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:heavy_check_mark: Src/GeometryR2/CommonArea/CircleAndPolygon.hpp

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#pragma once

#include "../Circle.hpp"
#include "../Polygon.hpp"
#include "../CrossPoint/CircleAndSegment.hpp"
#include "../Distance/PointAndPoint.hpp"

namespace zawa {

namespace geometryR2 {

Real CommonArea(const Circle& c, const Polygon& poly) {
    auto theta = [&](const Point& p, const Point& q) -> Real {
        const Real cosT = (p.normSquare() + q.normSquare() - DistanceSquare(p, q)) / (2 * p.norm() * q.norm());
        return acosl(cosT);
    };
    // signed common area of c and triangle(c.center(), p, q)
    auto common_area = [&](Point p, Point q) -> Real {
        p -= c.center();
        q -= c.center(); 
        if (Zero(Cross(p, q))) return 0;
        const bool pin = Smaller(p.normSquare(), Square(c.radius())), qin = Smaller(q.normSquare(), Square(c.radius()));
        if (pin and qin) return Cross(p, q) / 2;
        const auto cps = CrossPoint(c, Segment{p, q});
        if (cps.empty()) return Sign(Cross(p, q)) * c.sectorArea(theta(p, q));
        Real res = 0;
        if (pin) res += Cross(p, cps.front()) / 2;
        else {
            const Vector dir = p.normalized();
            const Point v = c.center() + dir * c.radius();
            res += Sign(Cross(v, cps.front())) * c.sectorArea(theta(v, cps.front()));
        }
        if (qin) res += Cross(cps.back(), q) / 2;
        else {
            const Vector dir = q.normalized();
            const Point v = c.center() + dir * c.radius();
            res += Sign(Cross(cps.back(), v)) * c.sectorArea(theta(cps.back(), v));
        }
        if (std::ssize(cps) == 2) res += Cross(cps[0], cps[1]) / 2;
        return res;
    };
    Real res = 0;
    for (usize i = 0 ; i < poly.size() ; i++) {
        const Real kiyo = common_area(poly[i], poly[i+1==poly.size()?0u:i+1]);
        res += kiyo;
    }
    return res;
}

} // namespace geometryR2

} // namespace zawa
#line 2 "Src/GeometryR2/CommonArea/CircleAndPolygon.hpp"

#line 2 "Src/GeometryR2/Circle.hpp"

#line 2 "Src/Template/TypeAlias.hpp"

#include <cstdint>
#include <cstddef>

namespace zawa {

using i16 = std::int16_t;
using i32 = std::int32_t;
using i64 = std::int64_t;
using i128 = __int128_t;

using u8 = std::uint8_t;
using u16 = std::uint16_t;
using u32 = std::uint32_t;
using u64 = std::uint64_t;

using usize = std::size_t;

} // namespace zawa
#line 2 "Src/GeometryR2/Real.hpp"

#line 4 "Src/GeometryR2/Real.hpp"

#include <cmath>
#include <cassert>

namespace zawa {

namespace geometryR2 {

using Real = long double;

namespace internal {

Real EPS{1e-12};
constexpr i32 negative{-1};
constexpr i32 zero{};
constexpr i32 positive{1};

} // namespace internal

Real& Eps() {
    return internal::EPS;
}

i32 Sign(Real value) {
    if (value < -Eps()) return internal::negative;
    if (value > Eps()) return internal::positive;
    return internal::zero;
}

bool Zero(Real value) {
    return Sign(value) == internal::zero;
}

bool Positive(Real value) {
    return Sign(value) == internal::positive;
}

bool Negative(Real value) {
    return Sign(value) == internal::negative;
}

bool Equal(Real a, Real b) {
    return Zero(a - b);
}

bool Smaller(Real a, Real b) {
    return Negative(a - b);
}

bool Bigger(Real a, Real b) {
    return Positive(a - b);
}

Real Square(Real value) {
    return (Zero(value) ? value : value * value);
}

Real Sqrt(Real value) {
    assert(!Negative(value));
    return (Zero(value) ? value : sqrtl(value));
}

