Tifa's CP Library

:warning: min_cover_c (src/code/geo2d/min_cover_c.hpp)

Depends on

Code

#ifndef TIFALIBS_GEO2D_MIN_COVER_C
#define TIFALIBS_GEO2D_MIN_COVER_C

#include "make_c_ppp.hpp"
#include "rel_cp.hpp"

namespace tifa_libs::geo {

// min coverage circle of a set of points
//! accuracy maybe reduced without shuffling `vp` first
template <class FP>
constexpr circle<FP> min_cover_C(vec<point<FP>> const &vp) {
  circle ret{vp.front(), 0};
  u32 sz = (u32)vp.size();
  for (u32 i = 1; i < sz; ++i) {
    if (relation_CP(ret, vp[i]) != RELCP::outside_cp) continue;
    ret = circle{vp[i], 0};
    for (u32 j = 0; j < i; ++j) {
      if (relation_CP(ret, vp[j]) != RELCP::outside_cp) continue;
      ret = circle{mid_point(vp[i], vp[j]), dist_PP(vp[i], vp[j]) / 2};
      for (u32 k = 0; k < j; ++k) {
        if (relation_CP(ret, vp[k]) != RELCP::outside_cp) continue;
        ret = make_C_PPP(vp[i], vp[j], vp[k]);
      }
    }
  }
  return ret;
}

}  // namespace tifa_libs::geo

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



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



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



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



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

namespace tifa_libs::geo {

// judge if two lines are intersected or not
template <class FP>
constexpr bool is_ins_LL(line<FP> const &l1, line<FP> const &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>
constexpr point<FP> ins_LL(line<FP> const &l1, line<FP> const &l2) {
  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>
constexpr point<FP> ins_LL(line<FP> const &l, FP a, FP b, FP c) {
  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/triangle.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/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 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 8 "src/code/geo2d/triangle.hpp"

namespace tifa_libs::geo {

template <class FP>
struct triangle {
  point<FP> A, B, C;

  constexpr triangle() {}
  constexpr triangle(point<FP> const &a, point<FP> const &b, point<FP> const &c) : A(a), B(b), C(c) {}
  constexpr triangle(FP a_x, FP a_y, FP b_x, FP b_y, FP c_x, FP c_y) : A(a_x, a_y), B(b_x, b_y), C(c_x, c_y) {}

  friend std::istream &operator>>(std::istream &is, triangle &t) { return is >> t.A >> t.B >> t.C; }
  friend std::ostream &operator<<(std::ostream &os, triangle const &t) { return os << t.A << ' ' << t.B << ' ' << t.C; }

  friend constexpr bool operator==(triangle const &l, triangle const &r) { return l.A == r.A && l.B == r.B && l.C == r.C; }

  // (a, b, c)
  constexpr pt3<FP> edges() const { return {dist_PP(B, C), dist_PP(C, A), dist_PP(A, B)}; }
  // (A, B, C)
  constexpr pt3<FP> angles() const { return {std::abs(ang_PP(C - A, B - A)), std::abs(ang_PP(A - B, C - B)), std::abs(ang_PP(A - C, B - C))}; }

  constexpr point<FP> trilinears(FP x, FP y, FP z) const {
    auto [a, b, c] = edges();
    x *= a, y *= b, z *= c;
    return (A * x + B * y + C * z) / (x + y + z);
  }
  constexpr point<FP> barycentrics(FP u, FP v, FP w) const { return (A * u + B * v + C * w) / (u + v + w); }
  constexpr FP area() const { return std::abs(cross(A, B, C)) / 2; }
  constexpr bool is_acute() const { return is_pos(dot(A, B, C)) && is_pos(dot(B, C, A)) && is_pos(dot(C, A, B)); }
  constexpr bool is_right() const { return is_zero(dot(A, B, C)) || is_zero(dot(B, C, A)) || is_zero(dot(C, A, B)); }
  constexpr bool is_obtuse() const { return is_neg(dot(A, B, C)) || is_neg(dot(B, C, A)) || is_neg(dot(C, A, B)); }
};

}  // namespace tifa_libs::geo


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

namespace tifa_libs::geo {

// radius of circumscribed circle
template <class FP>
constexpr FP radius_O(triangle<FP> const &t) { return dist_PP(t.B, t.C) / std::sin(std::abs(ang_PP(t.B - t.A, t.C - t.A))) / 2; }

// circumcenter (X3)
template <class FP>
constexpr point<FP> center_O(triangle<FP> const &t) {
  // auto [A, B, C] = t.angles();
  // return t.trilinears(std::cos(A), std::cos(B), std::cos(C));
  point<FP> p1 = mid_point(t.B, t.C), p2 = mid_point(t.C, t.A);
  return ins_LL<FP>({p1, p1 + (t.B - t.C).do_rot90()}, {p2, p2 + (t.C - t.A).do_rot90()});
}

}  // namespace tifa_libs::geo


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

namespace tifa_libs::geo {

// make circle by 3 point passed through
template <class FP>
constexpr circle<FP> make_C_PPP(point<FP> const &p1, point<FP> const &p2, point<FP> const &p3) {
  point o = center_O(triangle{p1, p2, p3});
  return {o, dist_PP(o, p1)};
}

}  // namespace tifa_libs::geo


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



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

namespace tifa_libs::geo {

// relation between circle and point
enum RELCP {
  outside_cp,
  onborder_cp,
  inside_cp,
};

template <class FP>
constexpr RELCP relation_CP(circle<FP> const &c, point<FP> const &p) {
  FP d = dist_PP(c.o, p);
  if (is_lt(d, c.r)) return inside_cp;
  if (is_eq(d, c.r)) return onborder_cp;
  return outside_cp;
}

}  // namespace tifa_libs::geo


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

namespace tifa_libs::geo {

// min coverage circle of a set of points
//! accuracy maybe reduced without shuffling `vp` first
template <class FP>
constexpr circle<FP> min_cover_C(vec<point<FP>> const &vp) {
  circle ret{vp.front(), 0};
  u32 sz = (u32)vp.size();
  for (u32 i = 1; i < sz; ++i) {
    if (relation_CP(ret, vp[i]) != RELCP::outside_cp) continue;
    ret = circle{vp[i], 0};
    for (u32 j = 0; j < i; ++j) {
      if (relation_CP(ret, vp[j]) != RELCP::outside_cp) continue;
      ret = circle{mid_point(vp[i], vp[j]), dist_PP(vp[i], vp[j]) / 2};
      for (u32 k = 0; k < j; ++k) {
        if (relation_CP(ret, vp[k]) != RELCP::outside_cp) continue;
        ret = make_C_PPP(vp[i], vp[j], vp[k]);
      }
    }
  }
  return ret;
}

}  // namespace tifa_libs::geo


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