Tifa's CP Library

:heavy_check_mark: ins_hps (src/code/geo2d/ins_hps.hpp)

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#ifndef TIFALIBS_GEO2D_INS_HPS
#define TIFALIBS_GEO2D_INS_HPS

#include "../edh/discretization.hpp"
#include "cvh.hpp"

namespace tifa_libs::geo {

template <class FP>
CEXP cvh<FP> ins_hPs(vec<line<FP>> vl) {
  auto check = [](line<FP> CR u, line<FP> CR v, line<FP> CR w) -> bool { return w.is_include_strict(ins_LL(u, v)); };
  if ((vl = uniq(vl)).size() < 3) return {};
  std::deque<line<FP>> q;
  for (auto it = vl.begin(); it != vl.end(); ++it) {
    if (it != vl.begin() && is_same_dir(*it, *(it - 1))) continue;
    while (q.size() > 1 && !check(*(q.rbegin() + 1), q.back(), *it)) q.pop_back();
    while (q.size() > 1 && !check(*(q.begin() + 1), q.front(), *it)) q.pop_front();
    q.push_back(*it);
  }
  while (q.size() > 2 && !check(*(q.rbegin() + 1), q.back(), q.front())) q.pop_back();
  while (q.size() > 2 && !check(*(q.begin() + 1), q.front(), q.back())) q.pop_front();
  vec<point<FP>> ret;
  flt_ (u32, i, 0, (u32)q.size()) ret.push_back(ins_LL(q[i], q[(i + 1) % q.size()]));
  return cvh<FP>{ret, true};
}

}  // namespace tifa_libs::geo

#endif
#line 1 "src/code/geo2d/ins_hps.hpp"



#line 1 "src/code/edh/discretization.hpp"



#line 1 "src/code/util/traits.hpp"



#line 1 "src/code/util/util.hpp"



#include <bits/extc++.h>

#define CEXP constexpr
#define CEXPE constexpr explicit
#define TPN typename
#define CR const&

#define cT_(...) std::conditional_t<sizeof(__VA_ARGS__) <= sizeof(size_t), __VA_ARGS__, __VA_ARGS__ CR>
#define fle_(T, i, l, r, ...) for (T i = (l), i##e = (r)__VA_OPT__(, ) __VA_ARGS__; i <= i##e; ++i)
#define flt_(T, i, l, r, ...) for (T i = (l), i##e = (r)__VA_OPT__(, ) __VA_ARGS__; i < i##e; ++i)

#ifdef ONLINE_JUDGE
#undef assert
#define assert(x) 42
#endif

using i8 = int8_t;
using i16 = int16_t;
using i32 = int32_t;
using i64 = int64_t;
using i128 = __int128_t;
using isz = ptrdiff_t;
using u8 = uint8_t;
using u16 = uint16_t;
using u32 = uint32_t;
using u64 = uint64_t;
using u128 = __uint128_t;
using usz = size_t;
using f32 = float;
using f64 = double;
using f128 = long double;
using strn = std::string;
using strnv = std::string_view;

// clang-format off
template <class T, T v> using ic = std::integral_constant<T, v>;
template <class T> using ptt = std::pair<T, T>;
template <class T> struct edge_t {
  T w; u32 u, v;
  CEXP auto operator<=>(edge_t CR) const = default;
};
template <class T> struct pt3 {
  T _0, _1, _2;
  CEXP auto operator<=>(pt3 CR) const = default;
};
template <class T> struct pt4 {
  T _0, _1, _2, _3;
  CEXP auto operator<=>(pt4 CR) const = default;
};
template <class E> using itl = std::initializer_list<E>;
template <class T> using vec = std::vector<T>;
template <class T> using vvec = vec<vec<T>>;
template <class T> using v3ec = vec<vvec<T>>;
template <class T> using vecpt = vec<ptt<T>>;
template <class T> using vvecpt = vvec<ptt<T>>;
template <class T> using ptvec = ptt<vec<T>>;
template <class T> using ptvvec = ptt<vvec<T>>;

template <class T, usz ext = std::dynamic_extent> using spn = std::span<T const, ext>;
template <class T, usz N> using arr = std::array<T, N>;
template <class U, class T> using vecp = vec<std::pair<U, T>>;
template <class U, class T> using vvecp = vvec<std::pair<U, T>>;

template <class T, class C = std::less<T>> using pq = std::priority_queue<T, vec<T>, C>;
template <class T> using pqg = std::priority_queue<T, vec<T>, std::greater<T>>;
// clang-format on

