#line 1 "verify/geometry/circle_coverage_areas.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#line 1 "geometry/circle_coverage_areas.hpp"
#line 1 "geometry/circle.hpp"
#include <algorithm>
#include <array>
#include <cassert>
#include <cmath>
#include <cstddef>
#include <numbers>
#include <optional>
#include <type_traits>
#include <vector>
#line 1 "geometry/linear.hpp"
#line 7 "geometry/linear.hpp"
#line 1 "geometry/point.hpp"
#line 5 "geometry/point.hpp"
#include <concepts>
#line 8 "geometry/point.hpp"
#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 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 15 "geometry/circle.hpp"
namespace m1une {
namespace geometry {
template <Coordinate T>
struct Circle {
Point<T> center;
T radius;
bool filled = true;
};
enum class PointInCircle {
Outside = 0,
Boundary = 1,
Inside = 2,
};
enum class CircleRelation {
Separate,
ExternallyTangent,
Intersecting,
InternallyTangent,
Contained,
Coincident,
};
enum class AngularCoverageKind {
Empty,
Point,
Arc,
Full,
};
struct AngularCoverage {
AngularCoverageKind kind = AngularCoverageKind::Empty;
long double begin = 0.0L;
long double end = 0.0L;
};
struct CircleContact {
Point<long double> point;
long double first_argument = 0.0L;
long double second_argument = 0.0L;
};
enum class CircleContactKind {
Empty,
Point,
TwoPoints,
Coincident,
};
struct CircleCircleIntersection {
CircleRelation relation = CircleRelation::Separate;
CircleContactKind contact_kind = CircleContactKind::Empty;
std::array<CircleContact, 2> contacts;
AngularCoverage first_inside_second;
AngularCoverage second_inside_first;
constexpr int contact_count() const noexcept {
if (contact_kind == CircleContactKind::Point) return 1;
if (contact_kind == CircleContactKind::TwoPoints) return 2;
return 0;
}
};
struct CircleLinearContact {
Point<long double> point;
long double circle_argument = 0.0L;
long double linear_parameter = 0.0L;
};
struct CircleLinearIntersection {
int contact_count = 0;
std::array<CircleLinearContact, 2> contacts;
};
namespace circle_detail {
inline int compare(long double first, long double second, long double eps) {
if (first < second - eps) return -1;
if (first > second + eps) return 1;
return 0;
}
inline bool close(
const Point<long double>& first,
const Point<long double>& second,
long double eps
) {
return geometry::distance(first, second) <= eps;
}
inline void push_unique(
std::vector<Point<long double>>& points,
const Point<long double>& point,
long double eps
) {
for (const Point<long double>& existing : points) {
if (close(existing, point, eps)) return;
}
points.push_back(point);
}
inline bool same_line(
const Line<long double>& first,
const Line<long double>& second,
long double eps
) {
Point<long double> first_direction = first.b - first.a;
Point<long double> second_direction = second.b - second.a;
if (std::fabs(cross(first_direction, second_direction)) > eps) {
return false;
}
return std::fabs(cross(first_direction, second.a - first.a)) <= eps;
}
inline Line<long double> tangent_line(
const Point<long double>& contact,
Point<long double> normal,
long double eps
) {
Point<long double> direction(-normal.y, normal.x);
if (
direction.x < -eps ||
(std::fabs(direction.x) <= eps && direction.y < 0)
) {
direction = -direction;
}
return Line<long double>{contact, contact + direction};
}
inline long double circular_segment_angle_term(
long double angle,
long double sine,
long double cosine
) {
if (angle >= 0.01L) return angle - sine * cosine;
const long double squared = angle * angle;
return angle * squared * (
2.0L / 3.0L +
squared * (
-2.0L / 15.0L +
squared * (4.0L / 315.0L - squared * 2.0L / 2835.0L)
)
);
}
inline long double segment_disk_signed_area(
const Point<long double>& first,
const Point<long double>& second,
long double radius,
long double eps
) {
const Point<long double> direction = second - first;
const long double quadratic = dot(direction, direction);
if (quadratic == 0.0L || radius == 0.0L) return 0.0L;
std::vector<long double> cuts = {0.0L, 1.0L};
const long double linear = 2.0L * dot(first, direction);
const long double constant = dot(first, first) - radius * radius;
const long double discriminant =
linear * linear - 4.0L * quadratic * constant;
const long double tolerance = eps * std::max({
1.0L,
std::fabs(linear * linear),
std::fabs(4.0L * quadratic * constant)
});
if (discriminant >= -tolerance) {
const long double root = std::sqrt(std::max(0.0L, discriminant));
const long double first_ratio =
(-linear - root) / (2.0L * quadratic);
const long double second_ratio =
(-linear + root) / (2.0L * quadratic);
if (eps < first_ratio && first_ratio < 1.0L - eps) {
cuts.push_back(first_ratio);
}
if (eps < second_ratio && second_ratio < 1.0L - eps) {
cuts.push_back(second_ratio);
}
}
std::sort(cuts.begin(), cuts.end());
cuts.erase(
std::unique(
cuts.begin(),
cuts.end(),
[eps](long double left, long double right) {
return std::fabs(left - right) <= eps;
}
),
cuts.end()
);
long double result = 0.0L;
for (std::size_t index = 1; index < cuts.size(); ++index) {
const long double left = cuts[index - 1];
const long double right = cuts[index];
const Point<long double> a = first + direction * left;
const Point<long double> b = first + direction * right;
const Point<long double> middle =
first + direction * ((left + right) / 2.0L);
if (norm(middle) <= radius + eps) {
result += cross(a, b) / 2.0L;
} else {
result +=
radius * radius * std::atan2(cross(a, b), dot(a, b)) /
2.0L;
}
}
return result;
}
} // namespace circle_detail
template <Coordinate T>
constexpr Point<long double> centroid(const Circle<T>& circle) {
assert(circle.radius >= 0);
return Point<long double>(circle.center);
}
template <Coordinate T>
constexpr long double circle_circumference(const Circle<T>& circle) {
assert(circle.radius >= 0);
return
2.0L * std::numbers::pi_v<long double> *
static_cast<long double>(circle.radius);
}
template <Coordinate T>
constexpr long double circle_area(const Circle<T>& circle) {
assert(circle.radius >= 0);
const long double radius = static_cast<long double>(circle.radius);
return std::numbers::pi_v<long double> * radius * radius;
}
inline long double normalize_circle_argument(long double argument) {
const long double full = 2.0L * std::numbers::pi_v<long double>;
argument = std::fmod(argument, full);
if (argument < 0.0L) argument += full;
if (argument == full) argument = 0.0L;
return argument;
}
template <Coordinate T>
Point<long double> circle_point_at(
const Circle<T>& circle,
long double argument
) {
assert(circle.radius >= 0);
const long double radius = static_cast<long double>(circle.radius);
return Point<long double>(circle.center) + Point<long double>(
radius * std::cos(argument),
radius * std::sin(argument)
);
}
inline long double angular_measure(const AngularCoverage& coverage) {
if (
coverage.kind == AngularCoverageKind::Empty ||
coverage.kind == AngularCoverageKind::Point
) {
return 0.0L;
}
if (coverage.kind == AngularCoverageKind::Full) {
return 2.0L * std::numbers::pi_v<long double>;
}
assert(coverage.kind == AngularCoverageKind::Arc);
assert(coverage.begin <= coverage.end);
return coverage.end - coverage.begin;
}
template <Coordinate T>
long double circle_arc_length(
const Circle<T>& circle,
const AngularCoverage& coverage
) {
assert(circle.radius >= 0);
return static_cast<long double>(circle.radius) * angular_measure(coverage);
}
template <Coordinate C, Coordinate P>
PointInCircle point_in_circle(
const Circle<C>& circle,
const Point<P>& point,
long double eps = 1e-12L
) {
assert(circle.radius >= 0);
assert(eps >= 0.0L);
if constexpr (ExactCoordinate<C> && ExactCoordinate<P>) {
using W = std::common_type_t<wide_type<C>, wide_type<P>>;
const W dx = W(point.x) - W(circle.center.x);
const W dy = W(point.y) - W(circle.center.y);
const W radius = W(circle.radius);
const W squared_distance = dx * dx + dy * dy;
const W squared_radius = radius * radius;
if (squared_distance < squared_radius) return PointInCircle::Inside;
if (squared_distance > squared_radius) return PointInCircle::Outside;
return PointInCircle::Boundary;
} else {
const int relation = circle_detail::compare(
geometry::distance(
Point<long double>(circle.center),
Point<long double>(point)
),
static_cast<long double>(circle.radius),
eps
);
if (relation < 0) return PointInCircle::Inside;
if (relation > 0) return PointInCircle::Outside;
return PointInCircle::Boundary;
}
}
template <Coordinate C, Coordinate P>
bool contains(
const Circle<C>& circle,
const Point<P>& point,
long double eps = 1e-12L
) {
const PointInCircle relation = point_in_circle(circle, point, eps);
return circle.filled
? relation != PointInCircle::Outside
: relation == PointInCircle::Boundary;
}
template <Coordinate C, Coordinate P>
bool on_circle(
const Circle<C>& circle,
const Point<P>& point,
long double eps = 1e-12L
) {
assert(circle.radius >= 0);
assert(eps >= 0.0L);
