Tifa's CP Library

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

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#ifndef TIFALIBS_GEO2D_AREA_POC
#define TIFALIBS_GEO2D_AREA_POC

#include "circle.hpp"
#include "polygon.hpp"
#include "sarea_ct.hpp"

namespace tifa_libs::geo {

template <class FP>
constexpr FP area_PoC(polygon<FP> const &po, circle<FP> const &c) {
  FP ans{};
  u32 sz = (u32)po.vs.size();
  if (sz < 3) return ans;
  for (u32 i = 0; i < sz; ++i) ans += sarea_CT(c, po[i], po[po.next(i)]);
  return std::abs(ans);
}

}  // namespace tifa_libs::geo

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



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



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



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



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



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



#include <bits/stdc++.h>

template <class T>
constexpr T abs(T x) { return x < 0 ? -x : x; }

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;

template <class T>
using ptt = std::pair<T, T>;
template <class T>
using pt3 = std::tuple<T, T, T>;
template <class T>
using pt4 = std::tuple<T, T, T, T>;

template <class T, usz N>
using arr = std::array<T, N>;
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 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>
using vecpt = vec<ptt<T>>;
template <class T>
using vvecpt = vvec<ptt<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>>;

using strn = std::string;
using strnv = std::string_view;

using vecu = vec<u32>;
using vvecu = vvec<u32>;
using v3ecu = v3ec<u32>;
using vecu64 = vec<u64>;
using vecb = vec<bool>;
using vvecb = vvec<bool>;

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

using namespace std::literals;

constexpr i8 operator""_i8(unsigned long long x) { return (i8)x; }
constexpr i16 operator""_i16(unsigned long long x) { return (i16)x; }
constexpr i32 operator""_i32(unsigned long long x) { return (i32)x; }
constexpr i64 operator""_i64(unsigned long long x) { return (i64)x; }
constexpr isz operator""_iz(unsigned long long x) { return (isz)x; }

constexpr u8 operator""_u8(unsigned long long x) { return (u8)x; }
constexpr u16 operator""_u16(unsigned long long x) { return (u16)x; }
constexpr u32 operator""_u32(unsigned long long x) { return (u32)x; }
constexpr u64 operator""_u64(unsigned long long x) { return (u64)x; }
constexpr usz operator""_uz(unsigned long long x) { return (usz)x; }

inline const auto fn_0 = [](auto&&...) {};


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

namespace tifa_libs {

using namespace std::numbers;

// std::sqrt(std::numeric_limits<FP>::epsilon())
template <std::floating_point FP>
constexpr inline FP eps_v = FP(1e-8L);

}  // namespace tifa_libs


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



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

namespace tifa_libs {

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

template <class T>
concept container_c = iterable_c<T> && !std::derived_from<T, std::basic_string<typename T::value_type>>;

template <class T>
constexpr 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>
constexpr 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>
constexpr 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>
constexpr 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>
constexpr 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>
constexpr 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>
constexpr 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>
constexpr 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 = typename 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 = typename to_uint<T>::type;

}  // namespace tifa_libs


#line 6 "src/code/util/fp_comp.hpp"

namespace tifa_libs {

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

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

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

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

}  // namespace tifa_libs


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

namespace tifa_libs::geo {

template <class FP>
struct point {
  FP x, y;
  explicit constexpr 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 const &p) { return os << p.x << ' ' << p.y; }

  // s * r + t * (1 - r)
  friend constexpr point lerp(point const &s, point const &t, FP r) { return s * r + t * (1 - r); }
  friend constexpr point mid_point(point const &s, point const &t) { return lerp(s, t, .5); }

  constexpr point &operator+=(FP n) {
    this->x += n;
    this->y += n;
    return *this;
  }
  constexpr point &operator-=(FP n) {
    this->x -= n;
    this->y -= n;
    return *this;
  }
  constexpr point &operator*=(FP n) {
    this->x *= n;
    this->y *= n;
    return *this;
  }
  constexpr point &operator/=(FP n) {
    this->x /= n;
    this->y /= n;
    return *this;
  }
  friend constexpr point operator+(point x, FP n) { return x += n; }
  friend constexpr point operator+(FP n, point x) { return x += n; }
  friend constexpr point operator-(point x, FP n) { return x -= n; }
  friend constexpr point operator-(FP n, point x) { return x -= n; }
  friend constexpr point operator*(point x, FP n) { return x *= n; }
  friend constexpr point operator*(FP n, point x) { return x *= n; }
  friend constexpr point operator/(point x, FP n) { return x /= n; }
  friend constexpr point operator/(FP n, point x) { return x /= n; }