Real Abs(Real value) {
    return (Negative(value) ? -value : value);
}

} // namespace geometryR2
 
} // namespace zawa
#line 2 "Src/GeometryR2/Angle.hpp"

#line 4 "Src/GeometryR2/Angle.hpp"

#line 6 "Src/GeometryR2/Angle.hpp"

namespace zawa {

namespace geometryR2 {

constexpr Real PI{acosl(-1)};
constexpr Real TAU{static_cast<Real>(2) * PI};

constexpr Real ArcToRadian(Real arc) {
    return (arc * PI) / static_cast<Real>(180);
}

constexpr Real RadianToArc(Real radian) {
    return (radian * static_cast<Real>(180)) / PI;
}

} // namespace geometryR2

} // namespace zawa
#line 2 "Src/GeometryR2/Point.hpp"

#line 5 "Src/GeometryR2/Point.hpp"

#line 7 "Src/GeometryR2/Point.hpp"
#include <iostream>
#line 9 "Src/GeometryR2/Point.hpp"

namespace zawa {

namespace geometryR2 {

class Point {
private:
    Real x_{}, y_{};
public:
    /* constructor */
    Point() = default;
    Point(Real x, Real y) : x_{x}, y_{y} {}

    /* getter, setter */
    Real x() const {
        return x_;
    }
    Real& x() {
        return x_;
    }
    Real y() const {
        return y_;
    }
    Real& y() {
        return y_;
    }

    /* operator */
    Point& operator+=(const Point& rhs) {
        x_ += rhs.x();
        y_ += rhs.y();
        return *this;
    }
    friend Point operator+(const Point& lhs, const Point& rhs) {
        return Point{lhs} += rhs;
    }
    Point operator+() const {
        return *this;
    }
    Point& operator-=(const Point& rhs) {
        x_ -= rhs.x();
        y_ -= rhs.y();
        return *this;
    }
    friend Point operator-(const Point& lhs, const Point& rhs) {
        return Point{lhs} -= rhs;
    }
    Point operator-() const {
        return Point{} - *this;
    }
    Point& operator*=(Real k) {
        x_ *= k;
        y_ *= k;
        return *this;
    }
    friend Point operator*(Real k, const Point& p) {
        return Point{p} *= k;
    }
    friend Point operator*(const Point& p, Real k) {
        return Point{p} *= k;
    }
    Point& operator/=(Real k) {
        assert(!Zero(k));
        x_ /= k;
        y_ /= k;
        return *this;
    }
    friend Point operator/(Real k, const Point& p) {
        return Point{p} /= k;
    }
    friend Point operator/(const Point& p, Real k) {
        return Point{p} /= k;
    }
    friend bool operator==(const Point& lhs, const Point& rhs) {
        return Equal(lhs.x(), rhs.x()) and Equal(lhs.y(), rhs.y());
    }
    friend bool operator!=(const Point& lhs, const Point& rhs) {
        return !Equal(lhs.x(), rhs.x()) or !Equal(lhs.y(), rhs.y());
    }
    friend bool operator<(const Point& lhs, const Point& rhs) {
        return Smaller(lhs.x(), rhs.x()) or 
            (Equal(lhs.x(), rhs.x()) and Smaller(lhs.y(), rhs.y()));
    }
    friend bool operator<=(const Point& lhs, const Point& rhs) {
        return Smaller(lhs.x(), rhs.x()) or 
            (Equal(lhs.x(), rhs.x()) and (Smaller(lhs.y(), rhs.y()) or Equal(lhs.y(), rhs.y())));
    }
    friend bool operator>(const Point& lhs, const Point& rhs) {
        return Bigger(lhs.x(), rhs.x()) or
            (Equal(lhs.x(), rhs.x()) and Bigger(lhs.y(), rhs.y()));
    }
    friend bool operator>=(const Point& lhs, const Point& rhs) {
        return Bigger(lhs.x(), rhs.x()) or
            (Equal(lhs.x(), rhs.x()) and (Bigger(lhs.y(), rhs.y()) or Equal(lhs.y(), rhs.y())));
    }
    friend std::istream& operator>>(std::istream& is, Point& p) {
        is >> p.x_ >> p.y_;
        return is;
    }
    friend std::ostream& operator<<(std::ostream& os, const Point& p) {
        os << '(' << p.x_ << ',' << p.y_ << ')';
        return os;
    }
    