if constexpr (ExactCoordinate<C> && ExactCoordinate<P>) {
using W = std::common_type_t<wide_type<C>, wide_type<P>>;
const W dx = W(point.x) - W(circle.center.x);
const W dy = W(point.y) - W(circle.center.y);
const W radius = W(circle.radius);
return dx * dx + dy * dy == radius * radius;
} else {
return circle_detail::compare(
geometry::distance(
Point<long double>(circle.center),
Point<long double>(point)
),
static_cast<long double>(circle.radius),
eps
) == 0;
}
}
template <Coordinate C, Coordinate P>
long double circle_argument(
const Circle<C>& circle,
const Point<P>& point
) {
assert(circle.radius >= 0);
return normalize_circle_argument(std::atan2(
static_cast<long double>(point.y) -
static_cast<long double>(circle.center.y),
static_cast<long double>(point.x) -
static_cast<long double>(circle.center.x)
));
}
template <Coordinate C, Coordinate P>
bool intersects(
const Circle<C>& circle,
const Point<P>& point,
long double eps = 1e-12L
) {
return contains(circle, point, eps);
}
template <Coordinate P, Coordinate C>
bool intersects(
const Point<P>& point,
const Circle<C>& circle,
long double eps = 1e-12L
) {
return intersects(circle, point, eps);
}
template <Coordinate A, Coordinate B>
Circle<long double> circle_from_diameter(
const Point<A>& first,
const Point<B>& second
) {
Point<long double> a(first);
Point<long double> b(second);
Point<long double> center = (a + b) / 2.0L;
return Circle<long double>{center, geometry::distance(a, b) / 2.0L};
}
template <Coordinate T>
std::optional<Circle<long double>> incircle(
const Point<T>& first,
const Point<T>& second,
const Point<T>& third,
long double eps = 1e-12L
) {
assert(eps >= 0.0L);
if (orientation(first, second, third, eps) == 0) return std::nullopt;
long double opposite_first = geometry::distance(second, third);
long double opposite_second = geometry::distance(third, first);
long double opposite_third = geometry::distance(first, second);
long double perimeter =
opposite_first + opposite_second + opposite_third;
Point<long double> center =
(Point<long double>(first) * opposite_first +
Point<long double>(second) * opposite_second +
Point<long double>(third) * opposite_third) /
perimeter;
long double doubled_area = std::fabs(
static_cast<long double>(cross(first, second, third))
);
return Circle<long double>{center, doubled_area / perimeter};
}
template <Coordinate T>
std::optional<Circle<long double>> circumcircle(
const Point<T>& first,
const Point<T>& second,
const Point<T>& third,
long double eps = 1e-12L
) {
assert(eps >= 0.0L);
if (orientation(first, second, third, eps) == 0) return std::nullopt;
Point<long double> origin(first);
Point<long double> u = Point<long double>(second) - origin;
Point<long double> v = Point<long double>(third) - origin;
long double denominator = 2.0L * cross(u, v);
long double u_norm = norm2(u);
long double v_norm = norm2(v);
Point<long double> offset(
(u_norm * v.y - v_norm * u.y) / denominator,
(u.x * v_norm - v.x * u_norm) / denominator
);
Point<long double> center = origin + offset;
return Circle<long double>{center, norm(offset)};
}
template <Coordinate A, Coordinate B>
CircleRelation circle_relation(
const Circle<A>& first,
const Circle<B>& second,
long double eps = 1e-12L
) {
assert(first.radius >= 0);
assert(second.radius >= 0);
assert(eps >= 0.0L);
if constexpr (ExactCoordinate<A> && ExactCoordinate<B>) {
using W = std::common_type_t<wide_type<A>, wide_type<B>>;
W dx = W(second.center.x) - W(first.center.x);
W dy = W(second.center.y) - W(first.center.y);
W squared_distance = dx * dx + dy * dy;
W first_radius = W(first.radius);
W second_radius = W(second.radius);
W sum = first_radius + second_radius;
W difference = first_radius - second_radius;
if (difference < 0) difference = -difference;
if (squared_distance == 0 && difference == 0) {
return CircleRelation::Coincident;
}
if (squared_distance > sum * sum) return CircleRelation::Separate;
if (squared_distance == sum * sum) {
return CircleRelation::ExternallyTangent;
}
if (squared_distance < difference * difference) {
return CircleRelation::Contained;
}
if (squared_distance == difference * difference) {
return CircleRelation::InternallyTangent;
}
return CircleRelation::Intersecting;
} else {
long double center_distance = geometry::distance(
Point<long double>(first.center),
Point<long double>(second.center)
);
long double first_radius = static_cast<long double>(first.radius);
long double second_radius = static_cast<long double>(second.radius);
long double sum = first_radius + second_radius;
long double difference = std::fabs(first_radius - second_radius);
if (
center_distance <= eps &&
difference <= eps
) {
return CircleRelation::Coincident;
}
int outer = circle_detail::compare(center_distance, sum, eps);
if (outer > 0) return CircleRelation::Separate;
if (outer == 0) return CircleRelation::ExternallyTangent;
int inner = circle_detail::compare(center_distance, difference, eps);
if (inner < 0) return CircleRelation::Contained;
if (inner == 0) return CircleRelation::InternallyTangent;
return CircleRelation::Intersecting;
}
}
template <Coordinate C, Coordinate L>
CircleLinearIntersection circle_boundary_intersection(
const Circle<C>& circle,
const Line<L>& line,
long double eps = 1e-12L
) {
assert(circle.radius >= 0);
assert(line.a != line.b);
assert(eps >= 0.0L);
const Point<long double> center(circle.center);
const Point<long double> origin(line.a);
const Point<long double> direction =
Point<long double>(line.b) - origin;
const long double squared_length = dot(direction, direction);
const long double length = std::sqrt(squared_length);
const long double foot_parameter =
dot(center - origin, direction) / squared_length;
const Point<long double> foot =
origin + direction * foot_parameter;
const long double distance_to_line = geometry::distance(center, foot);
const long double radius = static_cast<long double>(circle.radius);
const int relation =
circle_detail::compare(distance_to_line, radius, eps);
CircleLinearIntersection result;
if (relation > 0) return result;
if (relation == 0) {
result.contact_count = 1;
result.contacts[0] = CircleLinearContact{
foot,
circle_argument(circle, foot),
foot_parameter
};
return result;
}
const long double offset = std::sqrt(std::max(
0.0L,
radius * radius - distance_to_line * distance_to_line
));
const long double parameter_offset = offset / length;
result.contact_count = 2;
for (int index = 0; index < 2; ++index) {
const long double parameter = foot_parameter +
(index == 0 ? -parameter_offset : parameter_offset);
const Point<long double> point = origin + direction * parameter;
result.contacts[index] = CircleLinearContact{
point,
circle_argument(circle, point),
parameter
};
}
return result;
}
template <Coordinate L, Coordinate C>
CircleLinearIntersection circle_boundary_intersection(
const Line<L>& line,
const Circle<C>& circle,
long double eps = 1e-12L
) {
return circle_boundary_intersection(circle, line, eps);
}
template <Coordinate C, Coordinate R>
CircleLinearIntersection circle_boundary_intersection(
const Circle<C>& circle,
const Ray<R>& ray,
long double eps = 1e-12L
) {
assert(ray.origin != ray.through);
const Line<R> line{ray.origin, ray.through};
const CircleLinearIntersection line_result =
circle_boundary_intersection(circle, line, eps);
CircleLinearIntersection result;
for (int index = 0; index < line_result.contact_count; ++index) {
CircleLinearContact contact = line_result.contacts[index];
if (contact.linear_parameter < -eps) continue;
if (std::fabs(contact.linear_parameter) <= eps) {
contact.linear_parameter = 0.0L;
contact.point = Point<long double>(ray.origin);
contact.circle_argument = circle_argument(circle, contact.point);
}
result.contacts[result.contact_count++] = contact;
}
return result;
}
template <Coordinate R, Coordinate C>
CircleLinearIntersection circle_boundary_intersection(
const Ray<R>& ray,
const Circle<C>& circle,
long double eps = 1e-12L
) {
return circle_boundary_intersection(circle, ray, eps);
}
template <Coordinate C, Coordinate S>
CircleLinearIntersection circle_boundary_intersection(
const Circle<C>& circle,
const Segment<S>& segment,
long double eps = 1e-12L
) {
assert(circle.radius >= 0);
assert(eps >= 0.0L);
CircleLinearIntersection result;
if (segment.a == segment.b) {
if (on_circle(circle, segment.a, eps)) {
const Point<long double> point(segment.a);
result.contact_count = 1;
result.contacts[0] = CircleLinearContact{
point,
circle_argument(circle, point),
0.0L
};
}
return result;
}
const Line<S> line{segment.a, segment.b};
const CircleLinearIntersection line_result =
circle_boundary_intersection(circle, line, eps);
for (int index = 0; index < line_result.contact_count; ++index) {
CircleLinearContact contact = line_result.contacts[index];
if (
contact.linear_parameter < -eps ||
contact.linear_parameter > 1.0L + eps
) {
continue;
}
if (std::fabs(contact.linear_parameter) <= eps) {
contact.linear_parameter = 0.0L;
