular);
    VoronoiDiagram eight = m1une::geometry::voronoi_diagram(cocircular);
    assert(eight.vertices.size() == 1);
    assert(eight.edges.size() == 8);
    assert(count_kind(eight, VoronoiEdgeKind::Ray) == 8);

    std::vector<Site> square_with_center = square;
    square_with_center.emplace_back(1, 1);
    check_diagram(square_with_center);
    VoronoiDiagram five =
        m1une::geometry::voronoi_diagram(square_with_center);
    assert(five.vertices.size() == 4);
    assert(five.edges.size() == 8);
    assert(count_kind(five, VoronoiEdgeKind::Segment) == 4);
    assert(count_kind(five, VoronoiEdgeKind::Ray) == 4);

    std::vector<Site> collinear{
        Site(-5, 3),
        Site(-1, 3),
        Site(2, 3),
        Site(9, 3),
    };
    check_diagram(collinear);
    VoronoiDiagram line = m1une::geometry::voronoi_diagram(collinear);
    assert(line.vertices.empty());
    assert(line.edges.size() == 3);
    assert(count_kind(line, VoronoiEdgeKind::Line) == 3);
}

void test_randomized() {
    std::uint64_t state = 0x243f6a8885a308d3ULL;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int trial = 0; trial < 1500; ++trial) {
        int size = int(random() % 11);
        std::set<std::pair<long long, long long>> used;
        std::vector<Site> sites;
        sites.reserve(size);
        while (int(sites.size()) < size) {
            long long x = static_cast<long long>(random() % 21) - 10;
            long long y = static_cast<long long>(random() % 21) - 10;
            if (used.emplace(x, y).second) sites.emplace_back(x, y);
        }
        check_diagram(sites);
    }
}

long double polygon_area(const std::vector<RealPoint>& polygon) {
    long double twice_area = 0;
    for (int index = 0; index < int(polygon.size()); ++index) {
        twice_area += m1une::geometry::cross(
            polygon[index],
            polygon[(index + 1) % polygon.size()]
        );
    }
    return std::fabs(twice_area) / 2;
}

}  // namespace

int main() {
    test_fixed();
    test_randomized();

    std::cout << std::fixed << std::setprecision(10);
    while (true) {
        int island_size, site_count;
        std::cin >> island_size >> site_count;
        if (island_size == 0 && site_count == 0) break;

        std::vector<Site> island(island_size);
        for (Site& point : island) std::cin >> point.x >> point.y;
        std::vector<Site> sites(site_count);
        for (Site& point : sites) std::cin >> point.x >> point.y;

        VoronoiDiagram diagram = m1une::geometry::voronoi_diagram(sites);
        for (int site = 0; site < site_count; ++site) {
            std::vector<m1une::geometry::Line<long double>> half_planes;
            half_planes.reserve(island_size + diagram.cell_edges[site].size());
            for (int index = 0; index < island_size; ++index) {
                half_planes.push_back(m1une::geometry::Line<long double>{
                    RealPoint(island[index]),
                    RealPoint(island[(index + 1) % island_size]),
                });
            }
            for (int edge_index : diagram.cell_edges[site]) {
                const VoronoiEdge& edge = diagram.edges[edge_index];
                int other = edge.first_site == site
                    ? edge.second_site
                    : edge.first_site;
                RealPoint first(sites[site]);
                RealPoint second(sites[other]);
                RealPoint midpoint = (first + second) / 2.0L;
                RealPoint difference = second - first;
                RealPoint direction(-difference.y, difference.x);
                half_planes.push_back(m1une::geometry::Line<long double>{
                    midpoint,
                    midpoint + direction,
                });
            }

            auto intersection =
                m1une::geometry::half_plane_intersection(half_planes);
            long double area = 0;
            if (
                intersection.status ==
                m1une::geometry::HalfPlaneIntersectionStatus::Bounded
            ) {
                area = polygon_area(intersection.polygon);
            }
            std::cout << area << '\n';
        }
    }
}
#line 1 "verify/geometry/voronoi_diagram.test.cpp"
#define PROBLEM "https://onlinejudge.u-aizu.ac.jp/problems/2160"
#define ERROR "1e-4"

#line 1 "geometry/voronoi_diagram.hpp"



#include <algorithm>
#include <array>
#include <cassert>
#include <cmath>
#include <concepts>
#include <cstddef>
#include <limits>
#include <numeric>
#include <utility>
#include <vector>

#line 1 "geometry/euclidean_mst.hpp"



#line 10 "geometry/euclidean_mst.hpp"
#include <tuple>
#line 13 "geometry/euclidean_mst.hpp"

#line 1 "ds/dsu/dsu.hpp"



#line 8 "ds/dsu/dsu.hpp"

namespace m1une {
namespace ds {

struct Dsu {
   private:
    int _n;
    // parent_or_size[i] is the parent of i if it's >= 0.
    // If it's < 0, then i is a root and -parent_or_size[i] is the size of the group.
    std::vector<int> parent_or_size;

    // Returns {new leader, absorbed leader}. The absorbed leader is -1 when
    // both vertices already belong to the same component.
    std::pair<int, int> merge_leaders(int a, int b) {
        int x = leader(a), y = leader(b);
        if (x == y) return {x, -1};
        if (-parent_or_size[x] < -parent_or_size[y]) std::swap(x, y);
        parent_or_size[x] += parent_or_size[y];
        parent_or_size[y] = x;
        return {x, y};
    }

   public:
    Dsu() : _n(0) {}
    explicit Dsu(int n) : _n(n), parent_or_size(n, -1) {}

    // Merges the group containing 'a' with the group containing 'b'.
    // Returns the leader of the merged group.
    int merge(int a, int b) {
        return merge_leaders(a, b).first;
    }

    // Invokes callback(new_leader, absorbed_leader) after an actual merge.
    // Returns the leader of the merged group.
    template <class Callback>
    int merge(int a, int b, Callback&& callback) {
        std::pair<int, int> merged = merge_leaders(a, b);
        if (merged.second != -1) callback(merged.first, merged.second);
        return merged.first;
    }

    // Returns true if 'a' and 'b' belong to the same group.
    bool same(int a, int b) {
        return leader(a) == leader(b);
    }

    // Returns the leader (representative) of the group containing 'a'.
    int leader(int a) {
        if (parent_or_size[a] < 0) return a;
        // Path compression
        return parent_or_size[a] = leader(parent_or_size[a]);
    }

    // Returns the size of the group containing 'a'.
    int size(int a) {
        return -parent_or_size[leader(a)];
    }

    // Returns a list of all groups, where each group is a vector of its elements.
    std::vector<std::vector<int>> groups() {
        std::vector<int> leader_buf(_n), group_size(_n);
        for (int i = 0; i < _n; i++) {
            leader_buf[i] = leader(i);
            group_size[leader_buf[i]]++;
        }
        std::vector<std::vector<int>> result(_n);
        for (int i = 0; i < _n; i++) {
            result[i].reserve(group_size[i]);
        }
        for (int i = 0; i < _n; i++) {
            result[leader_buf[i]].push_back(i);
        }
        result.erase(std::remove_if(result.begin(), result.end(), [&](const std::vector<int>& v) { return v.empty(); }),
                     result.end());
        return result;
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "geometry/point.hpp"



#line 7 "geometry/point.hpp"
#include <type_traits>

#line 1 "geometry/detail/floating_predicate.hpp"



namespace m1une {
namespace geometry {
namespace predicate_detail {

template <typename T>
constexpr T absolute(T value) {
    return value < T(0) ? -value : value;
}

template <typename T>
constexpr T max_value(T first, T second) {
    return first < second ? second : first;
}

template <typename T>
constexpr T vector_scale(T x, T y) {
    return max_value(absolute(x), absolute(y));
}

template <bool Exact, typename T>
constexpr int scaled_sign(T value, T scale, long double eps) {
    if constexpr (Exact) {
        return (value > T(0)) - (value < T(0));
    } else {
        const T tolerance = T(eps) * scale;
        return (value > tolerance) - (value < -tolerance);
    }
}

template <bool Exact, typename T>
constexpr T determinant_scale(T ax, T ay, T bx, T by) {
    if constexpr (Exact) {
        return T(0);
    } else {
        return vector_scale(ax, ay) * vector_scale(bx, by);
    }
}

template <bool Exact, typename T>
constexpr int determinant_sign(
    T ax,
    T ay,
    T bx,
    T by,
    long double eps
) {
    const T determinant = ax * by - ay * bx;
    return scaled_sign<Exact>(
        determinant,
        determinant_scale<Exact>(ax, ay, bx, by),
        eps
    );
}

template <bool Exact, typename T>
constexpr int orientation_sign(
    T direction_x,
    T direction_y,
    T offset_x,
    T offset_y,
    long double eps
) {
    const T determinant =
        direction_x * offset_y - direction_y * offset_x;
    T scale = T(0);
    if constexpr (!Exact) {
        const T direction_scale =
            vector_scale(direction_x, direction_y);
        scale = direction_scale * max_value(
            direction_scale,
            vector_scale(offset_x, offset_y)
        );
    }
    return scaled_sign<Exact>(determinant, scale, eps);
}

template <bool Exact, typename T>
constexpr int dot_sign(
    T ax,
    T ay,
    T bx,
    T by,
    long double eps
) {
    const T value = ax * bx + ay * by;
    T scale = T(0);
    if constexpr (!Exact) {
        scale = vector_scale(ax, ay) * vector_scale(bx, by);
    }
    return scaled_sign<Exact>(value, scale, eps);
}

}  // namespace predicate_detail
}  // namespace geometry
}  // namespace m1une


#line 10 "geometry/point.hpp"

namespace m1une {
namespace geometry {

template <typename T>
concept Coordinate = !std::same_as<std::remove_cv_t<T>, bool> &&
    (std::is_arithmetic_v<T> ||
     (std::copyable<T> && std::totally_ordered<T> && requires(T a, T b) {
         T(0);
         T(1);
         static_cast<long double>(a);
         { +a } -> std::same_as<T>;
         { -a } -> std::same_as<T>;
         { a + b } -> std::same_as<T>;
         { a - b } -> std::same_as<T>;
         { a * b } -> std::same_as<T>;
         { a / b } -> std::same_as<T>;
         { a += b } -> std::same_as<T&>;
         { a -= b } -> std::same_as<T&>;
     }));

// Custom coordinate types keep their own exact arithmetic.
template <typename T>
concept ExactCoordinate = Coordinate<T> && !std::floating_point<T>;

template <Coordinate T>
using wide_type = std::conditional_t<std::integral<T>, __int128_t,
    std::conditional_t<std::floating_point<T>, long double, T>>;

template <Coordinate T>
struct Point {
    T x;
    T y;

    constexpr Point() : x(0), y(0) {}
    constexpr Point(T x_value, T y_value) : x(x_value), y(y_value) {}

    template <Coordinate U>
    explicit constexpr Point(const Point<U>& other)
        : x(static_cast<T>(other.x)), y(static_cast<T>(other.y)) {}

    constexpr Point& operator+=(const Point& other) {
        x += other.x;
        y += other.y;
        return *this;
    }

    constexpr Point& operator-=(const Point& other) {
        x -= other.x;
        y -= other.y;
        return *this;
    }

    constexpr Point operator+() const {
        return *this;
    }

    constexpr Point operator-() const {
        return Point(-x, -y);
    }

    friend constexpr Point operator+(Point left, const Point& right) {
        return left += right;
    }

    friend constexpr Point operator-(Point left, const Point& right) {
        return left -= right;
    }

    friend constexpr bool operator==(const Point&, const Point&) = default;

    friend constexpr bool operator<(const Point& left, const Point& right) {
        if (left.x != right.x) return left.x < right.x;
        return left.y < right.y;
    }
};

template <Coordinate T>
constexpr Point<long double> centroid(const Point<T>& point) {
    return Point<long double>(point);
}

template <Coordinate T, typename Scalar>
requires (std::is_arithmetic_v<Scalar> || Coordinate<Scalar>)
constexpr auto operator*(const Point<T>& point, Scalar scalar) {
    using Result = std::common_type_t<T, Scalar>;
    return Point<Result>(
        Result(point.x) * Result(scalar),
        Result(point.y) * Result(scalar)
    );
}

template <typename Scalar, Coordinate T>
requires (std::is_arithmetic_v<Scalar> || Coordinate<Scalar>)
constexpr auto operator*(Scalar scalar, const Point<T>& point) {
    return point * scalar;
}

template <Coordinate T, typename Scalar>
requires (std::is_arithmetic_v<Scalar> || Coordinate<Scalar>)
constexpr auto operator/(const Point<T>& point, Scalar scalar) {
    using Result = std::common_type_t<T, Scalar>;
    return Point<Result>(
        Result(point.x) / Result(scalar),
        Result(point.y) / Result(scalar)
    );
}

template <Coordinate T>
constexpr wide_type<T> dot(const Point<T>& a, const Point<T>& b) {
    using W = wide_type<T>;
    return W(a.x) * W(b.x) + W(a.y) * W(b.y);
}