#define mk_(V, A, T) using V##A = V<T>;
#define mk(A, T) mk_(edge_t, A, T) mk_(ptt, A, T) mk_(pt3, A, T) mk_(pt4, A, T) mk_(vec, A, T) mk_(vvec, A, T) mk_(v3ec, A, T) mk_(vecpt, A, T) mk_(vvecpt, A, T) mk_(ptvec, A, T) mk_(ptvvec, A, T) mk_(spn, A, T) mk_(itl, A, T)
mk(b, bool) mk(i, i32) mk(u, u32) mk(ii, i64) mk(uu, u64);
#undef mk
#undef mk_

using namespace std::literals;
CEXP i8 operator""_i8(unsigned long long x) { return (i8)x; }
CEXP i16 operator""_i16(unsigned long long x) { return (i16)x; }
CEXP i32 operator""_i32(unsigned long long x) { return (i32)x; }
CEXP i64 operator""_i64(unsigned long long x) { return (i64)x; }
CEXP isz operator""_iz(unsigned long long x) { return (isz)x; }
CEXP u8 operator""_u8(unsigned long long x) { return (u8)x; }
CEXP u16 operator""_u16(unsigned long long x) { return (u16)x; }
CEXP u32 operator""_u32(unsigned long long x) { return (u32)x; }
CEXP u64 operator""_u64(unsigned long long x) { return (u64)x; }
CEXP usz operator""_uz(unsigned long long x) { return (usz)x; }

using std::numbers::pi_v;
template <std::floating_point FP>
inline FP eps_v = std::sqrt(std::numeric_limits<FP>::epsilon());
template <std::floating_point FP>
CEXP void set_eps(FP v) { eps_v<FP> = v; }

inline const auto fn_0 = [](auto&&...) {};
inline const auto fn_is0 = [](auto x) { return x == 0; };

namespace tifa_libs {
using std::min, std::max, std::swap;
template <class T>
constexpr T abs(T x) { return x < 0 ? -x : x; }
}  // namespace tifa_libs


#line 5 "src/code/util/traits.hpp"

namespace tifa_libs {

template <class T>
concept iterable_c = requires(T v) {
  { v.begin() } -> std::same_as<TPN T::iterator>;
  { v.end() } -> std::same_as<TPN T::iterator>;
};

template <class T>
concept container_c = iterable_c<T> && !std::same_as<std::remove_cvref_t<T>, strn> && !std::same_as<std::remove_cvref_t<T>, strnv>;

template <class T>
CEXP bool is_char_v = std::is_same_v<T, char> || std::is_same_v<T, signed char> || std::is_same_v<T, unsigned char>;
template <class T>
concept char_c = is_char_v<T>;

template <class T>
CEXP bool is_s128_v = std::is_same_v<T, __int128_t> || std::is_same_v<T, __int128>;
template <class T>
concept s128_c = is_s128_v<T>;

template <class T>
CEXP bool is_u128_v = std::is_same_v<T, __uint128_t> || std::is_same_v<T, unsigned __int128>;
template <class T>
concept u128_c = is_u128_v<T>;

template <class T>
CEXP bool is_i128_v = is_s128_v<T> || is_u128_v<T>;
template <class T>
concept i128_c = is_u128_v<T>;

template <class T>
CEXP bool is_int_v = std::is_integral_v<T> || is_i128_v<T>;
template <class T>
concept int_c = is_int_v<T>;

template <class T>
CEXP bool is_sint_v = is_s128_v<T> || (is_int_v<T> && std::is_signed_v<T>);
template <class T>
concept sint_c = is_sint_v<T>;

template <class T>
CEXP bool is_uint_v = is_u128_v<T> || (is_int_v<T> && std::is_unsigned_v<T>);
template <class T>
concept uint_c = is_uint_v<T>;

template <class T>
concept mint_c = requires(T x) {
  { x.mod() } -> uint_c;
  { x.val() } -> uint_c;
};