  constexpr point &operator+=(point const &p) {
    this->x += p.x;
    this->y += p.y;
    return *this;
  }
  constexpr point &operator-=(point const &p) {
    this->x -= p.x;
    this->y -= p.y;
    return *this;
  }
  constexpr point operator+(point const &p) const { return point(*this) += p; }
  constexpr point operator-(point const &p) const { return point(*this) -= p; }

  constexpr point operator-() const { return point{-x, -y}; }
  constexpr auto operator<=>(point const &p) const {
    if (auto c = comp(x, p.x); c) return c;
    return comp(y, p.y);
  }
  constexpr bool operator==(point const &p) const { return is_eq(x, p.x) && is_eq(y, p.y); }

  constexpr FP operator*(point const &p) const { return x * p.x + y * p.y; }
  constexpr FP operator^(point const &p) const { return x * p.y - y * p.x; }

  constexpr auto arg() const { return std::atan2(y, x); }
  constexpr FP norm2() const { return x * x + y * y; }
  constexpr FP norm() const { return std::hypot(x, y); }
  constexpr point &do_unit() { return *this /= norm(); }

  static constexpr u32 QUAD__[9] = {6, 7, 8, 5, 0, 1, 4, 3, 2};
  // 4 3 2
  // 5 0 1
  // 6 7 8
  constexpr u32 quad() const { return QUAD__[(sgn(y) + 1) * 3 + sgn(x) + 1]; }

  constexpr point &do_rot(FP theta) {
    FP x0 = x, y0 = y, ct = std::cos(theta), st = std::sin(theta);
    x = x0 * ct - y0 * st;
    y = x0 * st + y0 * ct;
    return *this;
  }
  constexpr point &do_rot90() {
    FP tmp = x;
    x = -y;
    y = tmp;
    return *this;
  }
  friend constexpr point rot90(point p) { return p.do_rot90(); }
  constexpr point &do_rot270() {
    FP tmp = y;
    y = -x;
    x = tmp;
    return *this;
  }
  friend constexpr point rot270(point p) { return p.do_rot270(); }
};

}  // namespace tifa_libs::geo


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

namespace tifa_libs::geo {

template <class FP>
struct circle {
  point<FP> o;
  FP r;
  constexpr circle() {}

  constexpr circle(point<FP> const &c, FP r) : o(c), r(r) {}
  constexpr circle(FP c_x, FP c_y, FP r_) : o(c_x, c_y), r(r_) {}

  friend std::istream &operator>>(std::istream &is, circle &c) { return is >> c.o >> c.r; }
  friend std::ostream &operator<<(std::ostream &os, circle const &c) { return os << c.o << ' ' << c.r; }
  friend constexpr bool operator==(circle const &l, circle const &r) { return l.o == r.o && l.r == r.r; }

  constexpr FP area(FP angle = pi_v<FP> * 2) const { return angle * r * r / 2; }
  constexpr FP crown_area(FP angle = pi_v<FP> * 2) const { return (angle - std::sin(angle)) * r * r / 2; }
  constexpr FP arc(FP angle = pi_v<FP> * 2) const { return angle * r; }
};

}  // namespace tifa_libs::geo


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



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



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

namespace tifa_libs {

template <iterable_c T = vec<int>>
constexpr T uniq(T v) {
  std::ranges::sort(v);
  auto [f, l] = std::ranges::unique(v);
  v.erase(f, l);
  return v;
}
template <iterable_c T = vec<int>>
constexpr std::pair<T, vec<u32>> gen_id(T const &v) {
  T _ = uniq(v);
  vec<u32> _1;

  for (u32 i = 0; i < v.size(); ++i) _1.push_back(u32(std::ranges::lower_bound(_, v[i]) - _.begin()));
  return {_, _1};
}