    /* member function */
    Real normSquare() const {
        return Square(x_) + Square(y_);
    }
    Real norm() const {
        return Sqrt(normSquare());
    }
    void normalize() {
        assert((*this) != Point{});
        (*this) /= norm(); 
    }
    Point normalized() const {
        Point res{*this};
        res.normalize();
        return res;
    }
    Point rotated(Real radian) const {
        return Point{
            x_ * cosl(radian) - y_ * sinl(radian),
            x_ * sinl(radian) + y_ * cosl(radian)
        };
    }
    void rotate(Real radian) {
        *this = rotated(radian); 
    }
    Point rotatedByArc(Real arc) const {
        return rotated(ArcToRadian(arc));
    }
    void rotateByArc(Real arc) {
        *this = rotatedByArc(arc);
    }
    Real argument() const {
        return (Negative(y_) ? TAU : static_cast<Real>(0)) + atan2l(y_, x_);
    }
    Real argumentByArc() const {
        return RadianToArc(argument());
    }

    /* friend function */
    friend Real Dot(const Point& lhs, const Point& rhs) {
        return lhs.x() * rhs.x() + lhs.y() * rhs.y();
    }
    friend Real Cross(const Point& lhs, const Point& rhs) {
        return lhs.x() * rhs.y() - lhs.y() * rhs.x();
    }
    friend Real Argument(const Point& lhs, const Point& rhs) {
        return rhs.argument() - lhs.argument();
    }
    friend bool ArgComp(const Point& lhs, const Point& rhs) {
        return Smaller(lhs.argument(), rhs.argument());
    }
};

using Vector = Point;

} // namespace geometryR2

} // namespace zawa
#line 2 "Src/GeometryR2/Distance/PointAndPoint.hpp"

#line 4 "Src/GeometryR2/Distance/PointAndPoint.hpp"

namespace zawa {

namespace geometryR2 {

Real Distance(const Point& p0, const Point& p1) {
    return Point{p1 - p0}.norm();
}

Real DistanceSquare(const Point& p0, const Point& p1) {
    return Point{p1 - p0}.normSquare();
}

} // namespace geometryR2

} // namespace zawa
#line 8 "Src/GeometryR2/Circle.hpp"

#line 10 "Src/GeometryR2/Circle.hpp"
#include <utility>

namespace zawa {

namespace geometryR2 {

class Circle {
private:
    Point center_{};
    Real radius_{};
public:
    /* constructor */
    Circle() = default;
    Circle(const Point& center, Real radius) : center_{center}, radius_{radius} {
        assert(!Negative(radius));
    }
    Circle(Real x, Real y, Real r) : center_{x, y}, radius_{r} {
        assert(!Negative(r));
    }

    Circle(const Point& p0, const Point& p1) : center_{p0 + (p1 - p0) / 2}, radius_{Distance(p0, p1) / 2} {}

    /* getter setter */
    const Point& center() const {
        return center_;
    }
    Point& center() {
        return center_;
    }
    Real radius() const {
        return radius_;
    }
    Real& radius() {
        return radius_;
    }
    Real area() const {
        return PI * Square(radius_);
    }
    Real sectorArea(Real centerAngle) const {
        return Square(radius_) * centerAngle / 2;
    }