contact.point = Point<long double>(segment.a);
} else if (std::fabs(contact.linear_parameter - 1.0L) <= eps) {
contact.linear_parameter = 1.0L;
contact.point = Point<long double>(segment.b);
}
contact.circle_argument = circle_argument(circle, contact.point);
result.contacts[result.contact_count++] = contact;
}
return result;
}
template <Coordinate S, Coordinate C>
CircleLinearIntersection circle_boundary_intersection(
const Segment<S>& segment,
const Circle<C>& circle,
long double eps = 1e-12L
) {
return circle_boundary_intersection(circle, segment, eps);
}
template <Coordinate A, Coordinate B>
CircleCircleIntersection circle_boundary_intersection(
const Circle<A>& first,
const Circle<B>& second,
long double eps = 1e-12L
) {
assert(first.radius >= 0);
assert(second.radius >= 0);
assert(eps >= 0.0L);
const long double full = 2.0L * std::numbers::pi_v<long double>;
const long double first_radius = static_cast<long double>(first.radius);
const long double second_radius = static_cast<long double>(second.radius);
CircleCircleIntersection result;
result.relation = circle_relation(first, second, eps);
auto point_coverage = [](long double argument) {
return AngularCoverage{
AngularCoverageKind::Point,
argument,
argument
};
};
auto full_coverage = [full]() {
return AngularCoverage{AngularCoverageKind::Full, 0.0L, full};
};
if (result.relation == CircleRelation::Coincident) {
if (first_radius == 0.0L) {
const Point<long double> point(first.center);
result.contact_kind = CircleContactKind::Point;
result.contacts[0] = CircleContact{point, 0.0L, 0.0L};
result.first_inside_second = point_coverage(0.0L);
result.second_inside_first = point_coverage(0.0L);
} else {
result.contact_kind = CircleContactKind::Coincident;
result.first_inside_second = full_coverage();
result.second_inside_first = full_coverage();
}
return result;
}
if (result.relation == CircleRelation::Separate) return result;
if (result.relation == CircleRelation::Contained) {
if (first_radius < second_radius) {
result.first_inside_second = first_radius == 0.0L
? point_coverage(0.0L)
: full_coverage();
} else {
result.second_inside_first = second_radius == 0.0L
? point_coverage(0.0L)
: full_coverage();
}
return result;
}
const Point<long double> first_center(first.center);
const Point<long double> second_center(second.center);
const Point<long double> center_direction = second_center - first_center;
const long double center_distance = norm(center_direction);
const Point<long double> unit = center_direction / center_distance;
const long double along =
(first_radius * first_radius - second_radius * second_radius +
center_distance * center_distance) /
(2.0L * center_distance);
const Point<long double> base = first_center + unit * along;
if (
result.relation == CircleRelation::ExternallyTangent ||
result.relation == CircleRelation::InternallyTangent
) {
const long double first_argument =
circle_argument(first, base);
const long double second_argument =
circle_argument(second, base);
result.contact_kind = CircleContactKind::Point;
result.contacts[0] = CircleContact{
base,
first_argument,
second_argument
};
result.first_inside_second = point_coverage(first_argument);
result.second_inside_first = point_coverage(second_argument);
if (result.relation == CircleRelation::InternallyTangent) {
if (first_radius < second_radius && first_radius > 0.0L) {
result.first_inside_second = full_coverage();
} else if (
second_radius < first_radius && second_radius > 0.0L
) {
result.second_inside_first = full_coverage();
}
}
return result;
}
assert(result.relation == CircleRelation::Intersecting);
const long double height = std::sqrt(std::max(
0.0L,
first_radius * first_radius - along * along
));
const Point<long double> perpendicular(-unit.y, unit.x);
const Point<long double> first_point = base - perpendicular * height;
const Point<long double> second_point = base + perpendicular * height;
result.contact_kind = CircleContactKind::TwoPoints;
result.contacts[0] = CircleContact{
first_point,
circle_argument(first, first_point),
circle_argument(second, first_point)
};
result.contacts[1] = CircleContact{
second_point,
circle_argument(first, second_point),
circle_argument(second, second_point)
};
const long double first_begin = result.contacts[0].first_argument;
long double first_end = result.contacts[1].first_argument;
if (first_end <= first_begin) first_end += full;
result.first_inside_second = AngularCoverage{
AngularCoverageKind::Arc,
first_begin,
first_end
};
const long double second_begin = result.contacts[1].second_argument;
long double second_end = result.contacts[0].second_argument;
if (second_end <= second_begin) second_end += full;
result.second_inside_first = AngularCoverage{
AngularCoverageKind::Arc,
second_begin,
second_end
};
return result;
}
template <Coordinate C, Coordinate L>
bool intersects(
const Circle<C>& circle,
const Line<L>& line,
long double eps = 1e-12L
) {
if (circle.filled) {
const Line<long double> converted{
Point<long double>(line.a),
Point<long double>(line.b)
};
return contains(
circle,
projection(converted, Point<long double>(circle.center)),
eps
);
}
return circle_boundary_intersection(circle, line, eps).contact_count > 0;
}
template <Coordinate C, Coordinate L>
bool intersects(
const Line<L>& line,
const Circle<C>& circle,
long double eps = 1e-12L
) {
return intersects(circle, line, eps);
}
template <Coordinate C, Coordinate R>
bool intersects(
const Circle<C>& circle,
const Ray<R>& ray,
long double eps = 1e-12L
) {
if (circle.filled) {
const Ray<long double> converted{
Point<long double>(ray.origin),
Point<long double>(ray.through)
};
return contains(
circle,
projection(converted, Point<long double>(circle.center)),
eps
);
}
return circle_boundary_intersection(circle, ray, eps).contact_count > 0;
}
template <Coordinate C, Coordinate R>
bool intersects(
const Ray<R>& ray,
const Circle<C>& circle,
long double eps = 1e-12L
) {
return intersects(circle, ray, eps);
}
template <Coordinate C, Coordinate S>
bool intersects(
const Circle<C>& circle,
const Segment<S>& segment,
long double eps = 1e-12L
) {
if (circle.filled) {
const Segment<long double> converted{
Point<long double>(segment.a),
Point<long double>(segment.b)
};
return contains(
circle,
projection(converted, Point<long double>(circle.center)),
eps
);
}
return
circle_boundary_intersection(circle, segment, eps).contact_count > 0;
}
template <Coordinate C, Coordinate S>
bool intersects(
const Segment<S>& segment,
const Circle<C>& circle,
long double eps = 1e-12L
) {
return intersects(circle, segment, eps);
}
template <Coordinate A, Coordinate B>
bool intersects(
const Circle<A>& first,
const Circle<B>& second,
long double eps = 1e-12L
) {
assert(first.radius >= 0);
assert(second.radius >= 0);
assert(eps >= 0.0L);
if (first.filled && second.filled) {
if constexpr (ExactCoordinate<A> && ExactCoordinate<B>) {
using W = std::common_type_t<wide_type<A>, wide_type<B>>;
const W dx = W(second.center.x) - W(first.center.x);
const W dy = W(second.center.y) - W(first.center.y);
const W radius = W(first.radius) + W(second.radius);
return dx * dx + dy * dy <= radius * radius;
} else {
const long double center_distance = geometry::distance(
Point<long double>(first.center),
Point<long double>(second.center)
);
const long double radius_sum =
static_cast<long double>(first.radius) +
static_cast<long double>(second.radius);
return circle_detail::compare(
center_distance,
radius_sum,
eps
) <= 0;
}
}
if (first.filled != second.filled) {
const long double center_distance = geometry::distance(
Point<long double>(first.center),
Point<long double>(second.center)
);
const long double boundary_radius = first.filled
? static_cast<long double>(second.radius)
: static_cast<long double>(first.radius);
const long double filled_radius = first.filled
? static_cast<long double>(first.radius)
: static_cast<long double>(second.radius);
return circle_detail::compare(
std::fabs(center_distance - boundary_radius),
filled_radius,
eps
) <= 0;
}
CircleRelation relation = circle_relation(first, second, eps);
return
relation == CircleRelation::ExternallyTangent ||
relation == CircleRelation::Intersecting ||
relation == CircleRelation::InternallyTangent ||
relation == CircleRelation::Coincident;
}
template <Coordinate R, Coordinate H, Coordinate C>
Ray<long double> reflected_ray(
const Ray<R>& incoming,
const Point<H>& hit,
const Circle<C>& circle,
long double eps = 1e-12L
) {
assert(incoming.origin != incoming.through);
assert(eps >= 0.0L);
assert(static_cast<long double>(circle.radius) > eps);
assert(
std::fabs(
geometry::distance(
Point<long double>(hit),
Point<long double>(circle.center)
) -
static_cast<long double>(circle.radius)
) <= eps
);
Point<long double> hit_point(hit);
Point<long double> normal = normalized(
hit_point - Point<long double>(circle.center)
);
Point<long double> incoming_direction =