template <class T>
concept dft_c = requires(T x, vec<TPN T::data_t> v, u32 n) {
  { x.size() } -> std::same_as<u32>;
  x.bzr(n);
  x.dif(v, n);
  x.dit(v, n);
};

template <class T>
concept ntt_c = dft_c<T> && requires(T x) {
  T::max_size;
  T::G;
};

template <class T>
CEXP bool is_arithm_v = std::is_arithmetic_v<T> || is_int_v<T>;
template <class T>
concept arithm_c = is_arithm_v<T>;

template <class T>
struct to_sint : std::make_signed<T> {};
template <>
struct to_sint<u128> {
  using type = u128;
};
template <>
struct to_sint<i128> {
  using type = u128;
};
template <class T>
using to_sint_t = TPN to_sint<T>::type;

template <class T>
struct to_uint : std::make_unsigned<T> {};
template <>
struct to_uint<u128> {
  using type = u128;
};
template <>
struct to_uint<i128> {
  using type = u128;
};
template <class T>
using to_uint_t = TPN to_uint<T>::type;

}  // namespace tifa_libs


#line 5 "src/code/edh/discretization.hpp"

namespace tifa_libs {

template <iterable_c T>
CEXP T uniq(T v) { return std::sort(v.begin(), v.end()), v.erase(std::unique(v.begin(), v.end()), v.end()), v; }
template <iterable_c T>
CEXP std::pair<T, vecu> gen_id(T CR v) {
  const T _ = uniq(v);
  vecu _1;
  _1.reserve(v.size());
  flt_ (u32, i, 0, (u32)v.size()) _1.push_back(u32(std::ranges::lower_bound(_, v[i]) - _.begin()));
  return {_, _1};
}

}  // namespace tifa_libs


#line 1 "src/code/geo2d/cvh.hpp"



#line 1 "src/code/geo2d/dot.hpp"



#line 5 "src/code/geo2d/dot.hpp"

namespace tifa_libs::geo {

template <class P>
CEXP typename P::FP_t dot(P CR o, P CR a, P CR b) { return (a - o) * (b - o); }
template <class P>
CEXP int sgn_dot(P CR o, P CR a, P CR b) { return sgn(dot(o, a, b)); }

}  // namespace tifa_libs::geo


#line 1 "src/code/geo2d/ins_ll.hpp"



#line 1 "src/code/geo2d/line.hpp"



#line 1 "src/code/geo2d/cross.hpp"



#line 1 "src/code/util/fp_comp.hpp"



#line 5 "src/code/util/fp_comp.hpp"

namespace tifa_libs {

template <sint_c T>
CEXP int sgn(T x) { return (!!x) | (x >> (sizeof(T) * 8 - 1)); }
template <uint_c T>
CEXP int sgn(T x) { return !!x; }
template <std::floating_point FP>
CEXP int sgn(FP x) { return (x > eps_v<FP>)-(x < -eps_v<FP>); }

template <class FP>
CEXP bool is_neg(FP x) { return sgn(x) < 0; }
template <class FP>
CEXP bool is_zero(FP x) { return !sgn(x); }
template <class FP>
CEXP bool is_pos(FP x) { return sgn(x) > 0; }

template <int_c T>
CEXP int comp(T l, T r) { return sgn(l - r); }
template <std::floating_point FP>
CEXP int comp(FP l, FP r) { return sgn((l - r) / max({abs(l), abs(r), FP(1)})); }

template <class FP>
CEXP bool is_lt(FP l, FP r) { return comp(l, r) < 0; }
template <class FP>
CEXP bool is_eq(FP l, FP r) { return !comp(l, r); }
template <class FP>
CEXP bool is_gt(FP l, FP r) { return comp(l, r) > 0; }

}  // namespace tifa_libs


#line 5 "src/code/geo2d/cross.hpp"

namespace tifa_libs::geo {

template <class P>
CEXP auto cross(P CR o, P CR a, P CR b) { return (a - o) ^ (b - o); }
template <class P>
requires std::floating_point<typename P::FP_t>
CEXP auto cross_unit(P CR o, P CR a, P CR b) { return (a - o).do_unit() ^ (b - o).do_unit(); }
template <class P>
requires std::floating_point<typename P::FP_t>
CEXP int sgn_cross(P CR o, P CR a, P CR b) { return sgn(cross_unit(o, a, b)); }
template <class P>
CEXP int sgn_cross(P CR o, P CR a, P CR b) { return sgn(cross(o, a, b)); }