    /* operator */
    friend bool operator==(const Circle& lhs, const Circle& rhs) {
        return lhs.center() == rhs.center() and Equal(lhs.radius(), rhs.radius());
    }
    friend bool operator!=(const Circle& lhs, const Circle& rhs) {
        return lhs.center() != rhs.center() or !Equal(lhs.radius(), rhs.radius());
    }

    /* friend function */
    friend u32 NumberCommonTangent(const Circle& c0, const Circle& c1) {
        Real dist{DistanceSquare(c0.center(), c1.center())};
        Real down{Square(Abs(c0.radius() - c1.radius()))};
        if (Smaller(dist, down)) return 0;
        if (Equal(dist, down)) return 1;
        Real up{Square(c0.radius() + c1.radius())};
        if (Smaller(dist, up)) return 2;
        if (Equal(dist, up)) return 3;
        return 4;
    }
};

} // namespace geometryR2

} // namespace zawa
#line 2 "Src/GeometryR2/Polygon.hpp"

#line 2 "Src/GeometryR2/Relation.hpp"

#line 5 "Src/GeometryR2/Relation.hpp"

namespace zawa {

namespace geometryR2 {

enum RELATION {
    // p0 -> p1 -> p2の順で直線上に並んでいる
    ONLINE_FRONT = -2,
    // (p1 - p0) -> (p2 - p0)が時計回りになっている
    CLOCKWISE,
    // p0 -> p2 -> p1の順で直線上に並んでいる
    ON_SEGMENT,
    // (p1 - p0) -> (p2 - p0)が反時計回りになっている
    COUNTER_CLOCKWISE,
    // p2 -> p0 -> p1、またはp1 -> p0 -> p2の順で直線上に並んでいる
    ONLINE_BACK
};

RELATION Relation(const Point& p0, const Point& p1, const Point& p2) {
    Point a{p1 - p0}, b{p2 - p0};
    if (Positive(Cross(a, b))) return COUNTER_CLOCKWISE;
    if (Negative(Cross(a, b))) return CLOCKWISE;
    if (Negative(Dot(a, b))) return ONLINE_BACK;
    if (Smaller(a.normSquare(), b.normSquare())) return ONLINE_FRONT;
    return ON_SEGMENT;
};

} // namespace geometryR2

} // namespace zawa
#line 7 "Src/GeometryR2/Polygon.hpp"

#include <algorithm>
#line 10 "Src/GeometryR2/Polygon.hpp"
#include <concepts>
#include <vector>

namespace zawa {

namespace geometryR2 {

class Polygon {
private:
    std::vector<Point> data_;
public:
    /* member */
    usize size() const {
        return data_.size();
    }

    /* constructor */
    Polygon() = default;
    explicit Polygon(const std::vector<Point>& data) : data_{data} {}
    explicit Polygon(usize n) : data_(n) {}

    /* operator[] */
    Point& operator[](usize i) {
        assert(i < size());
        return data_[i];
    }
    const Point& operator[](usize i) const {
        assert(i < size());
        return data_[i];
    }
    friend std::istream& operator>>(std::istream& is, Polygon& polygon) {
        for (size_t i{} ; i < polygon.size() ; i++) {
            is >> polygon[i];
        }
        return is;
    }
    friend std::ostream& operator<<(std::ostream& os, const Polygon& polygon) {
        for (usize i{} ; i < polygon.size() ; i++) {
            std::cout << polygon[i] << (i + 1 == polygon.size() ? "" : " ");
        }
        return os;
    }