Point<long double>(incoming.through) -
Point<long double>(incoming.origin);
Point<long double> outgoing_direction =
incoming_direction - normal * (2.0L * dot(incoming_direction, normal));
return Ray<long double>{hit_point, hit_point + outgoing_direction};
}
template <Coordinate C, Coordinate P>
std::vector<Point<long double>> tangent_points(
const Circle<C>& circle,
const Point<P>& point,
long double eps = 1e-12L
) {
assert(circle.radius >= 0);
assert(eps >= 0.0L);
Point<long double> center(circle.center);
Point<long double> external(point);
Point<long double> direction = external - center;
long double squared_distance = dot(direction, direction);
long double radius = static_cast<long double>(circle.radius);
if (radius == 0.0L) return {center};
long double center_distance = std::sqrt(squared_distance);
int relation = circle_detail::compare(center_distance, radius, eps);
if (relation < 0) return {};
if (relation == 0) {
return {center + direction * (radius / center_distance)};
}
Point<long double> base =
center + direction * (radius * radius / squared_distance);
long double scale =
radius * std::sqrt(std::max(
0.0L,
squared_distance - radius * radius
)) /
squared_distance;
Point<long double> perpendicular(-direction.y, direction.x);
Point<long double> first = base - perpendicular * scale;
Point<long double> second = base + perpendicular * scale;
if (second < first) std::swap(first, second);
return {first, second};
}
template <Coordinate A, Coordinate B>
std::vector<Line<long double>> common_tangents(
const Circle<A>& first,
const Circle<B>& second,
long double eps = 1e-12L
) {
assert(first.radius >= 0);
assert(second.radius >= 0);
assert(eps >= 0.0L);
Point<long double> first_center(first.center);
Point<long double> second_center(second.center);
Point<long double> direction = second_center - first_center;
long double squared_distance = dot(direction, direction);
long double center_distance = std::sqrt(squared_distance);
if (center_distance <= eps) return {};
long double first_radius = static_cast<long double>(first.radius);
long double second_radius = static_cast<long double>(second.radius);
std::vector<Line<long double>> result;
for (int second_side : {1, -1}) {
long double difference =
first_radius - second_side * second_radius;
int relation = circle_detail::compare(
std::fabs(difference),
center_distance,
eps
);
if (relation > 0) continue;
long double perpendicular_length = relation == 0 ? 0.0L : std::sqrt(
std::max(0.0L, squared_distance - difference * difference)
);
int choices = perpendicular_length <= eps ? 1 : 2;
for (int choice = 0; choice < choices; ++choice) {
long double side = choice == 0 ? -1.0L : 1.0L;
Point<long double> normal =
direction * (difference / squared_distance) +
Point<long double>(-direction.y, direction.x) *
(side * perpendicular_length / squared_distance);
normal = normalized(normal);
Point<long double> contact =
first_center + normal * first_radius;
Line<long double> tangent =
circle_detail::tangent_line(contact, normal, eps);
bool duplicate = false;
for (const Line<long double>& existing : result) {
if (circle_detail::same_line(existing, tangent, eps)) {
duplicate = true;
break;
}
}
if (!duplicate) result.push_back(tangent);
}
}
std::sort(
result.begin(),
result.end(),
[](const Line<long double>& left, const Line<long double>& right) {
if (left.a != right.a) return left.a < right.a;
return left.b < right.b;
}
);
return result;
}
template <Coordinate A, Coordinate B>
std::vector<Point<long double>> common_tangent_points(
const Circle<A>& first,
const Circle<B>& second,
long double eps = 1e-12L
) {
std::vector<Point<long double>> result;
for (const Line<long double>& line : common_tangents(first, second, eps)) {
circle_detail::push_unique(result, line.a, eps);
}
std::sort(result.begin(), result.end());
return result;
}
// These area functions use the enclosed disks, independent of `filled`.
template <Coordinate A, Coordinate B>
long double circle_circle_intersection_area(
const Circle<A>& first,
const Circle<B>& second,
long double eps = 1e-12L
) {
assert(first.radius >= 0);
assert(second.radius >= 0);
assert(eps >= 0.0L);
const long double first_radius = static_cast<long double>(first.radius);
const long double second_radius = static_cast<long double>(second.radius);
const CircleRelation relation = circle_relation(first, second, eps);
if (
relation == CircleRelation::Separate ||
relation == CircleRelation::ExternallyTangent
) {
return 0.0L;
}
if (
relation == CircleRelation::Contained ||
relation == CircleRelation::InternallyTangent ||
relation == CircleRelation::Coincident
) {
const long double radius = std::min(first_radius, second_radius);
return std::numbers::pi_v<long double> * radius * radius;
}
const long double center_distance = geometry::distance(
Point<long double>(first.center),
Point<long double>(second.center)
);
const long double first_cosine = std::clamp(
(
(center_distance - second_radius) *
(center_distance + second_radius) +
first_radius * first_radius
) / (2.0L * center_distance * first_radius),
-1.0L,
1.0L
);
const long double second_cosine = std::clamp(
(
(center_distance - first_radius) *
(center_distance + first_radius) +
second_radius * second_radius
) / (2.0L * center_distance * second_radius),
-1.0L,
1.0L
);
const long double radicand =
(-center_distance + first_radius + second_radius) *
(center_distance + first_radius - second_radius) *
(center_distance - first_radius + second_radius) *
(center_distance + first_radius + second_radius);
const long double height =
std::sqrt(std::max(0.0L, radicand)) / (2.0L * center_distance);
const long double first_sine =
std::clamp(height / first_radius, 0.0L, 1.0L);
const long double second_sine =
std::clamp(height / second_radius, 0.0L, 1.0L);
const long double first_angle = std::atan2(first_sine, first_cosine);
const long double second_angle = std::atan2(second_sine, second_cosine);
return
first_radius * first_radius *
circle_detail::circular_segment_angle_term(
first_angle,
first_sine,
first_cosine
) +
second_radius * second_radius *
circle_detail::circular_segment_angle_term(
second_angle,
second_sine,
second_cosine
);
}
template <Coordinate C, Coordinate P>
long double circle_polygon_intersection_area(
const Circle<C>& circle,
const std::vector<Point<P>>& polygon,
long double eps = 1e-12L
) {
assert(circle.radius >= 0);
assert(eps >= 0.0L);
if (polygon.empty() || circle.radius == 0) return 0.0L;
const Point<long double> center(circle.center);
const long double radius = static_cast<long double>(circle.radius);
long double result = 0.0L;
for (std::size_t index = 0; index < polygon.size(); ++index) {
const Point<long double> first =
Point<long double>(polygon[index]) - center;
const Point<long double> second =
Point<long double>(polygon[(index + 1) % polygon.size()]) - center;
result += circle_detail::segment_disk_signed_area(
first,
second,
radius,
eps
);
}
return std::fabs(result);
}
namespace circle_detail {
template <Coordinate T>
Point<long double> point_toward(
const Circle<T>& circle,
const Point<long double>& target
) {
assert(circle.radius >= 0);
const Point<long double> center(circle.center);
const long double radius = static_cast<long double>(circle.radius);
const Point<long double> direction = target - center;
const long double length = norm(direction);
if (length == 0.0L) {
return center + Point<long double>(-radius, 0.0L);
}
return center + direction * (radius / length);
}
inline void consider(
ClosestPoints& best,
const ClosestPoints& candidate
) {
closest_points_detail::consider(best, candidate);
}
} // namespace circle_detail
template <Coordinate C, Coordinate P>
ClosestPoints closest_points(
const Circle<C>& circle,
const Point<P>& point,
long double eps = 1e-12L
) {
const Point<long double> converted(point);
if (circle.filled && contains(circle, converted, eps)) {
return ClosestPoints{converted, converted};
}
return ClosestPoints{
circle_detail::point_toward(circle, converted),
converted
};
}
template <Coordinate P, Coordinate C>
ClosestPoints closest_points(
const Point<P>& point,
const Circle<C>& circle,
long double eps = 1e-12L
) {
return closest_points_detail::reversed(
closest_points(circle, point, eps)
);
}
template <Coordinate C, Coordinate L>
ClosestPoints closest_points(
const Circle<C>& circle,
const Line<L>& line,
long double eps = 1e-12L
) {
const Line<long double> converted{
Point<long double>(line.a),
Point<long double>(line.b)
};
const Point<long double> point = projection(
converted,
Point<long double>(circle.center)
);
if (circle.filled) return closest_points(circle, point, eps);
const CircleLinearIntersection common =
circle_boundary_intersection(circle, line, eps);
if (common.contact_count > 0) {
return ClosestPoints{
common.contacts[0].point,
common.contacts[0].point
};
}
return ClosestPoints{
circle_detail::point_toward(circle, point),
point
};
}
template <Coordinate L, Coordinate C>