template <Coordinate T>
constexpr wide_type<T> cross(const Point<T>& a, const Point<T>& b) {
    using W = wide_type<T>;
    return W(a.x) * W(b.y) - W(a.y) * W(b.x);
}

template <Coordinate T>
constexpr wide_type<T> cross(
    const Point<T>& origin,
    const Point<T>& a,
    const Point<T>& b
) {
    using W = wide_type<T>;
    W ax = W(a.x) - W(origin.x);
    W ay = W(a.y) - W(origin.y);
    W bx = W(b.x) - W(origin.x);
    W by = W(b.y) - W(origin.y);
    return ax * by - ay * bx;
}

template <Coordinate T>
constexpr wide_type<T> norm2(const Point<T>& point) {
    return dot(point, point);
}

template <Coordinate T>
constexpr wide_type<T> distance2(const Point<T>& a, const Point<T>& b) {
    using W = wide_type<T>;
    W dx = W(a.x) - W(b.x);
    W dy = W(a.y) - W(b.y);
    return dx * dx + dy * dy;
}

template <Coordinate T>
long double norm(const Point<T>& point) {
    return std::hypot(
        static_cast<long double>(point.x),
        static_cast<long double>(point.y)
    );
}

template <Coordinate T>
long double distance(const Point<T>& a, const Point<T>& b) {
    return std::hypot(
        static_cast<long double>(a.x) - static_cast<long double>(b.x),
        static_cast<long double>(a.y) - static_cast<long double>(b.y)
    );
}

template <Coordinate T, typename M, typename N>
requires (std::is_arithmetic_v<M> || Coordinate<M>) &&
         (std::is_arithmetic_v<N> || Coordinate<N>)
constexpr Point<long double> internal_division_point(
    const Point<T>& a,
    const Point<T>& b,
    M m,
    N n
) {
    long double first_ratio = static_cast<long double>(m);
    long double second_ratio = static_cast<long double>(n);
    long double denominator = first_ratio + second_ratio;
    assert(denominator != 0);
    Point<long double> first(a);
    Point<long double> direction = Point<long double>(b) - first;
    return first + direction * (first_ratio / denominator);
}

template <Coordinate T, typename M, typename N>
requires (std::is_arithmetic_v<M> || Coordinate<M>) &&
         (std::is_arithmetic_v<N> || Coordinate<N>)
constexpr Point<long double> external_division_point(
    const Point<T>& a,
    const Point<T>& b,
    M m,
    N n
) {
    long double first_ratio = static_cast<long double>(m);
    long double second_ratio = static_cast<long double>(n);
    long double denominator = first_ratio - second_ratio;
    assert(denominator != 0);
    Point<long double> first(a);
    Point<long double> direction = Point<long double>(b) - first;
    return first + direction * (first_ratio / denominator);
}

template <Coordinate T>
constexpr int sign(wide_type<T> value, long double eps = 1e-12L) {
    return predicate_detail::scaled_sign<ExactCoordinate<T>>(
        value,
        wide_type<T>(1),
        eps
    );
}

template <Coordinate T>
constexpr int orientation(
    const Point<T>& a,
    const Point<T>& b,
    const Point<T>& c,
    long double eps = 1e-12L
) {
    using W = wide_type<T>;
    const W first_x = W(b.x) - W(a.x);
    const W first_y = W(b.y) - W(a.y);
    const W second_x = W(c.x) - W(a.x);
    const W second_y = W(c.y) - W(a.y);
    return predicate_detail::orientation_sign<ExactCoordinate<T>>(
        first_x,
        first_y,
        second_x,
        second_y,
        eps
    );
}

template <Coordinate T>
constexpr bool collinear(
    const Point<T>& a,
    const Point<T>& b,
    const Point<T>& c,
    long double eps = 1e-12L
) {
    return orientation(a, b, c, eps) == 0;
}

template <Coordinate T>
Point<long double> rotate(const Point<T>& point, long double angle) {
    long double cosine = std::cos(angle);
    long double sine = std::sin(angle);
    return Point<long double>(
        static_cast<long double>(point.x) * cosine -
            static_cast<long double>(point.y) * sine,
        static_cast<long double>(point.x) * sine +
            static_cast<long double>(point.y) * cosine
    );
}

template <Coordinate T>
Point<long double> normalized(const Point<T>& point) {
    long double length = norm(point);
    assert(length != 0);
    return Point<long double>(
        static_cast<long double>(point.x) / length,
        static_cast<long double>(point.y) / length
    );
}

}  // namespace geometry
}  // namespace m1une


#line 16 "geometry/euclidean_mst.hpp"

namespace m1une {
namespace geometry {

template <class T>
struct EuclideanMstEdge {
    int from;
    int to;
    T squared_distance;
};

template <class T>
struct EuclideanMst {
    long double cost;
    std::vector<EuclideanMstEdge<T>> edges;
};

namespace detail {

template <ExactCoordinate T>
class EuclideanDelaunay {
   private:
    using W = wide_type<T>;

    struct InternalPoint {
        W x;
        W y;

        friend bool operator==(const InternalPoint&, const InternalPoint&) = default;
    };

    struct Edge {
        int to;
        int ccw;
        int cw;
        int reverse;
        bool enabled = false;
    };

    std::vector<int> open_addresses;
    std::vector<InternalPoint> points;
    std::vector<Edge> edges;
    std::vector<int> duplicate_representative;

    static InternalPoint subtract(const InternalPoint& a, const InternalPoint& b) {
        return InternalPoint{a.x - b.x, a.y - b.y};
    }

    static W cross_product(const InternalPoint& a, const InternalPoint& b) {
        return a.x * b.y - a.y * b.x;
    }

    static W squared_norm(const InternalPoint& point) {
        return point.x * point.x + point.y * point.y;
    }

    static bool inside_circumcircle(
        InternalPoint a,
        InternalPoint b,
        InternalPoint c,
        const InternalPoint& d
    ) {
        a = subtract(a, d);
        b = subtract(b, d);
        c = subtract(c, d);
        W determinant = cross_product(b, c) * squared_norm(a)
                      + cross_product(c, a) * squared_norm(b)
                      + cross_product(a, b) * squared_norm(c);
        return determinant > 0;
    }

    int get_open_address() {
        if (open_addresses.empty()) {
            edges.push_back(Edge());
            return int(edges.size()) - 1;
        }
        int result = open_addresses.back();
        open_addresses.pop_back();
        return result;
    }

    std::pair<int, int> add_edge(int from, int to) {
        int forward = get_open_address();
        int backward = get_open_address();
        edges[forward].to = to;
        edges[forward].ccw = forward;
        edges[forward].cw = forward;
        edges[forward].reverse = backward;
        edges[forward].enabled = true;
        edges[backward].to = from;
        edges[backward].ccw = backward;
        edges[backward].cw = backward;
        edges[backward].reverse = forward;
        edges[backward].enabled = true;
        return {forward, backward};
    }

    void erase_directed_edge(int edge) {
        int ccw = edges[edge].ccw;
        int cw = edges[edge].cw;
        edges[ccw].cw = cw;
        edges[cw].ccw = ccw;
        edges[edge].enabled = false;
    }

    void erase_edge(int edge) {
        int reverse = edges[edge].reverse;
        erase_directed_edge(edge);
        erase_directed_edge(reverse);
        open_addresses.push_back(edge);
        open_addresses.push_back(reverse);
    }

    void insert_ccw_after(int edge, int position) {
        int next = edges[position].ccw;
        edges[edge].ccw = next;
        edges[next].cw = edge;
        edges[edge].cw = position;
        edges[position].ccw = edge;
    }

    void insert_cw_after(int edge, int position) {
        int next = edges[position].cw;
        edges[edge].cw = next;
        edges[next].ccw = edge;
        edges[edge].ccw = position;
        edges[position].cw = edge;
    }

    int orientation(int a, int b, int c) const {
        InternalPoint ab = subtract(points[b], points[a]);
        InternalPoint ac = subtract(points[c], points[a]);
        W value = cross_product(ab, ac);
        return (value > 0) - (value < 0);
    }

    std::pair<int, int> go_next(int edge) const {
        int vertex = edges[edge].to;
        int next_edge = edges[edges[edge].reverse].ccw;
        return {vertex, next_edge};
    }

    std::pair<int, int> go_previous(int edge) const {
        int vertex = edges[edges[edge].cw].to;
        int next_edge = edges[edges[edge].cw].reverse;
        return {vertex, next_edge};
    }

    std::tuple<int, int, int, int> lower_tangent(
        int left_vertex,
        int left_edge,
        int right_vertex,
        int right_edge
    ) const {
        while (true) {
            auto [next_left_vertex, next_left_edge] = go_previous(left_edge);
            if (orientation(right_vertex, left_vertex, next_left_vertex) > 0) {
                left_vertex = next_left_vertex;
                left_edge = next_left_edge;
                continue;
            }
            auto [next_right_vertex, next_right_edge] = go_next(right_edge);
            if (orientation(left_vertex, right_vertex, next_right_vertex) < 0) {
                right_vertex = next_right_vertex;
                right_edge = next_right_edge;
                continue;
            }
            break;
        }
        return {left_vertex, left_edge, right_vertex, right_edge};
    }

    std::pair<int, int> extreme_vertex(int vertex, int edge, bool minimum) const {
        std::pair<int, int> result = {vertex, edge};
        int current_vertex = vertex;
        int current_edge = edge;
        do {
            std::tie(current_vertex, current_edge) = go_next(current_edge);
            std::pair<int, int> candidate = {current_vertex, current_edge};
            if ((minimum && candidate < result) || (!minimum && result < candidate)) {
                result = candidate;
            }
        } while (current_edge != edge);
        return result;
    }

    bool inside_circumcircle(int a, int b, int c, int d) const {
        return inside_circumcircle(points[a], points[b], points[c], points[d]);
    }

    std::pair<int, int> merge_triangulations(
        int left_vertex,
        int left_edge,
        int right_vertex,
        int right_edge
    ) {
        std::tie(left_vertex, left_edge) = extreme_vertex(left_vertex, left_edge, false);
        std::tie(right_vertex, right_edge) = extreme_vertex(right_vertex, right_edge, true);

        auto [lower_left, lower_left_edge, lower_right, lower_right_edge]
            = lower_tangent(left_vertex, left_edge, right_vertex, right_edge);
        auto [upper_right, upper_right_edge, upper_left, upper_left_edge]
            = lower_tangent(right_vertex, right_edge, left_vertex, left_edge);
        lower_right_edge = edges[lower_right_edge].cw;
        upper_right_edge = edges[upper_right_edge].cw;

        auto [base, reverse_base] = add_edge(lower_left, lower_right);
        insert_cw_after(base, lower_left_edge);
        insert_ccw_after(reverse_base, lower_right_edge);
        if (lower_left == upper_left) upper_left_edge = base;
        if (lower_right == upper_right) upper_right_edge = reverse_base;

        int left = lower_left;
        int left_candidate = lower_left_edge;
        int right = lower_right;
        int right_candidate = lower_right_edge;
        while (left != upper_left || right != upper_right) {
            int next_left = edges[left_candidate].to;
            int next_right = edges[right_candidate].to;
            int next_left_candidate = edges[left_candidate].ccw;
            int next_right_candidate = edges[right_candidate].cw;

            if (left_candidate != upper_left_edge && next_left_candidate != base) {
                int second_left = edges[next_left_candidate].to;
                if (inside_circumcircle(left, right, next_left, second_left)) {
                    erase_edge(left_candidate);
                    left_candidate = next_left_candidate;
                    continue;
                }
            }

            if (right_candidate != upper_right_edge && next_right_candidate != reverse_base) {
                int second_right = edges[next_right_candidate].to;
                if (inside_circumcircle(next_right, left, right, second_right)) {
                    erase_edge(right_candidate);
                    right_candidate = next_right_candidate;
                    continue;
                }
            }

            bool choose_left = right_candidate == upper_right_edge;
            if (left_candidate != upper_left_edge && right_candidate != upper_right_edge) {
                if (orientation(left, right, next_right) < 0) {
                    choose_left = true;
                } else if (orientation(next_left, left, right) < 0) {
                    choose_left = false;
                } else {
                    choose_left = inside_circumcircle(left, right, next_right, next_left);
                }
            }

            if (choose_left) {
                next_left_candidate = edges[edges[left_candidate].reverse].ccw;
                auto [new_base, new_reverse_base] = add_edge(next_left, right);
                insert_cw_after(new_base, next_left_candidate);
                insert_ccw_after(new_reverse_base, right_candidate);
                left_candidate = next_left_candidate;
                left = next_left;
            } else {
                next_right_candidate = edges[edges[right_candidate].reverse].cw;
                auto [new_reverse_base, new_base] = add_edge(next_right, left);
                insert_ccw_after(new_reverse_base, next_right_candidate);
                insert_cw_after(new_base, left_candidate);
                right_candidate = next_right_candidate;
                right = next_right;
            }
        }
        return {lower_left, base};
    }