    /* member function */
    void orderRotate(usize i) {
        assert(i < size());
        std::rotate(data_.begin(), data_.begin() + i, data_.end());
    }
    void normalForm() {
        auto index{std::distance(data_.begin(), std::min_element(data_.begin(), data_.end()))};
        orderRotate(index);
    }
    Polygon normalFormed() const {
        Polygon res{*this};
        res.normalForm();
        return res;
    }
    bool isConvex() const {
        assert(size() >= static_cast<usize>(3));
        for (usize i{} ; i < size() ; i++) {
            if (Relation(data_[i], data_[i+1==size()?0:i+1], data_[i+2>=size()?i+2-size():i+2])
                    == CLOCKWISE) {
                return false;
            }
        }
        return true;
    }
    Real area() const {
        if (std::ssize(data_) <= 2) return 0;
        Real res{};
        for (usize i{1} ; i < size() ; i++) {
            res += Cross(data_[i] - data_[0], data_[i+1==size()?0:i+1] - data_[0]);
        }
        return res / static_cast<Real>(2);
    }
    void pushBack(const Point& p) {
        data_.push_back(p);
    }
    void emplaceBack(Real x, Real y) {
        data_.emplace_back(x, y);
    }
    void reserve(usize n) {
        data_.reserve(n);
    }
    template <std::input_iterator RandomAccessIterator>
    void insert(usize n, RandomAccessIterator first, RandomAccessIterator last) {
        assert(n <= size());
        data_.insert(std::next(data_.begin(), n), first, last);
    }
};

} // namespace geometryR2

} // namespace zawa

#line 2 "Src/GeometryR2/CrossPoint/CircleAndSegment.hpp"

#line 2 "Src/GeometryR2/Segment.hpp"

#line 6 "Src/GeometryR2/Segment.hpp"

#line 9 "Src/GeometryR2/Segment.hpp"

namespace zawa {

namespace geometryR2 {

class Segment {
private:
    Point p0_{}, p1_{};
public:
    /* constructor */
    Segment() = default;
    Segment(const Point& p0, const Point& p1) : p0_{p0}, p1_{p1} {}
    Segment(Real x0, Real y0, Real x1, Real y1) : p0_{x0, y0}, p1_{x1, y1} {}

    /* getter setter */
    const Point& p0() const {
        return p0_;
    }
    Point& p0() {
        return p0_;
    }
    const Point& p1() const {
        return p1_;
    }
    Point& p1() {
        return p1_;
    }

    /* member function */
    bool valid() const {
        return p0_ != p1_;
    }
    bool straddle(const Segment& s) const {
        return Relation(p0_, p1_, s.p0()) * Relation(p0_, p1_, s.p1()) <= 0;
    }
    Real length() const {
        assert(valid());
        return Distance(p0_, p1_);
    }
    Point midpoint() const {
        assert(valid());
        return p0_ + Vector{p1_ - p0_} / static_cast<Real>(2);
    }
};

} // namespace geometryR2

} // namespace zawa
#line 2 "Src/GeometryR2/Line.hpp"

#line 5 "Src/GeometryR2/Line.hpp"

#line 7 "Src/GeometryR2/Line.hpp"

namespace zawa {

namespace geometryR2 {

class Line {
private:
    Point p0_{}, p1_{};
public:
    /* constructor */
    Line() = default;
    Line(const Point& p0, const Point& p1) : p0_{p0}, p1_{p1} {}
    // y = ax + b 
    Line(Real a, Real b) : p0_{static_cast<Real>(0), b}, p1_{static_cast<Real>(1), a + b} {}

    /* getter, setter */
    const Point& p0() const {
        return p0_;
    }
    Point& p0() {
        return p0_;
    }
    const Point& p1() const {
        return p1_;
    }
    Point& p1() {
        return p1_;
    }

    /* operator */
    friend bool operator==(const Line& l0, const Line& l1) {
        return Zero(Cross(l0.p1() - l0.p0(), l1.p1() - l1.p0())) and Zero(Cross(l0.p1() - l0.p0(), l1.p1() - l0.p0()));
    }
    friend bool operator!=(const Line& l0, const Line& l1) {
        return !Zero(Cross(l0.p1() - l0.p0(), l1.p1() - l1.p0())) or !Zero(Cross(l0.p1() - l0.p0(), l1.p1() - l0.p0()));
    }