ClosestPoints closest_points(
const Line<L>& line,
const Circle<C>& circle,
long double eps = 1e-12L
) {
return closest_points_detail::reversed(closest_points(circle, line, eps));
}
template <Coordinate C, Coordinate R>
ClosestPoints closest_points(
const Circle<C>& circle,
const Ray<R>& ray,
long double eps = 1e-12L
) {
const Ray<long double> converted{
Point<long double>(ray.origin),
Point<long double>(ray.through)
};
const Point<long double> point = projection(
converted,
Point<long double>(circle.center)
);
if (circle.filled) return closest_points(circle, point, eps);
const CircleLinearIntersection common =
circle_boundary_intersection(circle, ray, eps);
if (common.contact_count > 0) {
return ClosestPoints{
common.contacts[0].point,
common.contacts[0].point
};
}
return ClosestPoints{
circle_detail::point_toward(circle, point),
point
};
}
template <Coordinate R, Coordinate C>
ClosestPoints closest_points(
const Ray<R>& ray,
const Circle<C>& circle,
long double eps = 1e-12L
) {
return closest_points_detail::reversed(closest_points(circle, ray, eps));
}
template <Coordinate C, Coordinate S>
ClosestPoints closest_points(
const Circle<C>& circle,
const Segment<S>& segment,
long double eps = 1e-12L
) {
const Segment<long double> converted{
Point<long double>(segment.a),
Point<long double>(segment.b)
};
const Point<long double> center(circle.center);
const Point<long double> projected = projection(converted, center);
if (circle.filled) return closest_points(circle, projected, eps);
const CircleLinearIntersection common =
circle_boundary_intersection(circle, segment, eps);
if (common.contact_count > 0) {
return ClosestPoints{
common.contacts[0].point,
common.contacts[0].point
};
}
ClosestPoints result{
circle_detail::point_toward(circle, projected),
projected
};
for (const Point<long double>& point : {converted.a, converted.b}) {
circle_detail::consider(
result,
ClosestPoints{circle_detail::point_toward(circle, point), point}
);
}
return result;
}
template <Coordinate S, Coordinate C>
ClosestPoints closest_points(
const Segment<S>& segment,
const Circle<C>& circle,
long double eps = 1e-12L
) {
return closest_points_detail::reversed(
closest_points(circle, segment, eps)
);
}
template <Coordinate A, Coordinate B>
ClosestPoints closest_points(
const Circle<A>& first,
const Circle<B>& second,
long double eps = 1e-12L
) {
if (first.filled && !second.filled) {
return closest_points_detail::reversed(
closest_points(second, first, eps)
);
}
if (!first.filled && second.filled) {
const ClosestPoints center_result =
closest_points(first, second.center, eps);
if (contains(second, center_result.first, eps)) {
return ClosestPoints{center_result.first, center_result.first};
}
const ClosestPoints filled_result =
closest_points(second, center_result.first, eps);
return ClosestPoints{center_result.first, filled_result.first};
}
if (first.filled && second.filled) {
assert(first.radius >= 0);
assert(second.radius >= 0);
const Point<long double> first_center(first.center);
const Point<long double> second_center(second.center);
Point<long double> direction = second_center - first_center;
const long double center_distance = norm(direction);
if (center_distance == 0.0L) {
return ClosestPoints{first_center, first_center};
}
direction = direction / center_distance;
const long double first_radius =
static_cast<long double>(first.radius);
const long double second_radius =
static_cast<long double>(second.radius);
if (intersects(first, second, eps)) {
const long double left = std::max(
-first_radius,
center_distance - second_radius
);
const long double right = std::min(
first_radius,
center_distance + second_radius
);
const Point<long double> common =
first_center + direction * ((left + right) / 2.0L);
return ClosestPoints{common, common};
}
return ClosestPoints{
first_center + direction * first_radius,
second_center - direction * second_radius
};
}
const CircleCircleIntersection common =
circle_boundary_intersection(first, second, eps);
if (common.contact_count() > 0) {
return ClosestPoints{
common.contacts[0].point,
common.contacts[0].point
};
}
const Point<long double> first_center(first.center);
const Point<long double> second_center(second.center);
if (circle_relation(first, second, eps) == CircleRelation::Coincident) {
const Point<long double> point =
circle_detail::point_toward(first, first_center);
return ClosestPoints{point, point};
}
Point<long double> direction = second_center - first_center;
const long double center_distance = norm(direction);
if (center_distance == 0.0L) {
const Point<long double> first_point =
circle_detail::point_toward(first, first_center);
const Point<long double> second_point =
circle_detail::point_toward(second, second_center);
return ClosestPoints{first_point, second_point};
}
direction = direction / center_distance;
const long double first_radius = static_cast<long double>(first.radius);
const long double second_radius = static_cast<long double>(second.radius);
ClosestPoints result{
first_center + direction * first_radius,
second_center + direction * second_radius
};
for (const long double first_sign : {-1.0L, 1.0L}) {
for (const long double second_sign : {-1.0L, 1.0L}) {
circle_detail::consider(
result,
ClosestPoints{
first_center + direction * (first_sign * first_radius),
second_center + direction * (second_sign * second_radius)
}
);
}
}
return result;
}
template <Coordinate C, Coordinate P>
long double distance(const Circle<C>& circle, const Point<P>& point) {
const ClosestPoints result = closest_points(circle, point);
return geometry::distance(result.first, result.second);
}
template <Coordinate P, Coordinate C>
long double distance(const Point<P>& point, const Circle<C>& circle) {
return distance(circle, point);
}
template <Coordinate C, Coordinate L>
long double distance(const Circle<C>& circle, const Line<L>& line) {
const ClosestPoints result = closest_points(circle, line);
return geometry::distance(result.first, result.second);
}
template <Coordinate L, Coordinate C>
long double distance(const Line<L>& line, const Circle<C>& circle) {
return distance(circle, line);
}
template <Coordinate C, Coordinate R>
long double distance(const Circle<C>& circle, const Ray<R>& ray) {
const ClosestPoints result = closest_points(circle, ray);
return geometry::distance(result.first, result.second);
}
template <Coordinate R, Coordinate C>
long double distance(const Ray<R>& ray, const Circle<C>& circle) {
return distance(circle, ray);
}
template <Coordinate C, Coordinate S>
long double distance(const Circle<C>& circle, const Segment<S>& segment) {
const ClosestPoints result = closest_points(circle, segment);
return geometry::distance(result.first, result.second);
}
template <Coordinate S, Coordinate C>
long double distance(const Segment<S>& segment, const Circle<C>& circle) {
return distance(circle, segment);
}
template <Coordinate A, Coordinate B>
long double distance(const Circle<A>& first, const Circle<B>& second) {
const ClosestPoints result = closest_points(first, second);
return geometry::distance(result.first, result.second);
}
} // namespace geometry
} // namespace m1une
#line 5 "geometry/circle_coverage_areas.hpp"
#line 10 "geometry/circle_coverage_areas.hpp"
#include <utility>
#line 12 "geometry/circle_coverage_areas.hpp"
namespace m1une {
namespace geometry {
namespace circle_coverage_areas_detail {
inline long double arc_integral(
long double center_x,
long double center_y,
long double radius,
long double first_angle,
long double second_angle
) {
return (
radius * center_x *
(std::sin(second_angle) - std::sin(first_angle)) -
radius * center_y *
(std::cos(second_angle) - std::cos(first_angle)) +
radius * radius * (second_angle - first_angle)
) / 2.0L;
}
} // namespace circle_coverage_areas_detail
template <Coordinate T>
std::vector<long double> circle_coverage_areas(
const std::vector<Circle<T>>& circles,
long double eps = 1e-12L
) {
assert(eps >= 0.0L);
const int count = int(circles.size());
const long double full_angle =
2.0L * std::numbers::pi_v<long double>;
std::vector<long double> at_least(count + 2, 0.0L);
for (int index = 0; index < count; ++index) {
const Circle<T>& circle = circles[index];
assert(circle.radius >= 0);
long double radius = static_cast<long double>(circle.radius);
if (radius == 0.0L) continue;
long double center_x = static_cast<long double>(circle.center.x);
long double center_y = static_cast<long double>(circle.center.y);
int coverage = 0;
int multiplicity = 1;
bool duplicate = false;
std::vector<std::pair<long double, int>> events;
events.reserve(2 * circles.size());
for (int other_index = 0; other_index < count; ++other_index) {
if (other_index == index) continue;
const Circle<T>& other = circles[other_index];
assert(other.radius >= 0);
long double other_radius =
static_cast<long double>(other.radius);
if (other_radius == 0.0L) continue;
long double difference_x =