}  // namespace tifa_libs::geo


#line 1 "src/code/geo2d/point.hpp"



#line 5 "src/code/geo2d/point.hpp"

namespace tifa_libs::geo {

template <class FP>
struct point {
  using FP_t = FP;

  FP x, y;
  CEXPE point(FP x = FP{}, FP y = FP{}) : x(x), y(y) {}

  friend std::istream &operator>>(std::istream &is, point &p) { return is >> p.x >> p.y; }
  friend std::ostream &operator<<(std::ostream &os, point CR p) { return os << p.x << ' ' << p.y; }
  // s * r + t * (1 - r)
  template <std::floating_point T>
  friend CEXP point lerp(point CR s, point CR t, T r) { return s * r + t * (1 - r); }
  friend CEXP point mid_point(point CR s, point CR t) { return lerp(s, t, .5); }
  template <arithm_c T>
  CEXP point &operator+=(T n) { return this->x += n, this->y += n, *this; }
  template <arithm_c T>
  CEXP point &operator-=(T n) { return this->x -= n, this->y -= n, *this; }
  template <arithm_c T>
  CEXP point &operator*=(T n) { return this->x *= n, this->y *= n, *this; }
  template <arithm_c T>
  CEXP point &operator/=(T n) { return this->x /= n, this->y /= n, *this; }
  template <arithm_c T>
  friend CEXP point operator+(point x, T n) { return x += n; }
  template <arithm_c T>
  friend CEXP point operator+(T n, point x) { return x += n; }
  template <arithm_c T>
  friend CEXP point operator-(point x, T n) { return x -= n; }
  template <arithm_c T>
  friend CEXP point operator-(T n, point x) { return x -= n; }
  template <arithm_c T>
  friend CEXP point operator*(point x, T n) { return x *= n; }
  template <arithm_c T>
  friend CEXP point operator*(T n, point x) { return x *= n; }
  template <arithm_c T>
  friend CEXP point operator/(point x, T n) { return x /= n; }
  template <arithm_c T>
  friend CEXP point operator/(T n, point x) { return x /= n; }
  CEXP point &operator+=(point CR p) { return this->x += p.x, this->y += p.y, *this; }
  CEXP point &operator-=(point CR p) { return this->x -= p.x, this->y -= p.y, *this; }
  CEXP point operator+(point CR p) const { return point(*this) += p; }
  CEXP point operator-(point CR p) const { return point(*this) -= p; }
  CEXP point operator-() const { return point{-x, -y}; }
  CEXP auto operator<=>(point CR p) const {
    if (auto CR c = comp(x, p.x); c) return c;
    return comp(y, p.y);
  }
  CEXP bool operator==(point CR p) const { return is_eq(x, p.x) && is_eq(y, p.y); }
  CEXP FP operator*(point CR p) const { return x * p.x + y * p.y; }
  CEXP FP operator^(point CR p) const { return x * p.y - y * p.x; }
  CEXP FP arg() const {
    static_assert(std::is_floating_point_v<FP>);
    return std::atan2(y, x);
  }
  CEXP FP arg_2pi() const {
    FP res = arg();
    return is_neg(res) ? res + 2 * pi_v<FP> : res;
  }
  CEXP FP norm2() const { return x * x + y * y; }
  CEXP FP norm() const {
    static_assert(std::is_floating_point_v<FP>);
    return std::hypot(x, y);
  }
  CEXP point &do_unit() {
    static_assert(std::is_floating_point_v<FP>);
    return *this /= norm();
  }
  static CEXP u32 QUAD__[9] = {6, 7, 8, 5, 0, 1, 4, 3, 2};
  // 4 3 2
  // 5 0 1
  // 6 7 8
  CEXP u32 quad() const { return QUAD__[(sgn(y) + 1) * 3 + sgn(x) + 1]; }
  CEXP int toleft(point CR p) const { return sgn(*this ^ p); }
  CEXP point &do_rot(FP theta) {
    const FP x0 = x, y0 = y, ct = std::cos(theta), st = std::sin(theta);
    return x = x0 * ct - y0 * st, y = x0 * st + y0 * ct, *this;
  }
  friend CEXP point rot(point p, FP theta) { return p.do_rot(theta); }
  CEXP point &do_rot90() {
    const FP _ = x;
    return x = -y, y = _, *this;
  }
  friend CEXP point rot90(point p) { return p.do_rot90(); }
  CEXP point &do_rot270() {
    const FP _ = y;
    return y = -x, x = _, *this;
  }
  friend CEXP point rot270(point p) { return p.do_rot270(); }
};