    /* member function */
    bool valid() const {
        return p0_ != p1_;
    }
    Vector slope() const {
        assert(valid());
        return Vector{p1() - p0()}.normalized();
    }
};

} // namespace geometryR2

} // namespace zawa
#line 2 "Src/GeometryR2/Projection.hpp"

#line 6 "Src/GeometryR2/Projection.hpp"

#line 8 "Src/GeometryR2/Projection.hpp"

namespace zawa {

namespace geometryR2 {

Point Projection(const Point& point, const Line& line) {
    assert(line.valid());
    Real coeff{Dot(line.p1() - line.p0(), point - line.p0()) / DistanceSquare(line.p0(), line.p1())};
    return coeff * line.p1() + (static_cast<Real>(1) - coeff) * line.p0();
}

} // namespace geometryR2

} // namespace zawa
#line 10 "Src/GeometryR2/CrossPoint/CircleAndSegment.hpp"

#line 14 "Src/GeometryR2/CrossPoint/CircleAndSegment.hpp"

namespace zawa {

namespace geometryR2 {

std::vector<Point> CrossPoint(const Circle& c, const Segment& s) {
    assert(s.valid());
    Point pr{Projection(c.center(), Line{s.p0(), s.p1()})};
    if (Bigger(DistanceSquare(c.center(), pr), Square(c.radius()))) return {};
    const Vector e = (s.p1() - s.p0()) / Distance(s.p0(), s.p1());
    const Real l = Sqrt(
            Square(c.radius()) - DistanceSquare(pr, c.center())
            );
    std::vector<Point> res;
    // s0に近い方からresに入る
    for (Point p : {pr - e * l, pr + e * l}) {
        if (Relation(s.p0(), s.p1(), p) == RELATION::ON_SEGMENT) {
            res.push_back(p);
        } 
    }
    return res;
}

} // namespace geometryR2

} // namespace zawa
#line 7 "Src/GeometryR2/CommonArea/CircleAndPolygon.hpp"

namespace zawa {

namespace geometryR2 {

Real CommonArea(const Circle& c, const Polygon& poly) {
    auto theta = [&](const Point& p, const Point& q) -> Real {
        const Real cosT = (p.normSquare() + q.normSquare() - DistanceSquare(p, q)) / (2 * p.norm() * q.norm());
        return acosl(cosT);
    };
    // signed common area of c and triangle(c.center(), p, q)
    auto common_area = [&](Point p, Point q) -> Real {
        p -= c.center();
        q -= c.center(); 
        if (Zero(Cross(p, q))) return 0;
        const bool pin = Smaller(p.normSquare(), Square(c.radius())), qin = Smaller(q.normSquare(), Square(c.radius()));
        if (pin and qin) return Cross(p, q) / 2;
        const auto cps = CrossPoint(c, Segment{p, q});
        if (cps.empty()) return Sign(Cross(p, q)) * c.sectorArea(theta(p, q));
        Real res = 0;
        if (pin) res += Cross(p, cps.front()) / 2;
        else {
            const Vector dir = p.normalized();
            const Point v = c.center() + dir * c.radius();
            res += Sign(Cross(v, cps.front())) * c.sectorArea(theta(v, cps.front()));
        }
        if (qin) res += Cross(cps.back(), q) / 2;
        else {
            const Vector dir = q.normalized();
            const Point v = c.center() + dir * c.radius();
            res += Sign(Cross(cps.back(), v)) * c.sectorArea(theta(cps.back(), v));
        }
        if (std::ssize(cps) == 2) res += Cross(cps[0], cps[1]) / 2;
        return res;
    };
    Real res = 0;
    for (usize i = 0 ; i < poly.size() ; i++) {
        const Real kiyo = common_area(poly[i], poly[i+1==poly.size()?0u:i+1]);
        res += kiyo;
    }
    return res;
}

} // namespace geometryR2

} // namespace zawa
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