    std::pair<int, int> solve_range(int left, int right) {
        if (right - left == 2) {
            auto [forward, backward] = add_edge(left, left + 1);
            (void)backward;
            return {left, forward};
        }
        if (right - left == 3) {
            int middle = left + 1;
            int last = left + 2;
            auto [first_middle, middle_first] = add_edge(left, middle);
            auto [middle_last, last_middle] = add_edge(middle, last);
            int direction = orientation(left, middle, last);
            if (direction == 0) {
                insert_ccw_after(middle_first, middle_last);
                return {left, first_middle};
            }

            auto [first_last, last_first] = add_edge(left, last);
            if (direction > 0) {
                insert_cw_after(first_middle, first_last);
                insert_cw_after(middle_last, middle_first);
                insert_cw_after(last_first, last_middle);
                return {left, first_middle};
            }
            insert_ccw_after(first_middle, first_last);
            insert_ccw_after(middle_last, middle_first);
            insert_ccw_after(last_first, last_middle);
            return {middle, middle_first};
        }

        int middle = (left + right) / 2;
        auto [left_vertex, left_edge] = solve_range(left, middle);
        auto [right_vertex, right_edge] = solve_range(middle, right);
        return merge_triangulations(left_vertex, left_edge, right_vertex, right_edge);
    }

    void solve() {
        int size = int(points.size());
        if (size <= 1) return;

        std::vector<int> order(size);
        for (int i = 0; i < size; i++) order[i] = i;
        std::stable_sort(order.begin(), order.end(), [&](int left, int right) {
            if (points[left].x != points[right].x) {
                return points[left].x < points[right].x;
            }
            return points[left].y < points[right].y;
        });

        std::vector<InternalPoint> original_points = points;
        duplicate_representative.assign(size, 0);
        int unique_size = 0;
        for (int i = 0; i < size; i++) {
            int vertex = order[i];
            if (i == 0 || !(original_points[order[unique_size - 1]] == original_points[vertex])) {
                order[unique_size] = vertex;
                points[unique_size] = original_points[vertex];
                unique_size++;
                duplicate_representative[vertex] = vertex;
            } else {
                duplicate_representative[vertex] = order[unique_size - 1];
            }
        }

        if (unique_size >= 2) solve_range(0, unique_size);
        points.swap(original_points);
        for (auto& edge : edges) edge.to = order[edge.to];
    }

   public:
    explicit EuclideanDelaunay(const std::vector<Point<T>>& input_points) {
        assert(input_points.size() <= std::size_t(std::numeric_limits<int>::max()));
        points.reserve(input_points.size());
        edges.reserve(std::size_t(6) * input_points.size());
        for (const auto& point : input_points) {
            points.push_back(InternalPoint{W(point.x), W(point.y)});
        }
        solve();
    }

    bool has_duplicates() const {
        for (
            int vertex = 0;
            vertex < int(duplicate_representative.size());
            ++vertex
        ) {
            if (duplicate_representative[vertex] != vertex) return true;
        }
        return false;
    }

    std::vector<std::pair<int, int>> get_edges() const {
        std::vector<std::pair<int, int>> result;
        result.reserve(edges.size() / 2 + duplicate_representative.size());
        for (int edge = 0; edge < int(edges.size()); edge++) {
            if (!edges[edge].enabled) continue;
            int reverse = edges[edge].reverse;
            if (edge < reverse) continue;
            result.emplace_back(edges[edge].to, edges[reverse].to);
        }
        for (int vertex = 0; vertex < int(duplicate_representative.size()); vertex++) {
            if (duplicate_representative[vertex] != vertex) {
                result.emplace_back(vertex, duplicate_representative[vertex]);
            }
        }
        return result;
    }
};

}  // namespace detail

// Returns O(n) Delaunay edges containing a Euclidean minimum spanning tree.
template <ExactCoordinate T>
std::vector<EuclideanMstEdge<wide_type<T>>> euclidean_mst_edges(
    const std::vector<Point<T>>& points
) {
    using W = wide_type<T>;
    auto delaunay_edges = detail::EuclideanDelaunay<T>(points).get_edges();
    std::vector<EuclideanMstEdge<W>> result;
    result.reserve(delaunay_edges.size());
    for (auto [from, to] : delaunay_edges) {
        result.push_back(EuclideanMstEdge<W>{from, to, distance2(points[from], points[to])});
    }
    return result;
}

// Returns a Euclidean minimum spanning tree.
template <ExactCoordinate T>
EuclideanMst<wide_type<T>> euclidean_mst(const std::vector<Point<T>>& points) {
    using W = wide_type<T>;
    auto candidates = euclidean_mst_edges(points);
    std::sort(candidates.begin(), candidates.end(), [](const auto& left, const auto& right) {
        if (left.squared_distance != right.squared_distance) {
            return left.squared_distance < right.squared_distance;
        }
        if (left.from != right.from) return left.from < right.from;
        return left.to < right.to;
    });

    m1une::ds::Dsu dsu(int(points.size()));
    EuclideanMst<W> result;
    result.cost = 0;
    result.edges.reserve(points.empty() ? 0 : points.size() - 1);
    for (const auto& edge : candidates) {
        if (dsu.same(edge.from, edge.to)) continue;
        dsu.merge(edge.from, edge.to);
        result.cost += std::sqrt(static_cast<long double>(edge.squared_distance));
        result.edges.push_back(edge);
        if (result.edges.size() + 1 == points.size()) break;
    }
    assert(points.empty() || result.edges.size() + 1 == points.size());
    return result;
}

}  // namespace geometry
}  // namespace m1une


#line 16 "geometry/voronoi_diagram.hpp"

namespace m1une {
namespace geometry {

enum class VoronoiEdgeKind {
    Segment,
    Ray,
    Line,
};

struct VoronoiEdge {
    VoronoiEdgeKind kind;
    int first_site;
    int second_site;
    int first_vertex;
    int second_vertex;
    Point<long double> point;
    Point<long double> direction;
};

struct VoronoiDiagram {
    std::vector<Point<long double>> vertices;
    std::vector<VoronoiEdge> edges;
    std::vector<std::vector<int>> cell_edges;
};

namespace voronoi_diagram_detail {

template <ExactCoordinate T>
int direction_half(
    const Point<T>& origin,
    const Point<T>& destination
) {
    using W = wide_type<T>;
    W x = W(destination.x) - W(origin.x);
    W y = W(destination.y) - W(origin.y);
    return y > 0 || (y == 0 && x >= 0) ? 0 : 1;
}

template <ExactCoordinate T>
bool direction_less(
    const std::vector<Point<T>>& sites,
    int origin,
    int first,
    int second
) {
    int first_half = direction_half(sites[origin], sites[first]);
    int second_half = direction_half(sites[origin], sites[second]);
    if (first_half != second_half) return first_half < second_half;

    using W = wide_type<T>;
    W first_x = W(sites[first].x) - W(sites[origin].x);
    W first_y = W(sites[first].y) - W(sites[origin].y);
    W second_x = W(sites[second].x) - W(sites[origin].x);
    W second_y = W(sites[second].y) - W(sites[origin].y);
    W product = first_x * second_y - first_y * second_x;
    if (product != 0) return product > 0;

    W first_norm = first_x * first_x + first_y * first_y;
    W second_norm = second_x * second_x + second_y * second_y;
    if (first_norm != second_norm) return first_norm < second_norm;
    return first < second;
}

template <ExactCoordinate T>
bool cocircular(
    const Point<T>& first,
    const Point<T>& second,
    const Point<T>& third,
    const Point<T>& fourth
) {
    using W = wide_type<T>;
    W ax = W(first.x) - W(fourth.x);
    W ay = W(first.y) - W(fourth.y);
    W bx = W(second.x) - W(fourth.x);
    W by = W(second.y) - W(fourth.y);
    W cx = W(third.x) - W(fourth.x);
    W cy = W(third.y) - W(fourth.y);
    W a_norm = ax * ax + ay * ay;
    W b_norm = bx * bx + by * by;
    W c_norm = cx * cx + cy * cy;
    W determinant =
        (bx * cy - by * cx) * a_norm +
        (cx * ay - cy * ax) * b_norm +
        (ax * by - ay * bx) * c_norm;
    return determinant == 0;
}

template <ExactCoordinate T>
Point<long double> circumcenter(
    const Point<T>& first,
    const Point<T>& second,
    const Point<T>& third
) {
    long double ax = static_cast<long double>(first.x);
    long double ay = static_cast<long double>(first.y);
    long double bx = static_cast<long double>(second.x);
    long double by = static_cast<long double>(second.y);
    long double cx = static_cast<long double>(third.x);
    long double cy = static_cast<long double>(third.y);
    long double denominator = 2 * (
        ax * (by - cy) +
        bx * (cy - ay) +
        cx * (ay - by)
    );
    assert(denominator != 0);
    long double first_norm = ax * ax + ay * ay;
    long double second_norm = bx * bx + by * by;
    long double third_norm = cx * cx + cy * cy;
    return Point<long double>(
        (first_norm * (by - cy) +
         second_norm * (cy - ay) +
         third_norm * (ay - by)) /
            denominator,
        (first_norm * (cx - bx) +
         second_norm * (ax - cx) +
         third_norm * (bx - ax)) /
            denominator
    );
}

inline Point<long double> unit(Point<long double> direction) {
    long double length = norm(direction);
    assert(length != 0);
    return direction / length;
}

inline int other_site(const VoronoiEdge& edge, int site) {
    assert(edge.first_site == site || edge.second_site == site);
    return edge.first_site == site ? edge.second_site : edge.first_site;
}

}  // namespace voronoi_diagram_detail

// Constructs the ordinary Euclidean Voronoi diagram of distinct exact-coordinate sites.
template <ExactCoordinate T>
VoronoiDiagram voronoi_diagram(const std::vector<Point<T>>& sites) {
    namespace detail = voronoi_diagram_detail;
    assert(sites.size() <= std::size_t(std::numeric_limits<int>::max()));

    const int size = int(sites.size());
    std::vector<int> site_order(size);
    std::iota(site_order.begin(), site_order.end(), 0);
    std::sort(site_order.begin(), site_order.end(), [&](int first, int second) {
        return sites[first] < sites[second];
    });
    for (int index = 1; index < size; ++index) {
        assert(sites[site_order[index - 1]] != sites[site_order[index]]);
    }

    std::vector<std::pair<int, int>> delaunay_edges =
        geometry::detail::EuclideanDelaunay<T>(sites).get_edges();
    for (auto& [first, second] : delaunay_edges) {
        if (first > second) std::swap(first, second);
    }
    std::sort(delaunay_edges.begin(), delaunay_edges.end());
    delaunay_edges.erase(
        std::unique(delaunay_edges.begin(), delaunay_edges.end()),
        delaunay_edges.end()
    );

    auto find_edge_index = [&](int first, int second) {
        if (first > second) std::swap(first, second);
        auto iterator = std::lower_bound(
            delaunay_edges.begin(),
            delaunay_edges.end(),
            std::pair(first, second)
        );
        if (
            iterator == delaunay_edges.end() ||
            *iterator != std::pair(first, second)
        ) {
            return -1;
        }
        return int(iterator - delaunay_edges.begin());
    };
    std::vector<std::vector<int>> neighbors(size);
    for (int index = 0; index < int(delaunay_edges.size()); ++index) {
        auto [first, second] = delaunay_edges[index];
        neighbors[first].push_back(second);
        neighbors[second].push_back(first);
    }
    for (int site = 0; site < size; ++site) {
        std::sort(
            neighbors[site].begin(),
            neighbors[site].end(),
            [&](int first, int second) {
                return detail::direction_less(sites, site, first, second);
            }
        );
    }

    std::vector<std::array<int, 3>> triangles;
    for (int site = 0; site < size; ++site) {
        int degree = int(neighbors[site].size());
        for (int index = 0; index < degree; ++index) {
            int first = neighbors[site][index];
            int second = neighbors[site][(index + 1) % degree];
            if (orientation(sites[site], sites[first], sites[second]) <= 0) {
                continue;
            }
            if (find_edge_index(first, second) == -1) continue;
            std::array<int, 3> triangle{site, first, second};
            std::sort(triangle.begin(), triangle.end());
            triangles.push_back(triangle);
        }
    }
    std::sort(triangles.begin(), triangles.end());
    triangles.erase(
        std::unique(triangles.begin(), triangles.end()),
        triangles.end()
    );
    for (auto& triangle : triangles) {
        if (orientation(
                sites[triangle[0]],
                sites[triangle[1]],
                sites[triangle[2]]
            ) < 0) {
            std::swap(triangle[1], triangle[2]);
        }
    }

    std::vector<std::array<int, 2>> incident_triangles(
        delaunay_edges.size(),
        std::array<int, 2>{-1, -1}
    );
    std::vector<int> incident_count(delaunay_edges.size(), 0);
    for (int triangle = 0; triangle < int(triangles.size()); ++triangle) {
        for (int side = 0; side < 3; ++side) {
            int first = triangles[triangle][side];
            int second = triangles[triangle][(side + 1) % 3];
            int edge = find_edge_index(first, second);
            assert(edge != -1);
            assert(incident_count[edge] < 2);
            incident_triangles[edge][incident_count[edge]++] = triangle;
        }
    }

    std::vector<int> parent(triangles.size());
    std::vector<int> component_size(triangles.size(), 1);
    std::iota(parent.begin(), parent.end(), 0);
    auto find_root = [&](auto&& self, int vertex) -> int {
        if (parent[vertex] == vertex) return vertex;
        return parent[vertex] = self(self, parent[vertex]);
    };
    auto merge = [&](int first, int second) {
        first = find_root(find_root, first);
        second = find_root(find_root, second);
        if (first == second) return;
        if (component_size[first] < component_size[second]) {
            std::swap(first, second);
        }
        parent[second] = first;
        component_size[first] += component_size[second];
    };
    for (int edge = 0; edge < int(delaunay_edges.size()); ++edge) {
        if (incident_count[edge] != 2) continue;
        int first_triangle = incident_triangles[edge][0];
        int second_triangle = incident_triangles[edge][1];
        const auto& first = triangles[first_triangle];
        const auto& second = triangles[second_triangle];
        int fourth = second[0];
        if (fourth == first[0] || fourth == first[1] || fourth == first[2]) {
            fourth = second[1];
        }
        if (fourth == first[0] || fourth == first[1] || fourth == first[2]) {
            fourth = second[2];
        }
        assert(
            fourth != first[0] &&
            fourth != first[1] &&
            fourth != first[2]
        );
        if (detail::cocircular(
                sites[first[0]],
                sites[first[1]],
                sites[first[2]],
                sites[fourth]
            )) {
            merge(first_triangle, second_triangle);
        }
    }