}  // namespace tifa_libs::geo


#line 6 "src/code/geo2d/line.hpp"

namespace tifa_libs::geo {

template <class FP>
struct line {
  point<FP> l, r;

  CEXP line() {}
  CEXP line(point<FP> CR s, point<FP> CR t) : l(s), r(t) {}
  CEXP line(point<FP> CR s, FP angle_x) : l(s), r(s + is_eq(angle_x, pi_v<FP> / 2) ? point<FP>{0, 1} : point<FP>{1, std::tan(angle_x)}) { assert(angle_x > 0 && angle_x < pi_v<FP>); }
  // ax + by + c = 0
  CEXP line(FP a, FP b, FP c) {
    if (is_zero(a)) l = {0, -c / b}, r = {1, -c / b};
    else if (is_zero(b)) l = {-c / a, 0}, r = {-c / a, 1};
    else l = {0, -c / b}, r = {1, -(c + a) / b};
  }
  CEXP line(FP s_x, FP s_y, FP t_x, FP t_y) : l(s_x, s_y), r(t_x, t_y) {}

  friend std::istream& operator>>(std::istream& is, line& l) { return is >> l.l >> l.r; }
  friend std::ostream& operator<<(std::ostream& os, line CR l) { return os << l.l << ' ' << l.r; }
  CEXP point<FP> direction() const { return r - l; }
  CEXP bool is_parallel(line CR r) const { return is_zero(direction() ^ r.direction()); }
  friend CEXP bool is_parallel(line CR l, line CR r) { return l.is_parallel(r); }
  CEXP bool is_same_dir(line CR r) const { return is_parallel(r) && is_pos(direction() * r.direction()); }
  friend CEXP bool is_same_dir(line CR l, line CR r) { return l.is_same_dir(r); }
  friend CEXP bool operator==(line CR l, line CR r) { return l.l == r.l && l.r == r.r; }
  friend CEXP auto operator<=>(line CR l, line CR r) {
    if (l == r) return 0;
    if (l.is_same_dir(r)) return r.is_include_strict(l.l) ? -1 : 1;
    if (const auto vl = l.direction(), vr = r.direction(); vl.quad() != vr.quad()) return (i32)vl.quad() - (i32)vr.quad();
    else return -sgn(vl ^ vr);
  }
  CEXP int toleft(point<FP> CR p) const { return sgn_cross(l, r, p); }
  // half plane
  CEXP bool is_include_strict(point<FP> CR p) const { return toleft(p) > 0; }
  // half plane
  CEXP bool is_include(point<FP> CR p) const { return toleft(p) >= 0; }
  // translate @dist along the direction of half plane
  CEXP line& do_push(FP dist) {
    const point delta = direction().do_rot90().do_unit() * dist;
    return l += delta, r += delta, *this;
  }
};

}  // namespace tifa_libs::geo


#line 5 "src/code/geo2d/ins_ll.hpp"

namespace tifa_libs::geo {

// judge if two lines are intersected or not
template <class FP>
CEXP bool is_ins_LL(line<FP> CR l1, line<FP> CR l2) { return !is_zero(cross(l2.l, l2.r, l1.l) - cross(l2.l, l2.r, l1.r)); }
// intersection point of two lines
template <class FP>
CEXP point<FP> ins_LL(line<FP> CR l1, line<FP> CR l2) {
  const FP a1 = cross(l2.l, l2.r, l1.l), a2 = -cross(l2.l, l2.r, l1.r);
  return (l1.l * a2 + l1.r * a1) / (a1 + a2);
}
template <class FP>
CEXP point<FP> ins_LL(line<FP> CR l, FP a, FP b, FP c) {
  const FP a1 = abs(a * l.l.x + b * l.l.y + c), a2 = abs(a * l.r.x + b * l.r.y + c);
  return (l.l * a2 + l.r * a1) / (a1 + a2);
}