}  // namespace tifa_libs


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



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

namespace tifa_libs::geo {

// (a - o) ^ (b - o)
template <class FP>
constexpr FP cross(point<FP> const &o, point<FP> const &a, point<FP> const &b) { return (a.x - o.x) * (b.y - o.y) - (b.x - o.x) * (a.y - o.y); }
template <class FP>
constexpr FP cross_unit(point<FP> const &o, point<FP> const &a, point<FP> const &b) { return (a - o).do_unit() ^ (b - o).do_unit(); }
template <class FP>
constexpr int sgn_cross(point<FP> const &o, point<FP> const &a, point<FP> const &b) { return sgn(cross_unit(o, a, b)); }

}  // namespace tifa_libs::geo


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



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

namespace tifa_libs::geo {

// distance of two points (Euclidian)
template <class FP>
constexpr FP dist_PP(point<FP> const &p1, point<FP> const &p2) { return (p1 - p2).norm(); }

}  // namespace tifa_libs::geo


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

namespace tifa_libs::geo {

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

  constexpr polygon() {}
  explicit constexpr polygon(u32 sz) : vs(sz) {}
  explicit constexpr polygon(vec<point<FP>> const &vs_) : vs(vs_) {}

  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 const &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();
  }
  constexpr point<FP> &operator[](u32 x) { return vs[x]; }
  constexpr point<FP> const &operator[](u32 x) const { return vs[x]; }

  constexpr polygon &resort() {
    std::ranges::sort(vs);
    return *this;
  }
  constexpr polygon &reunique() {
    vs = uniq(vs);
    return *this;
  }

  constexpr auto prev(typename vec<point<FP>>::const_iterator it) const { return --(it == vs.begin() ? it = vs.end() : it); }
  constexpr auto next(typename vec<point<FP>>::const_iterator it) const { return ++it == vs.end() ? vs.begin() : it; }
  constexpr u32 prev(u32 idx) const { return idx == 0 ? (u32)vs.size() - 1 : idx - 1; }
  constexpr u32 next(u32 idx) const { return idx + 1 == (u32)vs.size() ? 0 : idx + 1; }

  constexpr FP circum() const {
    FP ret = dist_PP(vs.back(), vs.front());
    for (u32 i = 0; i < (u32)vs.size() - 1; ++i) ret += dist_PP(vs[i], vs[i + 1]);
    return ret;
  }
  constexpr FP area() const {
    if (vs.size() < 3) return 0;
    FP ret = vs.back() ^ vs.front();
    for (u32 i = 0; i < (u32)vs.size() - 1; ++i) ret += vs[i] ^ vs[i + 1];
    return ret / 2;
  }
  constexpr bool is_convex() const {
    bool flag[2] = {false, false};
    u32 n = (u32)vs.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)) {
      auto sgn = sgn_cross(vs[i], vs[j], vs[k]);
      if (sgn) flag[(sgn + 1) / 2] = true;
      if (flag[0] && flag[1]) return false;
    }
    return true;
  }
};

}  // namespace tifa_libs::geo


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



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



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

namespace tifa_libs::geo {

// clamp angle of two points
template <class FP>
constexpr FP ang_PP(point<FP> const &p1, point<FP> const &p2) { return std::atan2(p1 ^ p2, p1 * p2); }

}  // namespace tifa_libs::geo


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



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

namespace tifa_libs::geo {

// (a - o) * (b - o)
template <class FP>
constexpr FP dot(point<FP> const &o, point<FP> const &a, point<FP> const &b) { return (a.x - o.x) * (b.x - o.x) + (a.y - o.y) * (b.y - o.y); }
template <class FP>
constexpr int sgn_dot(point<FP> const &o, point<FP> const &a, point<FP> const &b) { return sgn(dot(o, a, b)); }