    VoronoiDiagram result;
    result.cell_edges.resize(size);
    std::vector<int> root_vertex(triangles.size(), -1);
    std::vector<int> triangle_vertex(triangles.size(), -1);
    for (int triangle = 0; triangle < int(triangles.size()); ++triangle) {
        int root = find_root(find_root, triangle);
        if (root_vertex[root] == -1) {
            const auto& sites_on_circle = triangles[triangle];
            root_vertex[root] = int(result.vertices.size());
            result.vertices.push_back(detail::circumcenter(
                sites[sites_on_circle[0]],
                sites[sites_on_circle[1]],
                sites[sites_on_circle[2]]
            ));
        }
        triangle_vertex[triangle] = root_vertex[root];
    }

    result.edges.reserve(delaunay_edges.size());
    for (int edge = 0; edge < int(delaunay_edges.size()); ++edge) {
        auto [first_site, second_site] = delaunay_edges[edge];
        VoronoiEdge voronoi_edge;
        voronoi_edge.first_site = first_site;
        voronoi_edge.second_site = second_site;
        voronoi_edge.first_vertex = -1;
        voronoi_edge.second_vertex = -1;

        if (incident_count[edge] == 2) {
            int first_vertex =
                triangle_vertex[incident_triangles[edge][0]];
            int second_vertex =
                triangle_vertex[incident_triangles[edge][1]];
            if (first_vertex == second_vertex) continue;
            if (first_vertex > second_vertex) {
                std::swap(first_vertex, second_vertex);
            }
            voronoi_edge.kind = VoronoiEdgeKind::Segment;
            voronoi_edge.first_vertex = first_vertex;
            voronoi_edge.second_vertex = second_vertex;
            voronoi_edge.point = result.vertices[first_vertex];
            voronoi_edge.direction =
                result.vertices[second_vertex] - result.vertices[first_vertex];
        } else if (incident_count[edge] == 1) {
            int triangle = incident_triangles[edge][0];
            int third_site = triangles[triangle][0];
            if (third_site == first_site || third_site == second_site) {
                third_site = triangles[triangle][1];
            }
            if (third_site == first_site || third_site == second_site) {
                third_site = triangles[triangle][2];
            }
            assert(third_site != first_site && third_site != second_site);

            Point<long double> first(sites[first_site]);
            Point<long double> second(sites[second_site]);
            Point<long double> edge_direction = second - first;
            Point<long double> outward;
            if (orientation(
                    sites[first_site],
                    sites[second_site],
                    sites[third_site]
                ) > 0) {
                outward = Point<long double>(
                    edge_direction.y,
                    -edge_direction.x
                );
            } else {
                outward = Point<long double>(
                    -edge_direction.y,
                    edge_direction.x
                );
            }
            voronoi_edge.kind = VoronoiEdgeKind::Ray;
            voronoi_edge.first_vertex = triangle_vertex[triangle];
            voronoi_edge.point = result.vertices[voronoi_edge.first_vertex];
            voronoi_edge.direction = detail::unit(outward);
        } else {
            assert(incident_count[edge] == 0);
            Point<long double> first(sites[first_site]);
            Point<long double> second(sites[second_site]);
            Point<long double> edge_direction = second - first;
            voronoi_edge.kind = VoronoiEdgeKind::Line;
            voronoi_edge.point = (first + second) / 2.0L;
            voronoi_edge.direction = detail::unit(Point<long double>(
                edge_direction.y,
                -edge_direction.x
            ));
        }

        int voronoi_edge_index = int(result.edges.size());
        result.edges.push_back(voronoi_edge);
        result.cell_edges[first_site].push_back(voronoi_edge_index);
        result.cell_edges[second_site].push_back(voronoi_edge_index);
    }

    for (int site = 0; site < size; ++site) {
        std::sort(
            result.cell_edges[site].begin(),
            result.cell_edges[site].end(),
            [&](int first_edge, int second_edge) {
                int first_other =
                    detail::other_site(result.edges[first_edge], site);
                int second_other =
                    detail::other_site(result.edges[second_edge], site);
                return detail::direction_less(
                    sites,
                    site,
                    first_other,
                    second_other
                );
            }
        );
    }
    return result;
}

}  // namespace geometry
}  // namespace m1une


#line 1 "geometry/half_plane_intersection.hpp"



#line 8 "geometry/half_plane_intersection.hpp"
#include <deque>
#line 10 "geometry/half_plane_intersection.hpp"
#include <numbers>
#include <optional>
#include <random>
#line 15 "geometry/half_plane_intersection.hpp"

#line 1 "geometry/linear.hpp"



#line 7 "geometry/linear.hpp"

#line 9 "geometry/linear.hpp"

namespace m1une {
namespace geometry {

template <Coordinate T>
struct Line {
    Point<T> a;
    Point<T> b;
};

template <Coordinate T>
struct Segment {
    Point<T> a;
    Point<T> b;
};

template <Coordinate T>
struct Ray {
    Point<T> origin;
    Point<T> through;
};

enum class LinearIntersectionKind {
    Empty,
    Point,
    Segment,
    Ray,
    Line,
};

struct LinearIntersection {
    LinearIntersectionKind kind;
    Point<long double> first;
    Point<long double> second;
};

struct ClosestPoints {
    Point<long double> first;
    Point<long double> second;
};

namespace linear_intersection_detail {

inline LinearIntersection make_empty() {
    const Point<long double> zero;
    return LinearIntersection{
        LinearIntersectionKind::Empty,
        zero,
        zero,
    };
}

template <Coordinate T>
LinearIntersection make_point(const Point<T>& point) {
    const Point<long double> converted(point);
    return LinearIntersection{
        LinearIntersectionKind::Point,
        converted,
        converted,
    };
}

template <Coordinate T>
LinearIntersection make_object(
    LinearIntersectionKind kind,
    const Point<T>& first,
    const Point<T>& second
) {
    return LinearIntersection{
        kind,
        Point<long double>(first),
        Point<long double>(second),
    };
}

}  // namespace linear_intersection_detail

template <Coordinate T>
constexpr Point<long double> centroid(const Segment<T>& segment) {
    return Point<long double>(
        (
            static_cast<long double>(segment.a.x) +
            static_cast<long double>(segment.b.x)
        ) / 2,
        (
            static_cast<long double>(segment.a.y) +
            static_cast<long double>(segment.b.y)
        ) / 2
    );
}

template <Coordinate T>
bool on_line(
    const Line<T>& line,
    const Point<T>& point,
    long double eps = 1e-12L
) {
    assert(line.a != line.b);
    return orientation(line.a, line.b, point, eps) == 0;
}

template <Coordinate T>
bool parallel(const Line<T>& first, const Line<T>& second, long double eps = 1e-12L) {
    using W = wide_type<T>;
    W first_x = W(first.b.x) - W(first.a.x);
    W first_y = W(first.b.y) - W(first.a.y);
    W second_x = W(second.b.x) - W(second.a.x);
    W second_y = W(second.b.y) - W(second.a.y);
    return predicate_detail::determinant_sign<ExactCoordinate<T>>(
        first_x,
        first_y,
        second_x,
        second_y,
        eps
    ) == 0;
}

template <Coordinate T>
bool orthogonal(const Line<T>& first, const Line<T>& second, long double eps = 1e-12L) {
    using W = wide_type<T>;
    W first_x = W(first.b.x) - W(first.a.x);
    W first_y = W(first.b.y) - W(first.a.y);
    W second_x = W(second.b.x) - W(second.a.x);
    W second_y = W(second.b.y) - W(second.a.y);
    return predicate_detail::dot_sign<ExactCoordinate<T>>(
        first_x,
        first_y,
        second_x,
        second_y,
        eps
    ) == 0;
}

template <Coordinate T>
Point<long double> projection(const Line<T>& line, const Point<T>& point) {
    assert(line.a != line.b);
    Point<long double> a(line.a);
    Point<long double> direction(
        static_cast<long double>(line.b.x) - static_cast<long double>(line.a.x),
        static_cast<long double>(line.b.y) - static_cast<long double>(line.a.y)
    );
    Point<long double> offset(
        static_cast<long double>(point.x) - a.x,
        static_cast<long double>(point.y) - a.y
    );
    long double ratio = dot(offset, direction) / dot(direction, direction);
    return a + direction * ratio;
}

template <Coordinate T>
Point<long double> reflection(const Line<T>& line, const Point<T>& point) {
    Point<long double> projected = projection(line, point);
    return projected * 2.0L - Point<long double>(point);
}

template <Coordinate T>
bool intersects(
    const Line<T>& first,
    const Line<T>& second,
    long double eps = 1e-12L
) {
    return !parallel(first, second, eps) || on_line(first, second.a, eps);
}

template <Coordinate T>
bool on_segment(
    const Segment<T>& segment,
    const Point<T>& point,
    long double eps = 1e-12L
) {
    if (orientation(segment.a, segment.b, point, eps) != 0) return false;
    using W = wide_type<T>;
    const W direction_x = W(segment.b.x) - W(segment.a.x);
    const W direction_y = W(segment.b.y) - W(segment.a.y);
    if (direction_x == W(0) && direction_y == W(0)) {
        if constexpr (ExactCoordinate<T>) {
            return point == segment.a;
        } else {
            return
                predicate_detail::absolute(W(point.x) - W(segment.a.x)) <= eps &&
                predicate_detail::absolute(W(point.y) - W(segment.a.y)) <= eps;
        }
    }
    const W offset_x = W(point.x) - W(segment.a.x);
    const W offset_y = W(point.y) - W(segment.a.y);
    const W projection =
        offset_x * direction_x + offset_y * direction_y;
    const W length_squared =
        direction_x * direction_x + direction_y * direction_y;
    return
        predicate_detail::scaled_sign<ExactCoordinate<T>>(
            projection,
            length_squared,
            eps
        ) >= 0 &&
        predicate_detail::scaled_sign<ExactCoordinate<T>>(
            projection - length_squared,
            length_squared,
            eps
        ) <= 0;
}

template <Coordinate T>
Point<long double> projection(
    const Segment<T>& segment,
    const Point<T>& point
) {
    const Point<long double> first(segment.a);
    const Point<long double> direction =
        Point<long double>(segment.b) - first;
    const long double length_squared = dot(direction, direction);
    if (length_squared == 0) return first;
    const long double ratio = std::clamp(
        dot(Point<long double>(point) - first, direction) / length_squared,
        0.0L,
        1.0L
    );
    return first + direction * ratio;
}

template <Coordinate T>
bool intersects(
    const Segment<T>& first,
    const Segment<T>& second,
    long double eps = 1e-12L
) {
    int abc = orientation(first.a, first.b, second.a, eps);
    int abd = orientation(first.a, first.b, second.b, eps);
    int cda = orientation(second.a, second.b, first.a, eps);
    int cdb = orientation(second.a, second.b, first.b, eps);

    if (abc == 0 && on_segment(first, second.a, eps)) return true;
    if (abd == 0 && on_segment(first, second.b, eps)) return true;
    if (cda == 0 && on_segment(second, first.a, eps)) return true;
    if (cdb == 0 && on_segment(second, first.b, eps)) return true;
    return abc * abd < 0 && cda * cdb < 0;
}