}  // namespace tifa_libs::geo


#line 1 "src/code/geo2d/polygon.hpp"



#line 1 "src/code/math/kahan.hpp"



#line 5 "src/code/math/kahan.hpp"

namespace tifa_libs::math {

template <std::floating_point FP>
class kahan_fp {
  FP sum, c;

 public:
  CEXP kahan_fp(FP val = 0) : sum(val), c(0) {}

  CEXP kahan_fp& operator+=(FP x) {
    const FP y = x - c;
    volatile FP t = sum + y, z = t - sum;
    return c = z - y, sum = t, *this;
  }
  CEXP operator FP() const { return sum; }
};
template <class FP>
using kahan = std::conditional_t<std::is_floating_point_v<FP>, kahan_fp<FP>, FP>;

}  // namespace tifa_libs::math


#line 1 "src/code/geo2d/dist_pp.hpp"



#line 5 "src/code/geo2d/dist_pp.hpp"

namespace tifa_libs::geo {

// distance of two points (Manhattan)
template <class FP>
CEXP FP dist_PP_Ma(point<FP> CR p1, point<FP> CR p2) { return abs(p1.x - p2.x) + abs(p1.y - p2.y); }
// distance of two points (Euclidian)
template <class FP>
CEXP FP dist_PP(point<FP> CR p1, point<FP> CR p2) { return (p1 - p2).norm(); }
// distance of two points (Chebyshev)
template <class FP>
CEXP FP dist_PP_Ch(point<FP> CR p1, point<FP> CR p2) { return max(abs(p1.x - p2.x), abs(p1.y - p2.y)); }

}  // namespace tifa_libs::geo


#line 8 "src/code/geo2d/polygon.hpp"

namespace tifa_libs::geo {

template <class FP>
struct polygon {
  vec<point<FP>> vs;

  CEXP polygon() {}
  CEXPE polygon(u32 sz) : vs(sz) {}
  CEXP polygon(itl<point<FP>> vs_) : vs(vs_) {}
  CEXP polygon(spn<point<FP>> vs_) : vs(vs_.begin(), vs_.end()) {}

  friend std::istream &operator>>(std::istream &is, polygon &p) {
    for (auto &i : p.vs) is >> i;
    return is;
  }
  friend std::ostream &operator<<(std::ostream &os, polygon CR p) {
    if (p.vs.empty()) return os;
    for (auto it = p.vs.begin(); it != p.vs.end() - 1; ++it) os << *it << ' ';
    return os << p.vs.back();
  }
  CEXP u32 size() const { return (u32)vs.size(); }
  CEXP point<FP> &operator[](u32 x) { return vs[x]; }
  CEXP point<FP> CR operator[](u32 x) const { return vs[x]; }
  CEXP polygon &resort() { return std::ranges::sort(vs), *this; }
  CEXP polygon &reunique() { return vs = uniq(vs), *this; }
  CEXP auto prev(TPN vec<point<FP>>::const_iterator it) const { return --(it == vs.begin() ? it = vs.end() : it); }
  CEXP auto next(TPN vec<point<FP>>::const_iterator it) const { return ++it == vs.end() ? vs.begin() : it; }
  CEXP u32 prev(u32 idx) const { return idx == 0 ? size() - 1 : idx - 1; }
  CEXP u32 next(u32 idx) const { return idx + 1 == size() ? 0 : idx + 1; }
  CEXP FP circum() const {
    math::kahan<FP> ret = dist_PP(vs.back(), vs.front());
    flt_ (u32, i, 0, size() - 1) ret += dist_PP(vs[i], vs[i + 1]);
    return ret;
  }
  CEXP FP area2() const {
    if (size() < 3) return 0;
    math::kahan<FP> ret = vs.back() ^ vs.front();
    flt_ (u32, i, 0, size() - 1) ret += vs[i] ^ vs[i + 1];
    return ret;
  }
  CEXP FP area() const {
    static_assert(std::floating_point<FP>);
    return area2() * (FP).5;
  }
  CEXP bool is_convex() const {
    bool flag[2] = {false, false};
    const u32 n = size();
    if (n < 3) return true;
    for (u32 i = 0, j = next(i), k = next(j); i < n; ++i, j = next(j), k = next(k)) {
      if (auto sgn = sgn_cross(vs[i], vs[j], vs[k]); sgn) flag[(sgn + 1) / 2] = true;
      if (flag[0] && flag[1]) return false;
    }
    return true;
  }
  // @return nullopt if @p on board of polygon, otherwise winding number
  CEXP std::optional<u32> winding(point<FP> CR p) const {
    u32 cnt = 0;
    flt_ (u32, i, 0, size()) {
      auto &&u = vs[i], &&v = vs[next(i)];
      if (!sgn_cross(p, u, v) && !sgn_dot(p, u, v)) return {};
      if (is_zero(u.y - v.y)) continue;
      if (is_lt(u.y, v.y) && !is_pos(u, v, p)) continue;
      if (is_gt(u.y, v.y) && !is_neg(u, v, p)) continue;
      if (is_lt(u.y, p.y) && !is_lt(v.y, p.y)) ++cnt;
      if (!is_lt(u.y, p.y) && is_lt(v.y, p.y)) --cnt;
    }
    return cnt;
  }
};

