Geometry: Point
(geometry/point.hpp)

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• Last update: 2026-10-22 22:57:05+09:00
• Include: #include "geometry/point.hpp"

概要

• 2 次元の点。

使い方

• geo::point<T>: 座標の型 T。整数型と浮動小数点型を使える。
• geo::eps: 浮動小数点型の比較に使う誤差。初期値 $10^{-9}$。
• sgn(a): a の符号を返す。浮動小数点型では eps 以下を $0$ とみなす。
• dot(p, q): 内積を返す。
• cross(p, q): 外積 $p_x q_y - p_y q_x$ を返す。
• norm(p): $\lvert p \rvert^2$ を返す。
• abs(p): $\lvert p \rvert$ を返す。浮動小数点型のみ。
• arg(p): 偏角 $\mathrm{atan2}(p_y, p_x)$ を返す。浮動小数点型のみ。
• normal(p): p を $90^\circ$ 回転したベクトル $(-p_y, p_x)$ を返す。
• unit(p): p と同じ向きの単位ベクトルを返す。浮動小数点型のみ。
• rotate(p, theta): p を原点中心に theta 回転した点を返す。浮動小数点型のみ。
• ccw(a, b, c): cross(b - a, c - a) の符号を返す。
• arg_less: 偏角 $(-\pi, \pi]$ の昇順に並べる比較関数。原点の偏角は $0$。

制約

• 整数型では $\lvert x \rvert, \lvert y \rvert \le 10^9$

問題例

Library Checker - Sort Points by Argument（問題）

std::vector<geo::point<long long>> ps(n);
for (auto& p : ps) std::cin >> p;
std::sort(ps.begin(), ps.end(), geo::arg_less{});

Required by

• Geometry: Circle (geometry/circle.hpp)
• Geometry: Line (geometry/line.hpp)
• Geometry: Polygon (geometry/polygon.hpp)

Verified with

• test/aoj/CGL_1_A.test.cpp
• test/aoj/CGL_1_B.test.cpp
• test/aoj/CGL_1_C.test.cpp
• test/aoj/CGL_2_A.test.cpp
• test/aoj/CGL_2_B.test.cpp
• test/aoj/CGL_2_C.test.cpp
• test/aoj/CGL_2_D.test.cpp
• test/aoj/CGL_3_A.test.cpp
• test/aoj/CGL_3_B.test.cpp
• test/aoj/CGL_3_C.test.cpp
• test/aoj/CGL_4_A.test.cpp
• test/aoj/CGL_4_B.test.cpp
• test/aoj/CGL_4_C.test.cpp
• test/aoj/CGL_5_A.test.cpp
• test/aoj/CGL_7_A.test.cpp
• test/aoj/CGL_7_B.test.cpp
• test/aoj/CGL_7_C.test.cpp
• test/aoj/CGL_7_D.test.cpp
• test/aoj/CGL_7_E.test.cpp
• test/aoj/CGL_7_F.test.cpp
• test/aoj/CGL_7_G.test.cpp
• test/aoj/CGL_7_H.test.cpp
• test/aoj/CGL_7_I.test.cpp
• test/library-checker/closest_pair.test.cpp
• test/library-checker/sort_points_by_argument.test.cpp
• test/library-checker/static_convex_hull.test.cpp
• test/unit/geometry.test.cpp

Code

#pragma once
#include <cmath>
#include <concepts>
#include <iostream>
#include <type_traits>
namespace geo {
inline long double eps = 1e-9;
template <class T> int sgn(T a) {
    if constexpr (std::is_integral_v<T>) {
        return (a > 0) - (a < 0);
    } else {
        return a < -eps ? -1 : a > eps ? 1 : 0;
    }
}
template <class T> struct point {
    T x, y;
    point() : x(0), y(0) {
    }
    point(T x, T y) : x(x), y(y) {
    }
    point& operator+=(const point& p) {
        x += p.x, y += p.y;
        return *this;
    }
    point& operator-=(const point& p) {
        x -= p.x, y -= p.y;
        return *this;
    }
    point& operator*=(T k) {
        x *= k, y *= k;
        return *this;
    }
    point& operator/=(T k) requires std::floating_point<T> {
        x /= k, y /= k;
        return *this;
    }
    point operator-() const {
        return point(-x, -y);
    }
    friend point operator+(point a, const point& b) {
        return a += b;
    }
    friend point operator-(point a, const point& b) {
        return a -= b;
    }
    friend point operator*(point a, T k) {
        return a *= k;
    }
    friend point operator*(T k, point a) {
        return a *= k;
    }
    friend point operator/(point a, T k) requires std::floating_point<T> {
        return a /= k;
    }
    friend bool operator==(const point& a, const point& b) {
        return sgn(a.x - b.x) == 0 && sgn(a.y - b.y) == 0;
    }
    friend bool operator!=(const point& a, const point& b) {
        return !(a == b);
    }
    friend bool operator<(const point& a, const point& b) {
        if (sgn(a.x - b.x) != 0) return a.x < b.x;
        return sgn(a.y - b.y) < 0;
    }
    friend bool operator>(const point& a, const point& b) {
        return b < a;
    }
    friend std::istream& operator>>(std::istream& is, point& p) {
        return is >> p.x >> p.y;
    }
    friend std::ostream& operator<<(std::ostream& os, const point& p) {
        return os << p.x << ' ' << p.y;
    }
};
template <class T> T dot(const point<T>& a, const point<T>& b) {
    return a.x * b.x + a.y * b.y;
}
template <class T> T cross(const point<T>& a, const point<T>& b) {
    return a.x * b.y - a.y * b.x;
}
template <class T> T norm(const point<T>& p) {
    return p.x * p.x + p.y * p.y;
}
template <std::floating_point T> T abs(const point<T>& p) {
    return std::sqrt(norm(p));
}
template <std::floating_point T> T arg(const point<T>& p) {
    return std::atan2(p.y, p.x);
}
template <class T> point<T> normal(const point<T>& p) {
    return point<T>(-p.y, p.x);
}
template <std::floating_point T> point<T> unit(const point<T>& p) {
    return p / abs(p);
}
template <std::floating_point T> point<T> rotate(const point<T>& p, T theta) {
    T c = std::cos(theta), s = std::sin(theta);
    return point<T>(p.x * c - p.y * s, p.x * s + p.y * c);
}
template <class T> int ccw(const point<T>& a, const point<T>& b, const point<T>& c) {
    return sgn(cross(b - a, c - a));
}
struct arg_less {
    template <class T> static int group(const point<T>& p) {
        if (sgn(p.y) < 0) return 0;
        if (sgn(p.y) == 0 && sgn(p.x) >= 0) return 1;
        return 2;
    }
    template <class T> bool operator()(const point<T>& a, const point<T>& b) const {
        if (group(a) != group(b)) return group(a) < group(b);
        return sgn(cross(a, b)) > 0;
    }
};
}  // namespace geo