static_cast<long double>(other.center.x) - center_x;
long double difference_y =
static_cast<long double>(other.center.y) - center_y;
long double center_distance =
std::hypot(difference_x, difference_y);
long double tolerance = eps * std::max({
1.0L,
center_distance,
radius,
other_radius
});
if (
center_distance <= tolerance &&
std::fabs(radius - other_radius) <= tolerance
) {
if (other_index < index) duplicate = true;
multiplicity++;
continue;
}
if (
radius <= other_radius &&
center_distance + radius <= other_radius + tolerance
) {
coverage++;
continue;
}
if (
center_distance >= radius + other_radius - tolerance ||
center_distance <=
std::fabs(radius - other_radius) + tolerance
) {
continue;
}
long double direction =
std::atan2(difference_y, difference_x);
long double cosine = std::clamp(
(
center_distance * center_distance + radius * radius -
other_radius * other_radius
) / (2.0L * center_distance * radius),
-1.0L,
1.0L
);
long double half_width = std::acos(cosine);
long double left = std::fmod(
direction - half_width,
full_angle
);
if (left < 0.0L) left += full_angle;
long double right = std::fmod(
direction + half_width,
full_angle
);
if (right < 0.0L) right += full_angle;
if (left <= right) {
events.emplace_back(left, 1);
events.emplace_back(right, -1);
} else {
coverage++;
events.emplace_back(right, -1);
events.emplace_back(left, 1);
}
}
if (duplicate) continue;
std::sort(events.begin(), events.end());
long double previous_angle = 0.0L;
auto add_arc = [&](long double first_angle, long double second_angle) {
long double integral =
circle_coverage_areas_detail::arc_integral(
center_x,
center_y,
radius,
first_angle,
second_angle
);
for (int offset = 1; offset <= multiplicity; ++offset) {
at_least[coverage + offset] += integral;
}
};
int event_index = 0;
while (event_index < int(events.size())) {
long double angle = events[event_index].first;
add_arc(previous_angle, angle);
int next = event_index;
while (
next < int(events.size()) &&
events[next].first == angle
) {
coverage += events[next].second;
next++;
}
previous_angle = angle;
event_index = next;
}
add_arc(previous_angle, full_angle);
}
std::vector<long double> exact(count + 1, 0.0L);
for (int coverage = 1; coverage <= count; ++coverage) {
exact[coverage] = std::max(
0.0L,
at_least[coverage] - at_least[coverage + 1]
);
}
return exact;
}
} // namespace geometry
} // namespace m1une
#line 1 "geometry/circle_union_area.hpp"
#line 5 "geometry/circle_union_area.hpp"
#line 12 "geometry/circle_union_area.hpp"
namespace m1une {
namespace geometry {
namespace circle_union_area_detail {
inline long double arc_integral(
long double center_x,
long double center_y,
long double radius,
long double first_angle,
long double second_angle
) {
return (
radius * center_x *
(std::sin(second_angle) - std::sin(first_angle)) -
radius * center_y *
(std::cos(second_angle) - std::cos(first_angle)) +
radius * radius * (second_angle - first_angle)
) / 2.0L;
}
} // namespace circle_union_area_detail
template <Coordinate T>
long double circle_union_area(
const std::vector<Circle<T>>& circles,
long double eps = 1e-12L
) {
assert(eps >= 0.0L);
const long double pi = std::numbers::pi_v<long double>;
const long double full_angle = 2.0L * pi;
long double area = 0.0L;
for (int index = 0; index < int(circles.size()); ++index) {
const Circle<T>& circle = circles[index];
assert(circle.radius >= 0);
long double radius = static_cast<long double>(circle.radius);
if (radius == 0.0L) continue;
long double center_x = static_cast<long double>(circle.center.x);
long double center_y = static_cast<long double>(circle.center.y);
bool covered = false;
std::vector<std::pair<long double, long double>> intervals;
intervals.reserve(2 * circles.size());
for (int other_index = 0;
other_index < int(circles.size());
++other_index) {
if (other_index == index) continue;
const Circle<T>& other = circles[other_index];
assert(other.radius >= 0);
long double other_radius =
static_cast<long double>(other.radius);
if (other_radius == 0.0L) continue;
long double difference_x =
static_cast<long double>(other.center.x) - center_x;
long double difference_y =
static_cast<long double>(other.center.y) - center_y;
long double center_distance =
std::hypot(difference_x, difference_y);
long double tolerance = eps * std::max({
1.0L,
center_distance,
radius,
other_radius
});
if (
center_distance <= tolerance &&
std::fabs(radius - other_radius) <= tolerance
) {
if (other_index < index) covered = true;
continue;
}
if (
radius <= other_radius &&
center_distance + radius <= other_radius + tolerance
) {
covered = true;
break;
}
if (
center_distance >= radius + other_radius - tolerance ||
center_distance <=
std::fabs(radius - other_radius) + tolerance
) {
continue;
}
long double direction =
std::atan2(difference_y, difference_x);
long double cosine = std::clamp(
(
center_distance * center_distance + radius * radius -
other_radius * other_radius
) / (2.0L * center_distance * radius),
-1.0L,
1.0L
);
long double half_width = std::acos(cosine);
long double left = std::fmod(
direction - half_width,
full_angle
);
if (left < 0.0L) left += full_angle;
long double right = left + 2.0L * half_width;
if (right <= full_angle) {
intervals.emplace_back(left, right);
} else {
intervals.emplace_back(left, full_angle);
intervals.emplace_back(0.0L, right - full_angle);
}
}
if (covered) continue;
std::sort(intervals.begin(), intervals.end());
long double exposed_begin = 0.0L;
for (const auto& [left, right] : intervals) {
if (exposed_begin < left) {
area += circle_union_area_detail::arc_integral(
center_x,
center_y,
radius,
exposed_begin,
left
);
}
exposed_begin = std::max(exposed_begin, right);
}
if (exposed_begin < full_angle) {
area += circle_union_area_detail::arc_integral(
center_x,
center_y,
radius,
exposed_begin,
full_angle
);
}
}
return std::max(0.0L, area);
}
} // namespace geometry
} // namespace m1une
#line 5 "verify/geometry/circle_coverage_areas.test.cpp"
#line 1 "utilities/fast_io.hpp"
#line 6 "utilities/fast_io.hpp"
#include <cerrno>
#include <charconv>
#line 9 "utilities/fast_io.hpp"
#include <cstdio>
#include <cstdlib>
#include <cstdint>
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#line 18 "utilities/fast_io.hpp"
#include <unistd.h>
#line 20 "utilities/fast_io.hpp"
namespace m1une {
namespace utilities {
struct FastOutput;
namespace internal {
// Shared with the convenience helpers in template.hpp.
inline FastOutput* standard_output_instance = nullptr;
// Detect std::begin(x), std::end(x).
template <class T, class = void>
struct is_range : std::false_type {};
template <class T>
struct is_range<T, std::void_t<
decltype(std::begin(std::declval<T&>())),
decltype(std::end(std::declval<T&>()))
>> : std::true_type {};
template <class T>
inline constexpr bool is_range_v = is_range<T>::value;
template <class T>
using range_reference_t = decltype(*std::begin(std::declval<T&>()));
template <class T>
using range_value_t = std::remove_cv_t<std::remove_reference_t<range_reference_t<T>>>;
template <class T, class = void>
struct range_stored_value {
using type = range_value_t<T>;
};
template <class T>
struct range_stored_value<T, std::void_t<typename std::remove_cv_t<std::remove_reference_t<T>>::value_type>> {
using type = typename std::remove_cv_t<std::remove_reference_t<T>>::value_type;
};
template <class T>
using range_stored_value_t = typename range_stored_value<T>::type;
// Treat strings and C strings as scalar output objects, not as ranges.
template <class T>
struct is_char_array : std::false_type {};
template <class T, std::size_t N>
struct is_char_array<T[N]>
: std::bool_constant<std::is_same_v<std::remove_cv_t<T>, char>> {};
template <class T>
struct is_string_like
: std::bool_constant<
std::is_same_v<std::decay_t<T>, std::string>
|| std::is_same_v<std::decay_t<T>, const char*>
|| std::is_same_v<std::decay_t<T>, char*>
|| is_char_array<std::remove_reference_t<T>>::value
> {};
template <class T>
inline constexpr bool is_string_like_v = is_string_like<T>::value;
// ModInt-like type: x.val() is printable, and x can be assigned from long long.
template <class T, class = void>
struct has_val_method : std::false_type {};
template <class T>
struct has_val_method<T, std::void_t<decltype(std::declval<const T&>().val())>>
: std::true_type {};
template <class T>
inline constexpr bool has_val_method_v = has_val_method<T>::value;
template <class T, class = void>
struct has_static_mod_raw : std::false_type {};
template <class T>
struct has_static_mod_raw<
T, std::void_t<decltype(T::mod()), decltype(T::raw(std::declval<uint32_t>()))>>
: std::true_type {};
template <class T>
inline constexpr bool has_static_mod_raw_v = has_static_mod_raw<T>::value;
// libstdc++ before GCC 16 does not classify __int128 as an integral type in
// strict ISO modes such as -std=c++23. Keep the fast-I/O interface independent
// of that implementation detail.