}  // namespace tifa_libs::geo


#line 7 "src/code/geo2d/cvh.hpp"

namespace tifa_libs::geo {

template <class FP>
struct cvh : public polygon<FP> {
  CEXP cvh() {}
  CEXPE cvh(u32 sz) : polygon<FP>(sz) {}
  CEXP cvh(spn<point<FP>> vs_, bool inited = false, bool strict = true) : polygon<FP>(vs_) {
    if (!inited) strict ? init<true>() : init<false>();
  }

  friend std::istream &operator>>(std::istream &is, cvh &ch) {
    for (auto &i : ch.vs) is >> i;
    return is;
  }
  friend std::ostream &operator<<(std::ostream &os, cvh<FP> CR ch) {
    if (ch.vs.empty()) return os;
    for (auto it = ch.vs.begin(); it != ch.vs.end() - 1; ++it) os << *it << ' ';
    return os << ch.vs.back();
  }

  template <bool strict = true>
  CEXP cvh &init() {
    this->reunique();
    const u32 n = this->size();
    if (n <= 2) return *this;
    vec<point<FP>> cvh(n * 2);
    u32 m = 0;
    for (u32 i = 0; i < n; cvh[m++] = (*this)[i++])
      if CEXP (strict)
        while (m > 1 && sgn_cross(cvh[m - 2], cvh[m - 1], (*this)[i]) <= 0) --m;
      else
        while (m > 1 && sgn_cross(cvh[m - 2], cvh[m - 1], (*this)[i]) < 0) --m;
    for (u32 i = n - 2, t = m; ~i; cvh[m++] = (*this)[i--])
      if CEXP (strict)
        while (m > t && sgn_cross(cvh[m - 2], cvh[m - 1], (*this)[i]) <= 0) --m;
      else
        while (m > t && sgn_cross(cvh[m - 2], cvh[m - 1], (*this)[i]) < 0) --m;
    cvh.resize(m - 1);
    u32 p = 0;
    flt_ (u32, i, 1, m - 1)
      if (cvh[i] < cvh[p]) p = i;
    return std::ranges::rotate(cvh, cvh.begin() + p), this->vs = cvh, *this;
  }
  // @return true if @p in convex hull (include border)
  CEXP bool contains(point<FP> CR p) const {
    auto it = std::lower_bound(this->vs.begin() + 1, this->vs.end(), p, [&](point<FP> CR l, point<FP> CR r) { return is_pos(cross((*this)[0], l, r)); }) - 1;
    auto next_it = this->next(it);
    if (auto res = sgn_cross(p, *it, *next_it); res) return ~res;
    else return !res && !is_pos(dot(p, *it, *next_it));
  }
  template <bool get_index = false>
  CEXP auto diameter() const {
    const u32 n = this->size();
    if (n <= 1) return std::conditional_t<get_index, edge_t<FP>, FP>{};
    u32 is = 0, js = 0;
    flt_ (u32, k, 1, n)
      if ((*this)[js] < (*this)[k]) js = k;
    u32 i = is, j = js;
    std::conditional_t<get_index, edge_t<FP>, FP> ret;
    if CEXP (get_index) ret = {dist_PP((*this)[i], (*this)[j]), i, j};
    else ret = dist_PP((*this)[i], (*this)[j]);
    do {
      if CEXP ((++((((*this)[this->next(i)] - (*this)[i]) ^ ((*this)[this->next(j)] - (*this)[j])) >= 0 ? j : i)) %= n; get_index) ret = max(ret, edge_t<FP>{dist_PP((*this)[i], (*this)[j]), i, j});