template <Coordinate T>
bool intersects(
    const Line<T>& line,
    const Segment<T>& segment,
    long double eps = 1e-12L
) {
    int first_side = orientation(line.a, line.b, segment.a, eps);
    int second_side = orientation(line.a, line.b, segment.b, eps);
    return first_side == 0 || second_side == 0 || first_side != second_side;
}

template <Coordinate T>
bool intersects(
    const Segment<T>& segment,
    const Line<T>& line,
    long double eps = 1e-12L
) {
    return intersects(line, segment, eps);
}

namespace linear_parameter_detail {

template <Coordinate T>
struct Parameters {
    wide_type<T> denominator;
    wide_type<T> denominator_scale;
    wide_type<T> first_numerator;
    wide_type<T> second_numerator;
};

template <Coordinate T>
Parameters<T> parameters(
    const Point<T>& first_origin,
    const Point<T>& first_through,
    const Point<T>& second_origin,
    const Point<T>& second_through
) {
    using W = wide_type<T>;
    W first_x = W(first_through.x) - W(first_origin.x);
    W first_y = W(first_through.y) - W(first_origin.y);
    W second_x = W(second_through.x) - W(second_origin.x);
    W second_y = W(second_through.y) - W(second_origin.y);
    W offset_x = W(second_origin.x) - W(first_origin.x);
    W offset_y = W(second_origin.y) - W(first_origin.y);
    return Parameters<T>{
        first_x * second_y - first_y * second_x,
        predicate_detail::determinant_scale<ExactCoordinate<T>>(
            first_x,
            first_y,
            second_x,
            second_y
        ),
        offset_x * second_y - offset_y * second_x,
        offset_x * first_y - offset_y * first_x
    };
}

template <Coordinate T>
int denominator_sign(const Parameters<T>& values, long double eps) {
    return predicate_detail::scaled_sign<ExactCoordinate<T>>(
        values.denominator,
        values.denominator_scale,
        eps
    );
}

template <Coordinate T>
bool ratio_nonnegative(
    wide_type<T> numerator,
    wide_type<T> denominator,
    long double eps
) {
    const int numerator_sign =
        predicate_detail::scaled_sign<ExactCoordinate<T>>(
            numerator,
            predicate_detail::absolute(denominator),
            eps
        );
    const int denominator_direction =
        (denominator > 0) - (denominator < 0);
    return
        numerator_sign == 0 ||
        numerator_sign == denominator_direction;
}

template <Coordinate T>
bool ratio_in_unit_interval(
    wide_type<T> numerator,
    wide_type<T> denominator,
    long double eps
) {
    const auto scale = predicate_detail::absolute(denominator);
    const int start_sign =
        predicate_detail::scaled_sign<ExactCoordinate<T>>(
            numerator,
            scale,
            eps
        );
    const int finish_sign =
        predicate_detail::scaled_sign<ExactCoordinate<T>>(
            numerator - denominator,
            scale,
            eps
        );
    if (denominator > 0) {
        return start_sign >= 0 && finish_sign <= 0;
    }
    return start_sign <= 0 && finish_sign >= 0;
}

}  // namespace linear_parameter_detail

template <Coordinate T>
bool on_ray(
    const Ray<T>& ray,
    const Point<T>& point,
    long double eps = 1e-12L
) {
    assert(ray.origin != ray.through);
    if (orientation(ray.origin, ray.through, point, eps) != 0) return false;
    using W = wide_type<T>;
    W direction_x = W(ray.through.x) - W(ray.origin.x);
    W direction_y = W(ray.through.y) - W(ray.origin.y);
    W offset_x = W(point.x) - W(ray.origin.x);
    W offset_y = W(point.y) - W(ray.origin.y);
    const W projection =
        direction_x * offset_x + direction_y * offset_y;
    const W length_squared =
        direction_x * direction_x + direction_y * direction_y;
    return predicate_detail::scaled_sign<ExactCoordinate<T>>(
        projection,
        length_squared,
        eps
    ) >= 0;
}

template <Coordinate T>
Point<long double> projection(const Ray<T>& ray, const Point<T>& point) {
    assert(ray.origin != ray.through);
    Point<long double> origin(ray.origin);
    Point<long double> direction =
        Point<long double>(ray.through) - origin;
    Point<long double> offset = Point<long double>(point) - origin;
    long double ratio = dot(offset, direction) / dot(direction, direction);
    if (ratio < 0) ratio = 0;
    return origin + direction * ratio;
}

template <Coordinate T>
Ray<long double> reflection(const Line<T>& line, const Ray<T>& ray) {
    assert(ray.origin != ray.through);
    return Ray<long double>{
        reflection(line, ray.origin),
        reflection(line, ray.through)
    };
}

template <Coordinate T>
Ray<long double> reflected_ray(
    const Ray<T>& incoming,
    const Point<T>& hit,
    const Line<T>& mirror,
    long double eps = 1e-12L
) {
    assert(incoming.origin != incoming.through);
    assert(on_line(mirror, hit, eps));
    Point<T> translated = hit + (incoming.through - incoming.origin);
    return Ray<long double>{
        Point<long double>(hit),
        reflection(mirror, translated)
    };
}

template <Coordinate T>
bool intersects(
    const Ray<T>& ray,
    const Line<T>& line,
    long double eps = 1e-12L
) {
    assert(ray.origin != ray.through);
    assert(line.a != line.b);
    linear_parameter_detail::Parameters<T> values =
        linear_parameter_detail::parameters(
        ray.origin,
        ray.through,
        line.a,
        line.b
    );
    if (linear_parameter_detail::denominator_sign(values, eps) == 0) {
        return on_line(line, ray.origin, eps);
    }
    return linear_parameter_detail::ratio_nonnegative<T>(
        values.first_numerator,
        values.denominator,
        eps
    );
}

template <Coordinate T>
bool intersects(
    const Line<T>& line,
    const Ray<T>& ray,
    long double eps = 1e-12L
) {
    return intersects(ray, line, eps);
}

template <Coordinate T>
bool intersects(
    const Ray<T>& ray,
    const Segment<T>& segment,
    long double eps = 1e-12L
) {
    assert(ray.origin != ray.through);
    if (segment.a == segment.b) return on_ray(ray, segment.a, eps);

    linear_parameter_detail::Parameters<T> values =
        linear_parameter_detail::parameters(
        ray.origin,
        ray.through,
        segment.a,
        segment.b
    );
    if (linear_parameter_detail::denominator_sign(values, eps) == 0) {
        if (orientation(ray.origin, ray.through, segment.a, eps) != 0) {
            return false;
        }
        return on_ray(ray, segment.a, eps) ||
               on_ray(ray, segment.b, eps) ||
               on_segment(segment, ray.origin, eps);
    }
    return linear_parameter_detail::ratio_nonnegative<T>(
               values.first_numerator,
               values.denominator,
               eps
           ) &&
           linear_parameter_detail::ratio_in_unit_interval<T>(
               values.second_numerator,
               values.denominator,
               eps
           );
}

template <Coordinate T>
bool intersects(
    const Segment<T>& segment,
    const Ray<T>& ray,
    long double eps = 1e-12L
) {
    return intersects(ray, segment, eps);
}

template <Coordinate T>
bool intersects(
    const Ray<T>& first,
    const Ray<T>& second,
    long double eps = 1e-12L
) {
    assert(first.origin != first.through);
    assert(second.origin != second.through);
    linear_parameter_detail::Parameters<T> values =
        linear_parameter_detail::parameters(
        first.origin,
        first.through,
        second.origin,
        second.through
    );
    if (linear_parameter_detail::denominator_sign(values, eps) == 0) {
        if (orientation(first.origin, first.through, second.origin, eps) != 0) {
            return false;
        }
        return on_ray(first, second.origin, eps) ||
               on_ray(second, first.origin, eps);
    }
    return linear_parameter_detail::ratio_nonnegative<T>(
               values.first_numerator,
               values.denominator,
               eps
           ) &&
           linear_parameter_detail::ratio_nonnegative<T>(
               values.second_numerator,
               values.denominator,
               eps
           );
}

namespace linear_intersection_detail {

enum class Domain {
    Line,
    Segment,
    Ray,
};

template <Coordinate T>
struct ParametricObject {
    Point<T> origin;
    Point<T> through;
    Domain domain;
};

template <Coordinate T>
ParametricObject<T> parametric_object(const Line<T>& line) {
    assert(line.a != line.b);
    return ParametricObject<T>{line.a, line.b, Domain::Line};
}

template <Coordinate T>
ParametricObject<T> parametric_object(const Segment<T>& segment) {
    return ParametricObject<T>{segment.a, segment.b, Domain::Segment};
}

template <Coordinate T>
ParametricObject<T> parametric_object(const Ray<T>& ray) {
    assert(ray.origin != ray.through);
    return ParametricObject<T>{ray.origin, ray.through, Domain::Ray};
}

template <Coordinate T>
bool contains(
    const ParametricObject<T>& object,
    const Point<T>& point,
    long double eps
) {
    if (object.domain == Domain::Line) {
        return on_line(Line<T>{object.origin, object.through}, point, eps);
    }
    if (object.domain == Domain::Segment) {
        return on_segment(
            Segment<T>{object.origin, object.through},
            point,
            eps
        );
    }
    return on_ray(Ray<T>{object.origin, object.through}, point, eps);
}

template <Coordinate T>
bool accepts_parameter(
    Domain domain,
    wide_type<T> numerator,
    wide_type<T> denominator,
    long double eps
) {
    if (domain == Domain::Line) return true;
    if (domain == Domain::Ray) {
        return linear_parameter_detail::ratio_nonnegative<T>(
            numerator,
            denominator,
            eps
        );
    }
    return linear_parameter_detail::ratio_in_unit_interval<T>(
        numerator,
        denominator,
        eps
    );
}

template <Coordinate T>
Point<long double> point_at_ratio(
    const ParametricObject<T>& object,
    wide_type<T> numerator,
    wide_type<T> denominator
) {
    const long double ratio =
        static_cast<long double>(numerator) /
        static_cast<long double>(denominator);
    const Point<long double> origin(object.origin);
    const Point<long double> direction =
        Point<long double>(object.through) - origin;
    return origin + direction * ratio;
}

template <Coordinate T>
struct AxisProjection {
    bool use_x;
    bool negate;

    wide_type<T> operator()(const Point<T>& point) const {
        const wide_type<T> value = use_x
            ? wide_type<T>(point.x)
            : wide_type<T>(point.y);
        return negate ? -value : value;
    }
};

template <Coordinate T>
AxisProjection<T> axis_projection(const ParametricObject<T>& object) {
    using W = wide_type<T>;
    const W direction_x = W(object.through.x) - W(object.origin.x);
    const W direction_y = W(object.through.y) - W(object.origin.y);
    const bool use_x =
        predicate_detail::absolute(direction_x) >=
        predicate_detail::absolute(direction_y);
    const W component = use_x ? direction_x : direction_y;
    assert(component != W(0));
    return AxisProjection<T>{use_x, component < W(0)};
}

template <Coordinate T>
struct ParameterInterval {
    bool has_lower;
    bool has_upper;
    wide_type<T> lower;
    wide_type<T> upper;
};

template <Coordinate T>
ParameterInterval<T> parameter_interval(
    const ParametricObject<T>& object,
    const AxisProjection<T>& projection
) {
    using W = wide_type<T>;
    const W origin = projection(object.origin);
    const W through = projection(object.through);
    if (object.domain == Domain::Line) {
        return ParameterInterval<T>{false, false, W(0), W(0)};
    }
    if (object.domain == Domain::Segment) {
        return ParameterInterval<T>{
            true,
            true,
            std::min(origin, through),
            std::max(origin, through),
        };
    }
    if (origin < through) {
        return ParameterInterval<T>{true, false, origin, W(0)};
    }
    return ParameterInterval<T>{false, true, W(0), origin};
}

template <Coordinate T>
ParameterInterval<T> intersect_intervals(
    ParameterInterval<T> first,
    const ParameterInterval<T>& second
) {
    if (
        second.has_lower &&
        (!first.has_lower || first.lower < second.lower)
    ) {
        first.has_lower = true;
        first.lower = second.lower;
    }
    if (
        second.has_upper &&
        (!first.has_upper || second.upper < first.upper)
    ) {
        first.has_upper = true;
        first.upper = second.upper;
    }
    return first;
}

template <Coordinate T>
Point<long double> point_at_projection(
    const ParametricObject<T>& object,
    const AxisProjection<T>& projection,
    long double target
) {
    const long double origin =
        static_cast<long double>(projection(object.origin));
    const long double through =
        static_cast<long double>(projection(object.through));
    const long double ratio = (target - origin) / (through - origin);
    const Point<long double> point(object.origin);
    const Point<long double> direction =
        Point<long double>(object.through) - point;
    return point + direction * ratio;
}

template <Coordinate T>
LinearIntersection collinear_intersection(
    const ParametricObject<T>& first,
    const ParametricObject<T>& second,
    long double eps
) {
    using W = wide_type<T>;
    const AxisProjection<T> projection = axis_projection(first);
    const ParameterInterval<T> first_interval =
        parameter_interval(first, projection);
    const ParameterInterval<T> second_interval =
        parameter_interval(second, projection);
    const ParameterInterval<T> common =
        intersect_intervals(first_interval, second_interval);