template <class T>
inline constexpr bool is_integral_v =
std::is_integral_v<T>
|| std::is_same_v<std::remove_cv_t<T>, __int128_t>
|| std::is_same_v<std::remove_cv_t<T>, __uint128_t>;
template <class T>
inline constexpr bool is_signed_v =
std::is_signed_v<T>
|| std::is_same_v<std::remove_cv_t<T>, __int128_t>;
template <class T>
struct make_unsigned {
using type = std::make_unsigned_t<T>;
};
template <>
struct make_unsigned<__int128_t> {
using type = __uint128_t;
};
template <>
struct make_unsigned<__uint128_t> {
using type = __uint128_t;
};
template <class T>
using make_unsigned_t = typename make_unsigned<std::remove_cv_t<T>>::type;
} // namespace internal
struct FastInput {
static constexpr int buffer_size = 1 << 20;
private:
std::FILE* _stream;
char _buffer[buffer_size];
int _position;
int _length;
int _file_descriptor;
bool _streaming;
bool refill() {
_position = 0;
if (_streaming) {
ssize_t length;
do {
length = ::read(_file_descriptor, _buffer, buffer_size);
} while (length < 0 && errno == EINTR);
if (length <= 0) {
_length = 0;
return false;
}
_length = int(length);
} else {
_length = int(std::fread(_buffer, 1, buffer_size, _stream));
}
return _length != 0;
}
template <class T>
bool read_integer_from_stream(T& value) {
if (!skip_spaces()) return false;
int c = read_char_raw();
bool negative = false;
if (c == '-') {
negative = true;
c = read_char_raw();
}
if constexpr (internal::is_signed_v<T>) {
T result = 0;
while ('0' <= c && c <= '9') {
result = negative ? result * 10 - (c - '0')
: result * 10 + (c - '0');
c = read_char_raw();
}
value = result;
} else {
T result = 0;
while ('0' <= c && c <= '9') {
result = result * 10 + T(c - '0');
c = read_char_raw();
}
value = negative ? T(0) - result : result;
}
return true;
}
bool prepare_number() {
if (_length - _position >= 64) return true;
const int remaining = _length - _position;
if (remaining > 0) std::memmove(_buffer, _buffer + _position, remaining);
const int added = int(std::fread(_buffer + remaining, 1, buffer_size - remaining, _stream));
_position = 0;
_length = remaining + added;
if (_length < buffer_size) _buffer[_length] = '\0';
return _length != 0;
}
public:
explicit FastInput(std::FILE* stream = stdin)
: _stream(stream),
_position(0),
_length(0),
_file_descriptor(::fileno(stream)),
_streaming([&] {
struct stat status;
return _file_descriptor >= 0
&& ::fstat(_file_descriptor, &status) == 0
&& !S_ISREG(status.st_mode);
}()) {}
FastInput(const FastInput&) = delete;
FastInput& operator=(const FastInput&) = delete;
int read_char_raw() {
if (_position == _length && !refill()) return EOF;
return _buffer[_position++];
}
bool skip_spaces() {
int c = read_char_raw();
while (c != EOF && c <= ' ') c = read_char_raw();
if (c == EOF) return false;
--_position;
return true;
}
bool read(char& value) {
if (!skip_spaces()) return false;
value = char(read_char_raw());
return true;
}
bool read(std::string& value) {
if (!skip_spaces()) return false;
value.clear();
while (true) {
const int begin = _position;
while (_position < _length &&
static_cast<unsigned char>(_buffer[_position]) > ' ') {
++_position;
}
value.append(_buffer + begin, _position - begin);
if (_position < _length) {
++_position;
return true;
}
if (!refill()) return true;
}
}
bool read(bool& value) {
int x;
if (!read(x)) return false;
value = x != 0;
return true;
}
template <class T>
std::enable_if_t<
internal::is_integral_v<T>
&& !std::is_same_v<std::remove_cv_t<T>, bool>
&& !std::is_same_v<std::remove_cv_t<T>, char>,
bool
>
read(T& value) {
if (_streaming) return read_integer_from_stream(value);
if (!prepare_number()) return false;
int c = static_cast<unsigned char>(_buffer[_position++]);
while (c <= ' ') c = static_cast<unsigned char>(_buffer[_position++]);
bool negative = false;
if (c == '-') {
negative = true;
c = static_cast<unsigned char>(_buffer[_position++]);
}
if constexpr (internal::is_signed_v<T>) {
T result = 0;
while ('0' <= c && c <= '9') {
const int first = c - '0';
const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
if (0 <= second && second <= 9) {
result = negative ? result * 100 - (first * 10 + second)
: result * 100 + (first * 10 + second);
++_position;
} else {
result = negative ? result * 10 - first : result * 10 + first;
}
c = static_cast<unsigned char>(_buffer[_position++]);
}
value = result;
} else {
T result = 0;
while ('0' <= c && c <= '9') {
const unsigned first = unsigned(c - '0');
const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
if (0 <= second && second <= 9) {
result = result * 100 + T(first * 10 + unsigned(second));
++_position;
} else {
result = result * 10 + T(first);
}
c = static_cast<unsigned char>(_buffer[_position++]);
}
value = negative ? T(0) - result : result;
}
if (_position > _length) _position = _length;
return true;
}
template <class T>
std::enable_if_t<std::is_floating_point_v<T>, bool>
read(T& value) {
if (!skip_spaces()) return false;
int c = read_char_raw();
bool negative = false;
if (c == '-' || c == '+') {
negative = c == '-';
c = read_char_raw();
}
long double result = 0;
while ('0' <= c && c <= '9') {
result = result * 10 + (c - '0');
c = read_char_raw();
}
if (c == '.') {
long double place = 0.1L;
c = read_char_raw();
while ('0' <= c && c <= '9') {
result += (c - '0') * place;
place *= 0.1L;
c = read_char_raw();
}
}
if (c == 'e' || c == 'E') {
c = read_char_raw();
bool exponent_negative = false;
if (c == '-' || c == '+') {
exponent_negative = c == '-';
c = read_char_raw();
}
int exponent = 0;
while ('0' <= c && c <= '9') {
exponent = exponent * 10 + (c - '0');
c = read_char_raw();
}
long double scale = 1;
long double power = 10;
while (exponent > 0) {
if (exponent & 1) scale *= power;
power *= power;
exponent >>= 1;
}
result = exponent_negative ? result / scale : result * scale;
}
value = static_cast<T>(negative ? -result : result);
return true;
}
template <class T>
std::enable_if_t<
internal::has_val_method_v<T>
&& !internal::is_integral_v<T>
&& !internal::is_range_v<T>,
bool
>
read(T& value) {
long long x;
if (!read(x)) return false;
if constexpr (internal::has_static_mod_raw_v<T>) {
if (x >= 0 && uint64_t(x) < uint64_t(T::mod())) {
value = T::raw(uint32_t(x));
} else {
value = T(x);
}
} else {
value = T(x);
}
return true;
}
template <class First, class Second>
bool read(std::pair<First, Second>& value) {
if (!read(value.first)) return false;
return read(value.second);
}
template <class Range>
std::enable_if_t<
internal::is_range_v<Range>
&& !internal::is_string_like_v<Range>,
bool
>
read(Range& range) {
using StoredValue = internal::range_stored_value_t<Range>;
constexpr bool nested = internal::is_range_v<StoredValue>
&& !internal::is_string_like_v<StoredValue>;
for (auto&& value : range) {
if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
bool x;
if (!read(x)) return false;
value = x;
} else {
if (!read(value)) return false;
}
}
return true;
}
template <class First, class Second, class... Rest>
bool read(First& first, Second& second, Rest&... rest) {
if (!read(first)) return false;
return read(second, rest...);
}
template <class T>
FastInput& operator>>(T& value) {
if (!read(value)) std::abort();
return *this;
}
};
struct FastOutput {
static constexpr int buffer_size = 1 << 20;
private:
inline static const auto digit_quads = [] {
std::array<char, 40000> result{};
for (int i = 0; i < 10000; i++) {
int value = i;
for (int j = 3; j >= 0; j--) {
result[4 * i + j] = char('0' + value % 10);
value /= 10;
}
}
return result;
}();
std::FILE* _stream;
char _buffer[buffer_size];
int _position;
int _precision;
std::chars_format _float_format;
char _range_separator;
std::string* _capture = nullptr;
template <class T>
std::string format_cell(const T& value) {
std::string result;
struct CaptureGuard {
std::string*& target;
std::string* previous;
~CaptureGuard() { target = previous; }
} guard{_capture, _capture};
_capture = &result;
write(value);
return result;
}
template <class Matrix>
void write_aligned_matrix(const Matrix& matrix) {
std::vector<std::vector<std::string>> rows;
std::vector<std::size_t> widths;
for (const auto& row : matrix) {
auto& cells = rows.emplace_back();
std::size_t column = 0;
for (const auto& value : row) {
cells.push_back(format_cell(value));
if (column == widths.size()) widths.push_back(0);
widths[column] = std::max(widths[column], cells.back().size());
++column;
}
}
bool first = true;
for (const auto& row : rows) {
if (!first) write_char('\n');
first = false;
for (std::size_t column = 0; column < row.size(); ++column) {
if (column != 0) write_char(_range_separator);
for (std::size_t padding = row[column].size();
padding < widths[column]; ++padding) {
write_char(' ');
}
write(row[column]);
}
}
}
public:
explicit FastOutput(std::FILE* stream = stdout)
: _stream(stream),
_position(0),
_precision(6),
_float_format(std::chars_format::general),
_range_separator(' ') {
if (_stream == stdout
&& internal::standard_output_instance == nullptr) {
internal::standard_output_instance = this;