      else ret = max(ret, dist_PP((*this)[i], (*this)[j]));
    } while (i != is || j != js);
    return ret;
  }
  CEXP cvh &do_minkowski_sum(cvh<FP> CR r) {
    const u32 n = this->size(), m = r.size();
    if (!m) return *this;
    if (!n) return *this = r;
    vec<point<FP>> res{(*this)[0] + r[0]};
    res.reserve(n + m);
    u32 i = 0, j = 0;
    while (i < n && j < m) {
      auto dl = (*this)[this->next(i)] - (*this)[i], dr = r[r.next(j)] - r[j];
      bool f = !is_neg(dl ^ dr);
      res.push_back(res.back() + (f ? dl : dr)), ++(f ? i : j);
    }
    while (i < n) res.push_back(res.back() + ((*this)[this->next(i)] - (*this)[i])), ++i;
    while (j < m) res.push_back(res.back() + (r[r.next(j)] - r[j])), ++j;
    return this->vs = res, *this;
  }
  CEXP cvh &do_ins_CVHhP(line<FP> CR l) {
    const u32 n = this->size();
    vec<point<FP>> cvc;
    flt_ (u32, i, 0, n) {
      point p1 = (*this)[i], p2 = (*this)[this->next(i)];
      int d1 = l.toleft(p1), d2 = l.toleft(p2);
      if (d1 >= 0) cvc.push_back(p1);
      if (d1 * d2 < 0) cvc.push_back(ins_LL({p1, p2}, l));
    }
    return this->vs = cvc, *this;
  }
  template <class F>
  requires requires(F f, point<FP> i, point<FP> ni, point<FP> j) { f(i, ni, j); }
  CEXP void rotcaliper(F &&f) const {
    auto CR p = this->vs;
    for (u32 i = 0, j = 1; i < p.size(); ++i) {
      auto ni = this->next(i);
      f(p[i], p[ni], p[j]);
      while (!is_lt(cross(p[ni], p[this->next(j)], p[i]) - cross(p[ni], p[j], p[i]))) f(p[i], p[ni], p[j = this->next(j)]);
    }
  }
};

}  // namespace tifa_libs::geo


#line 6 "src/code/geo2d/ins_hps.hpp"

namespace tifa_libs::geo {

template <class FP>
CEXP cvh<FP> ins_hPs(vec<line<FP>> vl) {
  auto check = [](line<FP> CR u, line<FP> CR v, line<FP> CR w) -> bool { return w.is_include_strict(ins_LL(u, v)); };
  if ((vl = uniq(vl)).size() < 3) return {};
  std::deque<line<FP>> q;
  for (auto it = vl.begin(); it != vl.end(); ++it) {
    if (it != vl.begin() && is_same_dir(*it, *(it - 1))) continue;
    while (q.size() > 1 && !check(*(q.rbegin() + 1), q.back(), *it)) q.pop_back();
    while (q.size() > 1 && !check(*(q.begin() + 1), q.front(), *it)) q.pop_front();
    q.push_back(*it);
  }
  while (q.size() > 2 && !check(*(q.rbegin() + 1), q.back(), q.front())) q.pop_back();
  while (q.size() > 2 && !check(*(q.begin() + 1), q.front(), q.back())) q.pop_front();
  vec<point<FP>> ret;
  flt_ (u32, i, 0, (u32)q.size()) ret.push_back(ins_LL(q[i], q[(i + 1) % q.size()]));
  return cvh<FP>{ret, true};
}

}  // namespace tifa_libs::geo


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