    W scale = predicate_detail::absolute(
        projection(first.through) - projection(first.origin)
    );
    scale = std::max(
        scale,
        predicate_detail::absolute(
            projection(second.through) - projection(second.origin)
        )
    );

    if (common.has_lower && common.has_upper) {
        const int order = predicate_detail::scaled_sign<ExactCoordinate<T>>(
            common.lower - common.upper,
            scale,
            eps
        );
        if (order > 0) return make_empty();
        if (order == 0) {
            const long double coordinate =
                (
                    static_cast<long double>(common.lower) +
                    static_cast<long double>(common.upper)
                ) / 2.0L;
            return make_point(
                point_at_projection(first, projection, coordinate)
            );
        }
        return make_object(
            LinearIntersectionKind::Segment,
            point_at_projection(
                first,
                projection,
                static_cast<long double>(common.lower)
            ),
            point_at_projection(
                first,
                projection,
                static_cast<long double>(common.upper)
            )
        );
    }

    const Point<long double> direction =
        Point<long double>(first.through) -
        Point<long double>(first.origin);
    if (common.has_lower) {
        const Point<long double> origin = point_at_projection(
            first,
            projection,
            static_cast<long double>(common.lower)
        );
        return make_object(
            LinearIntersectionKind::Ray,
            origin,
            origin + direction
        );
    }
    if (common.has_upper) {
        const Point<long double> origin = point_at_projection(
            first,
            projection,
            static_cast<long double>(common.upper)
        );
        return make_object(
            LinearIntersectionKind::Ray,
            origin,
            origin - direction
        );
    }
    return make_object(
        LinearIntersectionKind::Line,
        first.origin,
        first.through
    );
}

template <Coordinate T>
LinearIntersection intersect(
    const ParametricObject<T>& first,
    const ParametricObject<T>& second,
    long double eps
) {
    const bool first_degenerate = first.origin == first.through;
    const bool second_degenerate = second.origin == second.through;
    if (first_degenerate) {
        assert(first.domain == Domain::Segment);
        if (contains(second, first.origin, eps)) {
            return make_point(first.origin);
        }
        return make_empty();
    }
    if (second_degenerate) {
        assert(second.domain == Domain::Segment);
        if (contains(first, second.origin, eps)) {
            return make_point(second.origin);
        }
        return make_empty();
    }

    const linear_parameter_detail::Parameters<T> values =
        linear_parameter_detail::parameters(
        first.origin,
        first.through,
        second.origin,
        second.through
    );
    if (linear_parameter_detail::denominator_sign(values, eps) != 0) {
        if (
            !accepts_parameter<T>(
                first.domain,
                values.first_numerator,
                values.denominator,
                eps
            ) ||
            !accepts_parameter<T>(
                second.domain,
                values.second_numerator,
                values.denominator,
                eps
            )
        ) {
            return make_empty();
        }
        return make_point(
            point_at_ratio(
                first,
                values.first_numerator,
                values.denominator
            )
        );
    }
    if (
        orientation(
            first.origin,
            first.through,
            second.origin,
            eps
        ) != 0
    ) {
        return make_empty();
    }
    return collinear_intersection(first, second, eps);
}

}  // namespace linear_intersection_detail

template <Coordinate T>
LinearIntersection linear_intersection(
    const Line<T>& first,
    const Line<T>& second,
    long double eps = 1e-12L
) {
    return linear_intersection_detail::intersect(
        linear_intersection_detail::parametric_object(first),
        linear_intersection_detail::parametric_object(second),
        eps
    );
}

template <Coordinate T>
LinearIntersection linear_intersection(
    const Line<T>& line,
    const Segment<T>& segment,
    long double eps = 1e-12L
) {
    return linear_intersection_detail::intersect(
        linear_intersection_detail::parametric_object(line),
        linear_intersection_detail::parametric_object(segment),
        eps
    );
}

template <Coordinate T>
LinearIntersection linear_intersection(
    const Segment<T>& segment,
    const Line<T>& line,
    long double eps = 1e-12L
) {
    return linear_intersection_detail::intersect(
        linear_intersection_detail::parametric_object(segment),
        linear_intersection_detail::parametric_object(line),
        eps
    );
}

template <Coordinate T>
LinearIntersection linear_intersection(
    const Segment<T>& first,
    const Segment<T>& second,
    long double eps = 1e-12L
) {
    return linear_intersection_detail::intersect(
        linear_intersection_detail::parametric_object(first),
        linear_intersection_detail::parametric_object(second),
        eps
    );
}

template <Coordinate T>
LinearIntersection linear_intersection(
    const Ray<T>& ray,
    const Line<T>& line,
    long double eps = 1e-12L
) {
    return linear_intersection_detail::intersect(
        linear_intersection_detail::parametric_object(ray),
        linear_intersection_detail::parametric_object(line),
        eps
    );
}

template <Coordinate T>
LinearIntersection linear_intersection(
    const Line<T>& line,
    const Ray<T>& ray,
    long double eps = 1e-12L
) {
    return linear_intersection_detail::intersect(
        linear_intersection_detail::parametric_object(line),
        linear_intersection_detail::parametric_object(ray),
        eps
    );
}

template <Coordinate T>
LinearIntersection linear_intersection(
    const Ray<T>& ray,
    const Segment<T>& segment,
    long double eps = 1e-12L
) {
    return linear_intersection_detail::intersect(
        linear_intersection_detail::parametric_object(ray),
        linear_intersection_detail::parametric_object(segment),
        eps
    );
}

template <Coordinate T>
LinearIntersection linear_intersection(
    const Segment<T>& segment,
    const Ray<T>& ray,
    long double eps = 1e-12L
) {
    return linear_intersection_detail::intersect(
        linear_intersection_detail::parametric_object(segment),
        linear_intersection_detail::parametric_object(ray),
        eps
    );
}

template <Coordinate T>
LinearIntersection linear_intersection(
    const Ray<T>& first,
    const Ray<T>& second,
    long double eps = 1e-12L
) {
    return linear_intersection_detail::intersect(
        linear_intersection_detail::parametric_object(first),
        linear_intersection_detail::parametric_object(second),
        eps
    );
}

namespace closest_points_detail {

inline ClosestPoints reversed(const ClosestPoints& result) {
    return ClosestPoints{result.second, result.first};
}

inline bool point_less(
    const Point<long double>& first,
    const Point<long double>& second
) {
    if (first.x != second.x) return first.x < second.x;
    return first.y < second.y;
}

inline ClosestPoints common_point(const LinearIntersection& intersection) {
    assert(intersection.kind != LinearIntersectionKind::Empty);
    Point<long double> point = intersection.first;
    if (intersection.kind == LinearIntersectionKind::Segment) {
        if (point_less(intersection.second, point)) {
            point = intersection.second;
        }
    } else if (intersection.kind == LinearIntersectionKind::Line) {
        const Line<long double> line{
            intersection.first,
            intersection.second
        };
        point = projection(line, Point<long double>(0, 0));
    }
    return ClosestPoints{point, point};
}

inline long double separation2(const ClosestPoints& result) {
    return distance2(result.first, result.second);
}

inline bool canonical_less(
    const ClosestPoints& first,
    const ClosestPoints& second
) {
    Point<long double> first_start = first.first;
    Point<long double> first_finish = first.second;
    if (point_less(first_finish, first_start)) {
        std::swap(first_start, first_finish);
    }
    Point<long double> second_start = second.first;
    Point<long double> second_finish = second.second;
    if (point_less(second_finish, second_start)) {
        std::swap(second_start, second_finish);
    }
    if (point_less(first_start, second_start)) return true;
    if (point_less(second_start, first_start)) return false;
    return point_less(first_finish, second_finish);
}

inline void consider(ClosestPoints& best, const ClosestPoints& candidate) {
    const long double best_distance = separation2(best);
    const long double candidate_distance = separation2(candidate);
    if (
        candidate_distance < best_distance ||
        (
            candidate_distance == best_distance &&
            canonical_less(candidate, best)
        )
    ) {
        best = candidate;
    }
}

}  // namespace closest_points_detail

template <Coordinate T>
ClosestPoints closest_points(
    const Point<T>& first,
    const Point<T>& second
) {
    return ClosestPoints{
        Point<long double>(first),
        Point<long double>(second),
    };
}

template <Coordinate T>
ClosestPoints closest_points(
    const Line<T>& line,
    const Point<T>& point
) {
    return ClosestPoints{
        projection(line, point),
        Point<long double>(point),
    };
}

template <Coordinate T>
ClosestPoints closest_points(
    const Point<T>& point,
    const Line<T>& line
) {
    return closest_points_detail::reversed(closest_points(line, point));
}

template <Coordinate T>
ClosestPoints closest_points(
    const Segment<T>& segment,
    const Point<T>& point
) {
    return ClosestPoints{
        projection(segment, point),
        Point<long double>(point),
    };
}

template <Coordinate T>
ClosestPoints closest_points(
    const Point<T>& point,
    const Segment<T>& segment
) {
    return closest_points_detail::reversed(closest_points(segment, point));
}

template <Coordinate T>
ClosestPoints closest_points(
    const Ray<T>& ray,
    const Point<T>& point
) {
    return ClosestPoints{
        projection(ray, point),
        Point<long double>(point),
    };
}

template <Coordinate T>
ClosestPoints closest_points(
    const Point<T>& point,
    const Ray<T>& ray
) {
    return closest_points_detail::reversed(closest_points(ray, point));
}

template <Coordinate T>
ClosestPoints closest_points(
    const Line<T>& first,
    const Line<T>& second,
    long double eps = 1e-12L
) {
    const LinearIntersection intersection =
        linear_intersection(first, second, eps);
    if (intersection.kind != LinearIntersectionKind::Empty) {
        return closest_points_detail::common_point(intersection);
    }
    ClosestPoints result = closest_points(first, second.a);
    closest_points_detail::consider(
        result,
        closest_points(first.a, second)
    );
    return result;
}

template <Coordinate T>
ClosestPoints closest_points(
    const Line<T>& line,
    const Segment<T>& segment,
    long double eps = 1e-12L
) {
    const LinearIntersection intersection =
        linear_intersection(line, segment, eps);
    if (intersection.kind != LinearIntersectionKind::Empty) {
        return closest_points_detail::common_point(intersection);
    }
    ClosestPoints result = closest_points(line, segment.a);
    closest_points_detail::consider(
        result,
        closest_points(line, segment.b)
    );
    return result;
}

template <Coordinate T>
ClosestPoints closest_points(
    const Segment<T>& segment,
    const Line<T>& line,
    long double eps = 1e-12L
) {
    return closest_points_detail::reversed(
        closest_points(line, segment, eps)
    );
}

template <Coordinate T>
ClosestPoints closest_points(
    const Segment<T>& first,
    const Segment<T>& second,
    long double eps = 1e-12L
) {
    const LinearIntersection intersection =
        linear_intersection(first, second, eps);
    if (intersection.kind != LinearIntersectionKind::Empty) {
        return closest_points_detail::common_point(intersection);
    }
    ClosestPoints result = closest_points(first, second.a);
    closest_points_detail::consider(
        result,
        closest_points(first, second.b)
    );
    closest_points_detail::consider(
        result,
        closest_points(first.a, second)
    );
    closest_points_detail::consider(
        result,
        closest_points(first.b, second)
    );
    return result;
}

template <Coordinate T>
ClosestPoints closest_points(
    const Line<T>& line,
    const Ray<T>& ray,
    long double eps = 1e-12L
) {
    const LinearIntersection intersection =
        linear_intersection(line, ray, eps);
    if (intersection.kind != LinearIntersectionKind::Empty) {
        return closest_points_detail::common_point(intersection);
    }
    return closest_points(line, ray.origin);
}

template <Coordinate T>
ClosestPoints closest_points(
    const Ray<T>& ray,
    const Line<T>& line,
    long double eps = 1e-12L
) {
    return closest_points_detail::reversed(closest_points(line, ray, eps));
}

template <Coordinate T>
ClosestPoints closest_points(
    const Ray<T>& ray,
    const Segment<T>& segment,
    long double eps = 1e-12L
) {
    const LinearIntersection intersection =
        linear_intersection(ray, segment, eps);
    if (intersection.kind != LinearIntersectionKind::Empty) {
        return closest_points_detail::common_point(intersection);
    }
    ClosestPoints result = closest_points(ray, segment.a);
    closest_points_detail::consider(
        result,
        closest_points(ray, segment.b)
    );
    closest_points_detail::consider(
        result,
        closest_points(ray.origin, segment)
    );
    return result;
}