}
}
FastOutput(const FastOutput&) = delete;
FastOutput& operator=(const FastOutput&) = delete;
~FastOutput() {
flush();
if (internal::standard_output_instance == this) {
internal::standard_output_instance = nullptr;
}
}
void flush() {
if (_position != 0) {
std::fwrite(_buffer, 1, _position, _stream);
_position = 0;
}
std::fflush(_stream);
}
void write_char(char c) {
if (_capture != nullptr) {
_capture->push_back(c);
return;
}
if (_position == buffer_size) flush();
_buffer[_position++] = c;
}
void write(const char* s) {
while (*s != '\0') write_char(*s++);
}
void write(const std::string& s) {
if (_capture != nullptr) {
_capture->append(s);
return;
}
std::size_t position = 0;
while (position < s.size()) {
if (_position == buffer_size) flush();
const std::size_t copied =
std::min<std::size_t>(buffer_size - _position, s.size() - position);
std::memcpy(_buffer + _position, s.data() + position, copied);
_position += int(copied);
position += copied;
}
}
void write(char c) {
write_char(c);
}
void write(bool value) {
write_char(value ? '1' : '0');
}
template <class T>
std::enable_if_t<std::is_floating_point_v<T>>
write(T value) {
char digits[128];
auto [end, error] = std::to_chars(
digits,
digits + sizeof(digits),
value,
_float_format,
_precision
);
if (error != std::errc()) std::abort();
for (const char* pointer = digits; pointer != end; pointer++) {
write_char(*pointer);
}
}
template <class T>
std::enable_if_t<
internal::is_integral_v<T>
&& !std::is_same_v<std::remove_cv_t<T>, bool>
&& !std::is_same_v<std::remove_cv_t<T>, char>
>
write(T value) {
using Raw = std::remove_cv_t<T>;
using Unsigned = internal::make_unsigned_t<Raw>;
Unsigned magnitude;
if constexpr (internal::is_signed_v<Raw>) {
if (value < 0) {
write_char('-');
magnitude = Unsigned(0) - Unsigned(value);
} else {
magnitude = Unsigned(value);
}
} else {
magnitude = value;
}
if (magnitude == 0) {
write_char('0');
return;
}
unsigned chunks[16];
int count = 0;
while (magnitude >= 10000) {
const Unsigned quotient = magnitude / 10000;
chunks[count++] = unsigned(magnitude - quotient * 10000);
magnitude = quotient;
}
if (_capture == nullptr && _position > buffer_size - 64) flush();
char captured[64];
char* const begin = _capture != nullptr ? captured : _buffer + _position;
char* destination = begin;
const unsigned leading = unsigned(magnitude);
const char* first = digit_quads.data() + 4 * leading;
int skip = leading < 10 ? 3 : leading < 100 ? 2 : leading < 1000 ? 1 : 0;
for (; skip < 4; skip++) *destination++ = first[skip];
while (count--) {
const char* digits = digit_quads.data() + 4 * chunks[count];
std::memcpy(destination, digits, 4);
destination += 4;
}
if (_capture != nullptr) {
_capture->append(begin, destination - begin);
} else {
_position += int(destination - begin);
}
}
template <class T>
std::enable_if_t<
internal::has_val_method_v<T>
&& !internal::is_integral_v<T>
&& !internal::is_range_v<T>
>
write(const T& value) {
write(value.val());
}
template <class First, class Second>
void write(const std::pair<First, Second>& value) {
write(value.first);
write_char(' ');
write(value.second);
}
template <class Range>
std::enable_if_t<
internal::is_range_v<Range>
&& !internal::is_string_like_v<Range>
>
write(const Range& range) {
using StoredValue = internal::range_stored_value_t<const Range>;
constexpr bool nested = internal::is_range_v<StoredValue>
&& !internal::is_string_like_v<StoredValue>;
bool first = true;
for (const auto& value : range) {
if (!first) write_char(nested ? '\n' : _range_separator);
first = false;
if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
write(static_cast<bool>(value));
} else {
write(value);
}
}
}
template <class First, class... Rest>
void print(const First& first, const Rest&... rest) {
write(first);
((write_char(' '), write(rest)), ...);
}
void println() {
write_char('\n');
}
void set_precision(int precision) {
_precision = precision;
}
void set_fixed(int precision = 6) {
_float_format = std::chars_format::fixed;
_precision = precision;
}
void set_general(int precision = 6) {
_float_format = std::chars_format::general;
_precision = precision;
}
void set_range_separator(char separator) {
_range_separator = separator;
}
template <class Matrix>
void write_aligned(const Matrix& matrix) {
using Row = internal::range_stored_value_t<const Matrix>;
using Cell = internal::range_stored_value_t<const Row>;
static_assert(internal::is_range_v<Row> && !internal::is_string_like_v<Row>,
"write_aligned requires a two-dimensional range");
static_assert(!internal::is_range_v<Cell> || internal::is_string_like_v<Cell>,
"write_aligned requires scalar cells");
write_aligned_matrix(matrix);
}
template <class Matrix>
void println_aligned(const Matrix& matrix) {
write_aligned(matrix);
write_char('\n');
}
template <class... Args>
void println(const Args&... args) {
print(args...);
write_char('\n');
}
template <class T>
FastOutput& operator<<(const T& value) {
write(value);
return *this;
}
};
} // namespace utilities
} // namespace m1une
#line 7 "verify/geometry/circle_coverage_areas.test.cpp"
#line 13 "verify/geometry/circle_coverage_areas.test.cpp"
#include <numeric>
#line 15 "verify/geometry/circle_coverage_areas.test.cpp"
namespace {
using m1une::geometry::Circle;
using m1une::geometry::Point;
using m1une::geometry::circle_area;
using m1une::geometry::circle_circle_intersection_area;
using m1une::geometry::circle_coverage_areas;
using m1une::geometry::circle_union_area;
bool close(long double first, long double second, long double eps = 1e-9L) {
return std::fabs(first - second) <=
eps * std::max({1.0L, std::fabs(first), std::fabs(second)});
}
void test_edge_cases() {
const long double pi = std::numbers::pi_v<long double>;
std::vector<Circle<long double>> circles;
assert(circle_coverage_areas(circles) == std::vector<long double>(1, 0));
circles.push_back(Circle<long double>{Point<long double>(0, 0), 3});
circles.push_back(Circle<long double>{Point<long double>(0, 0), 3});
circles.push_back(Circle<long double>{Point<long double>(1, 0), 1});
circles.push_back(Circle<long double>{Point<long double>(0, 0), 0});
std::vector<long double> area = circle_coverage_areas(circles);
assert(area.size() == 5);
assert(close(area[1], 0));
assert(close(area[2], 8.0L * pi));
assert(close(area[3], pi));
assert(close(area[4], 0));
circles.clear();
circles.push_back(Circle<long double>{Point<long double>(0, 0), 2});
circles.push_back(Circle<long double>{Point<long double>(4, 0), 2});
area = circle_coverage_areas(circles);
assert(close(area[1], 8.0L * pi));
assert(close(area[2], 0));
}
void test_two_circles_randomized() {
std::uint64_t state = 1618033988ULL;
auto random = [&]() {
state ^= state << 7;
state ^= state >> 9;
return state;
};
for (int trial = 0; trial < 5000; ++trial) {
Circle<long double> first;
Circle<long double> second;
first.center = Point<long double>(
static_cast<long double>(int(random() % 41) - 20) / 4.0L,
static_cast<long double>(int(random() % 41) - 20) / 4.0L
);
second.center = Point<long double>(
static_cast<long double>(int(random() % 41) - 20) / 4.0L,
static_cast<long double>(int(random() % 41) - 20) / 4.0L
);
first.radius = static_cast<long double>(random() % 20 + 1) / 4.0L;
second.radius = static_cast<long double>(random() % 20 + 1) / 4.0L;
std::vector<Circle<long double>> circles = {first, second};
std::vector<long double> area = circle_coverage_areas(circles);
long double intersection =
circle_circle_intersection_area(first, second);
assert(close(area[2], intersection));
assert(close(
area[1],
circle_area(first) + circle_area(second) - 2.0L * intersection
));
}
}
void test_randomized_invariants() {
std::uint64_t state = 5772156649ULL;
auto random = [&]() {
state ^= state << 7;
state ^= state >> 9;
return state;
};
for (int trial = 0; trial < 1000; ++trial) {
int count = int(random() % 9);
std::vector<Circle<long double>> circles(count);
long double area_with_multiplicity = 0.0L;
for (Circle<long double>& circle : circles) {
circle.center = Point<long double>(
static_cast<long double>(int(random() % 25) - 12) / 3.0L,
static_cast<long double>(int(random() % 25) - 12) / 3.0L
);
circle.radius =
static_cast<long double>(random() % 13) / 3.0L;
area_with_multiplicity += circle_area(circle);
}
std::vector<long double> area = circle_coverage_areas(circles);
long double union_area =
std::accumulate(area.begin() + 1, area.end(), 0.0L);
long double weighted_area = 0.0L;
for (int coverage = 1; coverage <= count; ++coverage) {
weighted_area += coverage * area[coverage];
}
assert(close(union_area, circle_union_area(circles), 1e-8L));
assert(close(weighted_area, area_with_multiplicity, 1e-8L));
}
}
} // namespace
int main() {
test_edge_cases();
test_two_circles_randomized();
test_randomized_invariants();
m1une::utilities::FastInput input;
m1une::utilities::FastOutput output;
long long first = 0, second = 0;
input.read(first, second);
output.println(first + second);
}
verify/geometry/circle_coverage_areas.test.cpp