}  // namespace tifa_libs::geo


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



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



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

namespace tifa_libs::geo {

template <class FP>
struct line {
  point<FP> l, r;
  constexpr line() {}
  constexpr line(point<FP> const &s, point<FP> const &t) : l(s), r(t) {}
  constexpr line(point<FP> const &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
  constexpr 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};
  }
  constexpr 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 const &l) { return os << l.l << ' ' << l.r; }

  constexpr point<FP> direction() const { return r - l; }
  constexpr bool is_parallel(line const &r) const { return is_zero(direction() ^ r.direction()); }
  friend constexpr bool is_parallel(line const &l, line const &r) { return l.is_parallel(r); }
  constexpr bool is_same_dir(line const &r) const { return is_parallel(r) && is_pos(direction() * r.direction()); }
  friend constexpr bool is_same_dir(line const &l, line const &r) { return l.is_same_dir(r); }

  friend constexpr bool operator==(line const &l, line const &r) { return l.l == r.l && l.r == r.r; }
  friend constexpr auto operator<=>(line const &l, line const &r) {
    if (l == r) return 0;
    if (l.is_same_dir(r)) return r.is_include_strict(l.l) ? -1 : 1;
    auto vl = l.direction(), vr = r.direction();
    if (vl.quad() != vr.quad()) return (i32)vl.quad() - (i32)vr.quad();
    return -sgn(vl ^ vr);
  }

  // half plane
  constexpr bool is_include_strict(point<FP> const &p) const { return is_pos(cross(l, r, p)); }
  // half plane
  constexpr bool is_include(point<FP> const &p) const { return !is_neg(cross(l, r, p)); }

  // translate @dist along the direction of half plane
  constexpr line &do_push(FP dist) {
    point delta = direction().do_rot90().do_unit() * dist;
    l += delta;
    r += delta;
    return *this;
  }
};

}  // namespace tifa_libs::geo


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

namespace tifa_libs::geo {

// intersection point of circle and line
// maybe duplicate
template <class FP>
constexpr std::optional<ptt<point<FP>>> ins_CL(circle<FP> const &c, line<FP> const &l) {
  if (is_gt(std::abs((c.o - l.l) ^ (l.r - l.l) / dist_PP(l.l, l.r)), c.r)) return {};
  FP x = (l.l - c.o) * (l.r - l.l), y = l.direction().norm2(), d = std::max(x * x - y * ((l.l - c.o).norm2() - c.r * c.r), FP{});
  point m = l.l - l.direction() * (x / y), dr = l.direction() * (std::sqrt(d) / y);
  return ptt<point<FP>>{m - dr, m + dr};
}

}  // namespace tifa_libs::geo


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

namespace tifa_libs::geo {

// area of intersection of circle and triangle
//! WITH DIRECTION, sgn is cross(c.o, p1, p2)
//! the center ot circle is also a endpoint of triangle
template <class FP>
constexpr FP sarea_CT(circle<FP> const &c, point<FP> const &p1, point<FP> const &p2) {
  if (is_zero(cross(c.o, p1, p2))) return FP{};
  auto is = ins_CL(c, {p1, p2});
  if (!is.has_value()) return c.area(ang_PP(p1 - c.o, p2 - c.o));
  bool b1 = is_gt(dist_PP(p1, c.o), c.r), b2 = is_gt(dist_PP(p2, c.o), c.r);
  if (b1 && b2) {
    FP res = c.area(ang_PP(p1 - c.o, p2 - c.o));
    if (!is_pos(dot(is.value().first, p1, p2))) res -= c.crown_area(ang_PP(is.value().first - c.o, is.value().second - c.o));
    return res;
  }
  if (b1) return c.area(ang_PP(p1 - c.o, is.value().first - c.o)) + cross(c.o, is.value().first, p2) / 2;
  if (b2) return c.area(ang_PP(is.value().second - c.o, p2 - c.o)) + cross(c.o, p1, is.value().second) / 2;
  return cross(c.o, p1, p2) / 2;
}

}  // namespace tifa_libs::geo


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

namespace tifa_libs::geo {

template <class FP>
constexpr FP area_PoC(polygon<FP> const &po, circle<FP> const &c) {
  FP ans{};
  u32 sz = (u32)po.vs.size();
  if (sz < 3) return ans;
  for (u32 i = 0; i < sz; ++i) ans += sarea_CT(c, po[i], po[po.next(i)]);
  return std::abs(ans);
}

}  // namespace tifa_libs::geo


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