template <Coordinate T>
ClosestPoints closest_points(
    const Segment<T>& segment,
    const Ray<T>& ray,
    long double eps = 1e-12L
) {
    return closest_points_detail::reversed(
        closest_points(ray, segment, eps)
    );
}

template <Coordinate T>
ClosestPoints closest_points(
    const Ray<T>& first,
    const Ray<T>& second,
    long double eps = 1e-12L
) {
    const LinearIntersection intersection =
        linear_intersection(first, second, eps);
    if (intersection.kind != LinearIntersectionKind::Empty) {
        return closest_points_detail::common_point(intersection);
    }
    ClosestPoints result = closest_points(first, second.origin);
    closest_points_detail::consider(
        result,
        closest_points(first.origin, second)
    );
    return result;
}

template <Coordinate T>
long double distance(const Line<T>& line, const Point<T>& point) {
    const ClosestPoints result = closest_points(line, point);
    return geometry::distance(result.first, result.second);
}

template <Coordinate T>
long double distance(const Point<T>& point, const Line<T>& line) {
    return distance(line, point);
}

template <Coordinate T>
long double distance(const Segment<T>& segment, const Point<T>& point) {
    const ClosestPoints result = closest_points(segment, point);
    return geometry::distance(result.first, result.second);
}

template <Coordinate T>
long double distance(const Point<T>& point, const Segment<T>& segment) {
    return distance(segment, point);
}

template <Coordinate T>
long double distance(const Ray<T>& ray, const Point<T>& point) {
    const ClosestPoints result = closest_points(ray, point);
    return geometry::distance(result.first, result.second);
}

template <Coordinate T>
long double distance(const Point<T>& point, const Ray<T>& ray) {
    return distance(ray, point);
}

template <Coordinate T>
long double distance(const Line<T>& first, const Line<T>& second) {
    const ClosestPoints result = closest_points(first, second);
    return geometry::distance(result.first, result.second);
}

template <Coordinate T>
long double distance(const Line<T>& line, const Segment<T>& segment) {
    const ClosestPoints result = closest_points(line, segment);
    return geometry::distance(result.first, result.second);
}

template <Coordinate T>
long double distance(const Segment<T>& segment, const Line<T>& line) {
    return distance(line, segment);
}

template <Coordinate T>
long double distance(const Segment<T>& first, const Segment<T>& second) {
    const ClosestPoints result = closest_points(first, second);
    return geometry::distance(result.first, result.second);
}

template <Coordinate T>
long double distance(const Line<T>& line, const Ray<T>& ray) {
    const ClosestPoints result = closest_points(line, ray);
    return geometry::distance(result.first, result.second);
}

template <Coordinate T>
long double distance(const Ray<T>& ray, const Line<T>& line) {
    return distance(line, ray);
}

template <Coordinate T>
long double distance(const Ray<T>& ray, const Segment<T>& segment) {
    const ClosestPoints result = closest_points(ray, segment);
    return geometry::distance(result.first, result.second);
}

template <Coordinate T>
long double distance(const Segment<T>& segment, const Ray<T>& ray) {
    return distance(ray, segment);
}

template <Coordinate T>
long double distance(const Ray<T>& first, const Ray<T>& second) {
    const ClosestPoints result = closest_points(first, second);
    return geometry::distance(result.first, result.second);
}

}  // namespace geometry
}  // namespace m1une


#line 17 "geometry/half_plane_intersection.hpp"

namespace m1une {
namespace geometry {

enum class HalfPlaneIntersectionStatus {
    Empty,
    Unbounded,
    Degenerate,
    Bounded,
};

struct HalfPlaneIntersectionResult {
    HalfPlaneIntersectionStatus status;
    std::vector<Point<long double>> polygon;
};

namespace half_plane_intersection_detail {

struct HalfPlane {
    Point<long double> point;
    Point<long double> direction;
    long double angle;

    HalfPlane(
        const Point<long double>& point_value,
        const Point<long double>& direction_value
    ) : point(point_value), direction(direction_value) {
        angle = std::atan2(direction.y, direction.x);
        if (angle < 0) angle += 2 * std::numbers::pi_v<long double>;
    }
};

inline bool direction_less(const HalfPlane& first, const HalfPlane& second) {
    return first.angle < second.angle;
}

inline bool parallel(
    const HalfPlane& first,
    const HalfPlane& second,
    long double eps
) {
    return std::fabs(cross(first.direction, second.direction)) <= eps;
}

inline bool same_direction(
    const HalfPlane& first,
    const HalfPlane& second,
    long double eps
) {
    return parallel(first, second, eps) &&
           dot(first.direction, second.direction) > 0;
}

inline bool outside(
    const HalfPlane& half_plane,
    const Point<long double>& point,
    long double eps
) {
    return cross(half_plane.direction, point - half_plane.point) < -eps;
}

inline bool more_restrictive(
    const HalfPlane& candidate,
    const HalfPlane& current,
    long double eps
) {
    return cross(
        current.direction,
        candidate.point - current.point
    ) > eps;
}

inline std::optional<Point<long double>> intersection(
    const HalfPlane& first,
    const HalfPlane& second,
    long double eps
) {
    long double denominator = cross(first.direction, second.direction);
    if (std::fabs(denominator) <= eps) return std::nullopt;
    long double ratio = cross(
        second.point - first.point,
        second.direction
    ) / denominator;
    return first.point + first.direction * ratio;
}

inline void merge_same_direction(
    std::vector<HalfPlane>& half_planes,
    const HalfPlane& half_plane,
    long double eps
) {
    if (
        half_planes.empty() ||
        !same_direction(half_planes.back(), half_plane, eps)
    ) {
        half_planes.push_back(half_plane);
        return;
    }
    if (more_restrictive(half_plane, half_planes.back(), eps)) {
        half_planes.back() = half_plane;
    }
}

inline void merge_cyclic_ends(
    std::vector<HalfPlane>& half_planes,
    long double eps
) {
    if (
        half_planes.size() < 2 ||
        !same_direction(half_planes.front(), half_planes.back(), eps)
    ) {
        return;
    }
    if (more_restrictive(half_planes.back(), half_planes.front(), eps)) {
        half_planes.front() = half_planes.back();
    }
    half_planes.pop_back();
}

inline bool has_feasible_point(
    std::vector<HalfPlane> half_planes,
    long double eps
) {
    std::mt19937_64 generator(0x6a09e667f3bcc909ULL);
    std::shuffle(half_planes.begin(), half_planes.end(), generator);

    Point<long double> feasible(0, 0);
    for (std::size_t index = 0; index < half_planes.size(); ++index) {
        const HalfPlane& current = half_planes[index];
        if (!outside(current, feasible, eps)) continue;

        Point<long double> normal(
            -current.direction.y,
            current.direction.x
        );
        Point<long double> base = normal * dot(normal, current.point);
        long double lower = -std::numeric_limits<long double>::infinity();
        long double upper = std::numeric_limits<long double>::infinity();
        for (std::size_t previous_index = 0;
             previous_index < index;
             ++previous_index) {
            const HalfPlane& previous = half_planes[previous_index];
            long double coefficient = cross(
                previous.direction,
                current.direction
            );
            long double constant = cross(
                previous.direction,
                base - previous.point
            );
            if (std::fabs(coefficient) <= eps) {
                if (constant < -eps) return false;
                continue;
            }

            long double bound = (-eps - constant) / coefficient;
            if (coefficient > 0) {
                lower = std::max(lower, bound);
            } else {
                upper = std::min(upper, bound);
            }
            if (lower > upper) return false;
        }

        long double parameter = 0;
        if (parameter < lower) parameter = lower;
        if (parameter > upper) parameter = upper;
        feasible = base + current.direction * parameter;
    }
    return true;
}

inline bool has_bounded_recession_cone(
    const std::vector<HalfPlane>& half_planes,
    long double eps
) {
    if (half_planes.empty()) return false;

    constexpr long double pi = std::numbers::pi_v<long double>;
    long double maximum_gap =
        half_planes.front().angle + 2 * pi - half_planes.back().angle;
    for (std::size_t index = 1; index < half_planes.size(); ++index) {
        maximum_gap = std::max(
            maximum_gap,
            half_planes[index].angle - half_planes[index - 1].angle
        );
    }
    return maximum_gap < pi - eps;
}

}  // namespace half_plane_intersection_detail

// Each directed line keeps its closed left half-plane. Returns the vertices of
// a bounded intersection with positive area in counterclockwise order. Empty,
// unbounded, and bounded zero-area intersections have distinct statuses.
template <Coordinate T>
HalfPlaneIntersectionResult half_plane_intersection(
    const std::vector<Line<T>>& half_planes,
    long double eps = 1e-12L
) {
    using half_plane_intersection_detail::HalfPlane;
    namespace detail = half_plane_intersection_detail;

    assert(eps >= 0);
    std::vector<HalfPlane> sorted;
    sorted.reserve(half_planes.size());
    for (const Line<T>& line : half_planes) {
        assert(line.a != line.b);
        Point<long double> point(line.a);
        Point<long double> direction = Point<long double>(line.b) - point;
        long double length = norm(direction);
        direction = direction / length;
        sorted.push_back(HalfPlane{point, direction});
    }
    if (!detail::has_feasible_point(sorted, eps)) {
        return HalfPlaneIntersectionResult{
            HalfPlaneIntersectionStatus::Empty,
            {},
        };
    }
    std::sort(sorted.begin(), sorted.end(), detail::direction_less);
    if (!detail::has_bounded_recession_cone(sorted, eps)) {
        return HalfPlaneIntersectionResult{
            HalfPlaneIntersectionStatus::Unbounded,
            {},
        };
    }
    if (sorted.size() < 3) {
        return HalfPlaneIntersectionResult{
            HalfPlaneIntersectionStatus::Degenerate,
            {},
        };
    }

    std::vector<HalfPlane> unique;
    unique.reserve(sorted.size());
    for (const HalfPlane& half_plane : sorted) {
        detail::merge_same_direction(unique, half_plane, eps);
    }
    detail::merge_cyclic_ends(unique, eps);
    if (unique.size() < 3) {
        return HalfPlaneIntersectionResult{
            HalfPlaneIntersectionStatus::Degenerate,
            {},
        };
    }

    std::deque<HalfPlane> deque;
    for (const HalfPlane& half_plane : unique) {
        while (deque.size() >= 2) {
            auto point = detail::intersection(
                deque[deque.size() - 2],
                deque.back(),
                eps
            );
            if (!point.has_value()) {
                return HalfPlaneIntersectionResult{
                    HalfPlaneIntersectionStatus::Degenerate,
                    {},
                };
            }
            if (!detail::outside(half_plane, *point, eps)) break;
            deque.pop_back();
        }
        while (deque.size() >= 2) {
            auto point = detail::intersection(deque[0], deque[1], eps);
            if (!point.has_value()) {
                return HalfPlaneIntersectionResult{
                    HalfPlaneIntersectionStatus::Degenerate,
                    {},
                };
            }
            if (!detail::outside(half_plane, *point, eps)) break;
            deque.pop_front();
        }
        deque.push_back(half_plane);
    }

    while (deque.size() >= 3) {
        auto point = detail::intersection(
            deque[deque.size() - 2],
            deque.back(),
            eps
        );
        if (!point.has_value()) {
            return HalfPlaneIntersectionResult{
                HalfPlaneIntersectionStatus::Degenerate,
                {},
            };
        }
        if (!detail::outside(deque.front(), *point, eps)) break;
        deque.pop_back();
    }
    while (deque.size() >= 3) {
        auto point = detail::intersection(deque[0], deque[1], eps);
        if (!point.has_value()) {
            return HalfPlaneIntersectionResult{
                HalfPlaneIntersectionStatus::Degenerate,
                {},
            };
        }
        if (!detail::outside(deque.back(), *point, eps)) break;
        deque.pop_front();
    }
    if (deque.size() < 3) {
        return HalfPlaneIntersectionResult{
            HalfPlaneIntersectionStatus::Degenerate,
            {},
        };
    }

    std::vector<Point<long double>> polygon;
    polygon.reserve(deque.size());
    for (std::size_t index = 0; index < deque.size(); ++index) {
        auto point = detail::intersection(
            deque[index],
            deque[(index + 1) % deque.size()],
            eps
        );
        if (!point.has_value()) {
            return HalfPlaneIntersectionResult{
                HalfPlaneIntersectionStatus::Degenerate,
                {},
            };
        }
        if (
            polygon.empty() ||
            distance(polygon.back(), *point) > eps
        ) {
            polygon.push_back(*point);
        }
    }
    if (
        polygon.size() >= 2 &&
        distance(polygon.front(), polygon.back()) <= eps
    ) {
        polygon.pop_back();
    }
    if (polygon.size() < 3) {
        return HalfPlaneIntersectionResult{
            HalfPlaneIntersectionStatus::Degenerate,
            {},
        };
    }

    long double signed_area2 = 0;
    Point<long double> origin = polygon.front();
    for (std::size_t index = 1; index + 1 < polygon.size(); ++index) {
        signed_area2 += cross(
            polygon[index] - origin,
            polygon[index + 1] - origin
        );
    }
    if (signed_area2 <= eps) {
        return HalfPlaneIntersectionResult{
            HalfPlaneIntersectionStatus::Degenerate,
            {},
        };
    }

    auto first = std::min_element(polygon.begin(), polygon.end());
    std::rotate(polygon.begin(), first, polygon.end());
    return HalfPlaneIntersectionResult{
        HalfPlaneIntersectionStatus::Bounded,
        std::move(polygon),
    };
}