#line 2 "geometry/point.hpp"
#include <cmath>
#include <concepts>
#include <iostream>
#include <type_traits>
namespace geo {
inline long double eps = 1e-9;
template <class T> int sgn(T a) {
    if constexpr (std::is_integral_v<T>) {
        return (a > 0) - (a < 0);
    } else {
        return a < -eps ? -1 : a > eps ? 1 : 0;
    }
}
template <class T> struct point {
    T x, y;
    point() : x(0), y(0) {
    }
    point(T x, T y) : x(x), y(y) {
    }
    point& operator+=(const point& p) {
        x += p.x, y += p.y;
        return *this;
    }
    point& operator-=(const point& p) {
        x -= p.x, y -= p.y;
        return *this;
    }
    point& operator*=(T k) {
        x *= k, y *= k;
        return *this;
    }
    point& operator/=(T k) requires std::floating_point<T> {
        x /= k, y /= k;
        return *this;
    }
    point operator-() const {
        return point(-x, -y);
    }
    friend point operator+(point a, const point& b) {
        return a += b;
    }
    friend point operator-(point a, const point& b) {
        return a -= b;
    }
    friend point operator*(point a, T k) {
        return a *= k;
    }
    friend point operator*(T k, point a) {
        return a *= k;
    }
    friend point operator/(point a, T k) requires std::floating_point<T> {
        return a /= k;
    }
    friend bool operator==(const point& a, const point& b) {
        return sgn(a.x - b.x) == 0 && sgn(a.y - b.y) == 0;
    }
    friend bool operator!=(const point& a, const point& b) {
        return !(a == b);
    }
    friend bool operator<(const point& a, const point& b) {
        if (sgn(a.x - b.x) != 0) return a.x < b.x;
        return sgn(a.y - b.y) < 0;
    }
    friend bool operator>(const point& a, const point& b) {
        return b < a;
    }
    friend std::istream& operator>>(std::istream& is, point& p) {
        return is >> p.x >> p.y;
    }
    friend std::ostream& operator<<(std::ostream& os, const point& p) {
        return os << p.x << ' ' << p.y;
    }
};
template <class T> T dot(const point<T>& a, const point<T>& b) {
    return a.x * b.x + a.y * b.y;
}
template <class T> T cross(const point<T>& a, const point<T>& b) {
    return a.x * b.y - a.y * b.x;
}
template <class T> T norm(const point<T>& p) {
    return p.x * p.x + p.y * p.y;
}
template <std::floating_point T> T abs(const point<T>& p) {
    return std::sqrt(norm(p));
}
template <std::floating_point T> T arg(const point<T>& p) {
    return std::atan2(p.y, p.x);
}
template <class T> point<T> normal(const point<T>& p) {
    return point<T>(-p.y, p.x);
}
template <std::floating_point T> point<T> unit(const point<T>& p) {
    return p / abs(p);
}
template <std::floating_point T> point<T> rotate(const point<T>& p, T theta) {
    T c = std::cos(theta), s = std::sin(theta);
    return point<T>(p.x * c - p.y * s, p.x * s + p.y * c);
}
template <class T> int ccw(const point<T>& a, const point<T>& b, const point<T>& c) {
    return sgn(cross(b - a, c - a));
}
struct arg_less {
    template <class T> static int group(const point<T>& p) {
        if (sgn(p.y) < 0) return 0;
        if (sgn(p.y) == 0 && sgn(p.x) >= 0) return 1;
        return 2;
    }
    template <class T> bool operator()(const point<T>& a, const point<T>& b) const {
        if (group(a) != group(b)) return group(a) < group(b);
        return sgn(cross(a, b)) > 0;
    }
};
}  // namespace geo

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