}  // namespace geometry
}  // namespace m1une


#line 6 "verify/geometry/voronoi_diagram.test.cpp"

#line 11 "verify/geometry/voronoi_diagram.test.cpp"
#include <cstdint>
#include <iomanip>
#include <iostream>
#line 15 "verify/geometry/voronoi_diagram.test.cpp"
#include <set>
#line 18 "verify/geometry/voronoi_diagram.test.cpp"

namespace {

using m1une::geometry::Point;
using m1une::geometry::VoronoiDiagram;
using m1une::geometry::VoronoiEdge;
using m1une::geometry::VoronoiEdgeKind;
using Site = Point<long long>;
using RealPoint = Point<long double>;

long double squared_distance(const RealPoint& first, const RealPoint& second) {
    long double x = first.x - second.x;
    long double y = first.y - second.y;
    return x * x + y * y;
}

bool close(long double first, long double second, long double eps = 1e-8L) {
    return std::fabs(first - second) <=
           eps * std::max(
               1.0L,
               std::max(std::fabs(first), std::fabs(second))
           );
}

void check_boundary_point(
    const std::vector<Site>& sites,
    const VoronoiEdge& edge,
    const RealPoint& point
) {
    RealPoint first(sites[edge.first_site]);
    RealPoint second(sites[edge.second_site]);
    long double first_distance = squared_distance(point, first);
    long double second_distance = squared_distance(point, second);
    assert(close(first_distance, second_distance, 1e-7L));
    for (const Site& site : sites) {
        long double candidate = squared_distance(point, RealPoint(site));
        long double tolerance =
            1e-7L * std::max(1.0L, std::max(first_distance, candidate));
        assert(first_distance <= candidate + tolerance);
    }
}

bool naive_has_voronoi_edge(
    const std::vector<Site>& sites,
    int first_site,
    int second_site
) {
    RealPoint first(sites[first_site]);
    RealPoint second(sites[second_site]);
    RealPoint midpoint = (first + second) / 2.0L;
    RealPoint difference = second - first;
    RealPoint direction(difference.y, -difference.x);
    long double lower = -std::numeric_limits<long double>::infinity();
    long double upper = std::numeric_limits<long double>::infinity();

    for (const Site& integer_site : sites) {
        RealPoint site(integer_site);
        long double constant =
            squared_distance(midpoint, first) - squared_distance(midpoint, site);
        RealPoint shifted = midpoint + direction;
        long double coefficient =
            squared_distance(shifted, first) - squared_distance(shifted, site) -
            constant;
        if (std::fabs(coefficient) <= 1e-14L) {
            if (constant > 1e-12L) return false;
        } else {
            long double bound = -constant / coefficient;
            if (coefficient > 0) {
                upper = std::min(upper, bound);
            } else {
                lower = std::max(lower, bound);
            }
        }
    }
    return lower + 1e-10L < upper;
}

void check_diagram(const std::vector<Site>& sites) {
    VoronoiDiagram diagram = m1une::geometry::voronoi_diagram(sites);
    assert(diagram.cell_edges.size() == sites.size());

    std::set<std::pair<int, int>> actual_pairs;
    std::vector<int> cell_occurrences(diagram.edges.size(), 0);
    for (int site = 0; site < int(sites.size()); ++site) {
        for (int edge_index : diagram.cell_edges[site]) {
            assert(0 <= edge_index && edge_index < int(diagram.edges.size()));
            const VoronoiEdge& edge = diagram.edges[edge_index];
            assert(edge.first_site == site || edge.second_site == site);
            ++cell_occurrences[edge_index];
        }
    }

    for (int edge_index = 0; edge_index < int(diagram.edges.size()); ++edge_index) {
        const VoronoiEdge& edge = diagram.edges[edge_index];
        assert(cell_occurrences[edge_index] == 2);
        assert(0 <= edge.first_site && edge.first_site < int(sites.size()));
        assert(0 <= edge.second_site && edge.second_site < int(sites.size()));
        assert(edge.first_site < edge.second_site);
        assert(actual_pairs.emplace(edge.first_site, edge.second_site).second);

        if (edge.kind == VoronoiEdgeKind::Segment) {
            assert(0 <= edge.first_vertex);
            assert(edge.first_vertex < int(diagram.vertices.size()));
            assert(0 <= edge.second_vertex);
            assert(edge.second_vertex < int(diagram.vertices.size()));
            assert(edge.first_vertex < edge.second_vertex);
            assert(close(edge.point.x, diagram.vertices[edge.first_vertex].x));
            assert(close(edge.point.y, diagram.vertices[edge.first_vertex].y));
            RealPoint expected_direction =
                diagram.vertices[edge.second_vertex] -
                diagram.vertices[edge.first_vertex];
            assert(close(edge.direction.x, expected_direction.x));
            assert(close(edge.direction.y, expected_direction.y));
            for (long double parameter : std::array<long double, 3>{0, 0.5L, 1}) {
                check_boundary_point(
                    sites,
                    edge,
                    edge.point + edge.direction * parameter
                );
            }
        } else if (edge.kind == VoronoiEdgeKind::Ray) {
            assert(0 <= edge.first_vertex);
            assert(edge.first_vertex < int(diagram.vertices.size()));
            assert(edge.second_vertex == -1);
            assert(close(edge.point.x, diagram.vertices[edge.first_vertex].x));
            assert(close(edge.point.y, diagram.vertices[edge.first_vertex].y));
            assert(close(m1une::geometry::norm(edge.direction), 1));
            for (long double parameter : std::array<long double, 3>{0, 1, 100}) {
                check_boundary_point(
                    sites,
                    edge,
                    edge.point + edge.direction * parameter
                );
            }
        } else {
            assert(edge.kind == VoronoiEdgeKind::Line);
            assert(edge.first_vertex == -1);
            assert(edge.second_vertex == -1);
            assert(close(m1une::geometry::norm(edge.direction), 1));
            for (long double parameter : std::array<long double, 3>{-100, 0, 100}) {
                check_boundary_point(
                    sites,
                    edge,
                    edge.point + edge.direction * parameter
                );
            }
        }
    }

    for (int first = 0; first < int(sites.size()); ++first) {
        for (int second = first + 1; second < int(sites.size()); ++second) {
            bool actual = actual_pairs.contains(std::pair(first, second));
            bool expected = naive_has_voronoi_edge(sites, first, second);
            assert(actual == expected);
        }
    }
}

int count_kind(const VoronoiDiagram& diagram, VoronoiEdgeKind kind) {
    return int(std::count_if(
        diagram.edges.begin(),
        diagram.edges.end(),
        [&](const VoronoiEdge& edge) { return edge.kind == kind; }
    ));
}

void test_fixed() {
    check_diagram({});
    check_diagram(std::vector<Site>{Site(4, -2)});

    std::vector<Site> two_sites{Site(0, 0), Site(4, 0)};
    check_diagram(two_sites);
    VoronoiDiagram two = m1une::geometry::voronoi_diagram(two_sites);
    assert(two.vertices.empty());
    assert(two.edges.size() == 1);
    assert(two.edges[0].kind == VoronoiEdgeKind::Line);
    assert(close(two.edges[0].point.x, 2));
    assert(close(two.edges[0].point.y, 0));

    std::vector<Site> triangle{
        Site(0, 0),
        Site(6, 0),
        Site(0, 8),
    };
    check_diagram(triangle);
    VoronoiDiagram three = m1une::geometry::voronoi_diagram(triangle);
    assert(three.vertices.size() == 1);
    assert(three.edges.size() == 3);
    assert(count_kind(three, VoronoiEdgeKind::Ray) == 3);
    assert(close(three.vertices[0].x, 3));
    assert(close(three.vertices[0].y, 4));

    std::vector<Site> square{
        Site(0, 0),
        Site(2, 0),
        Site(2, 2),
        Site(0, 2),
    };
    check_diagram(square);
    VoronoiDiagram four = m1une::geometry::voronoi_diagram(square);
    assert(four.vertices.size() == 1);
    assert(four.edges.size() == 4);
    assert(count_kind(four, VoronoiEdgeKind::Ray) == 4);

    std::vector<Site> cocircular{
        Site(5, 0),
        Site(3, 4),
        Site(0, 5),
        Site(-3, 4),
        Site(-5, 0),
        Site(-3, -4),
        Site(0, -5),
        Site(3, -4),
    };
    check_diagram(cocircular);
    VoronoiDiagram eight = m1une::geometry::voronoi_diagram(cocircular);
    assert(eight.vertices.size() == 1);
    assert(eight.edges.size() == 8);
    assert(count_kind(eight, VoronoiEdgeKind::Ray) == 8);

    std::vector<Site> square_with_center = square;
    square_with_center.emplace_back(1, 1);
    check_diagram(square_with_center);
    VoronoiDiagram five =
        m1une::geometry::voronoi_diagram(square_with_center);
    assert(five.vertices.size() == 4);
    assert(five.edges.size() == 8);
    assert(count_kind(five, VoronoiEdgeKind::Segment) == 4);
    assert(count_kind(five, VoronoiEdgeKind::Ray) == 4);

    std::vector<Site> collinear{
        Site(-5, 3),
        Site(-1, 3),
        Site(2, 3),
        Site(9, 3),
    };
    check_diagram(collinear);
    VoronoiDiagram line = m1une::geometry::voronoi_diagram(collinear);
    assert(line.vertices.empty());
    assert(line.edges.size() == 3);
    assert(count_kind(line, VoronoiEdgeKind::Line) == 3);
}

void test_randomized() {
    std::uint64_t state = 0x243f6a8885a308d3ULL;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int trial = 0; trial < 1500; ++trial) {
        int size = int(random() % 11);
        std::set<std::pair<long long, long long>> used;
        std::vector<Site> sites;
        sites.reserve(size);
        while (int(sites.size()) < size) {
            long long x = static_cast<long long>(random() % 21) - 10;
            long long y = static_cast<long long>(random() % 21) - 10;
            if (used.emplace(x, y).second) sites.emplace_back(x, y);
        }
        check_diagram(sites);
    }
}

long double polygon_area(const std::vector<RealPoint>& polygon) {
    long double twice_area = 0;
    for (int index = 0; index < int(polygon.size()); ++index) {
        twice_area += m1une::geometry::cross(
            polygon[index],
            polygon[(index + 1) % polygon.size()]
        );
    }
    return std::fabs(twice_area) / 2;
}

}  // namespace

int main() {
    test_fixed();
    test_randomized();

    std::cout << std::fixed << std::setprecision(10);
    while (true) {
        int island_size, site_count;
        std::cin >> island_size >> site_count;
        if (island_size == 0 && site_count == 0) break;

        std::vector<Site> island(island_size);
        for (Site& point : island) std::cin >> point.x >> point.y;
        std::vector<Site> sites(site_count);
        for (Site& point : sites) std::cin >> point.x >> point.y;

        VoronoiDiagram diagram = m1une::geometry::voronoi_diagram(sites);
        for (int site = 0; site < site_count; ++site) {
            std::vector<m1une::geometry::Line<long double>> half_planes;
            half_planes.reserve(island_size + diagram.cell_edges[site].size());
            for (int index = 0; index < island_size; ++index) {
                half_planes.push_back(m1une::geometry::Line<long double>{
                    RealPoint(island[index]),
                    RealPoint(island[(index + 1) % island_size]),
                });
            }
            for (int edge_index : diagram.cell_edges[site]) {
                const VoronoiEdge& edge = diagram.edges[edge_index];
                int other = edge.first_site == site
                    ? edge.second_site
                    : edge.first_site;
                RealPoint first(sites[site]);
                RealPoint second(sites[other]);
                RealPoint midpoint = (first + second) / 2.0L;
                RealPoint difference = second - first;
                RealPoint direction(-difference.y, difference.x);
                half_planes.push_back(m1une::geometry::Line<long double>{
                    midpoint,
                    midpoint + direction,
                });
            }

            auto intersection =
                m1une::geometry::half_plane_intersection(half_planes);
            long double area = 0;
            if (
                intersection.status ==
                m1une::geometry::HalfPlaneIntersectionStatus::Bounded
            ) {
                area = polygon_area(intersection.polygon);
            }
            std::cout << area << '\n';
        }
    }
}
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