Convex Hull
(geometry/convex_hull.hpp)

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• Last update: 2024-01-08 01:08:59+09:00
• Include: #include "geometry/convex_hull.hpp"

Description

与えられた点の凸包を求める．この実装では Graham scan アルゴリズムを用いている．

Operations

• vector<Vec> convex_hull(vector<Vec> pts)
  • pts の凸包の境界の頂点を返す
  • 時間計算量: $O(n\log n)$

Depends on

• Geometry (geometry/geometry.hpp)

Verified with

• test/aoj/1298.test.cpp
• test/aoj/CGL_4_A.test.cpp
• test/aoj/CGL_4_B.test.cpp

Code

#pragma once
#include <vector>

#include "geometry.hpp"

std::vector<Vec> convex_hull(std::vector<Vec>& pts) {
    const int n = pts.size();
    if (n == 1) return pts;
    std::ranges::sort(pts, {}, [](const Vec& v) {
        return std::make_pair(v.imag(), v.real());
    });
    int k = 0;  // the number of vertices in the convex hull
    std::vector<Vec> ch(2 * n);
    // right
    for (int i = 0; i < n; ++i) {
        while (k > 1 && lt(cross(ch[k - 1] - ch[k - 2], pts[i] - ch[k - 1]), 0))
            --k;
        ch[k++] = pts[i];
    }
    int t = k;
    // left
    for (int i = n - 2; i >= 0; --i) {
        while (k > t && lt(cross(ch[k - 1] - ch[k - 2], pts[i] - ch[k - 1]), 0))
            --k;
        ch[k++] = pts[i];
    }
    ch.resize(k - 1);
    return ch;
}

#line 2 "geometry/convex_hull.hpp"
#include <vector>

#line 2 "geometry/geometry.hpp"
#include <algorithm>
#include <cassert>
#include <cmath>
#include <complex>
#include <iostream>
#include <numbers>
#include <numeric>
#line 10 "geometry/geometry.hpp"

// note that if T is of an integer type, std::abs does not work
using T = double;
using Vec = std::complex<T>;

std::istream& operator>>(std::istream& is, Vec& p) {
    T x, y;
    is >> x >> y;
    p = {x, y};
    return is;
}

T dot(const Vec& a, const Vec& b) { return (std::conj(a) * b).real(); }

T cross(const Vec& a, const Vec& b) { return (std::conj(a) * b).imag(); }

constexpr T PI = std::numbers::pi_v<T>;
constexpr T eps = 1e-10;
inline bool eq(T a, T b) { return std::abs(a - b) <= eps; }
inline bool eq(Vec a, Vec b) { return std::abs(a - b) <= eps; }
inline bool lt(T a, T b) { return a < b - eps; }
inline bool leq(T a, T b) { return a <= b + eps; }

struct Line {
    Vec p1, p2;
    Line() = default;
    Line(const Vec& p1, const Vec& p2) : p1(p1), p2(p2) {}
    Vec dir() const { return p2 - p1; }
};

struct Segment : Line {
    using Line::Line;
};

struct Circle {
    Vec c;
    T r;
    Circle() = default;
    Circle(const Vec& c, T r) : c(c), r(r) {}
};

using Polygon = std::vector<Vec>;

Vec rot(const Vec& a, T ang) { return a * Vec(std::cos(ang), std::sin(ang)); }

Vec perp(const Vec& a) { return Vec(-a.imag(), a.real()); }

Vec projection(const Line& l, const Vec& p) {
    return l.p1 + dot(p - l.p1, l.dir()) * l.dir() / std::norm(l.dir());
}

Vec reflection(const Line& l, const Vec& p) {
    return T(2) * projection(l, p) - p;
}

// 0: collinear
// 1: counter-clockwise
// -1: clockwise
int ccw(const Vec& a, const Vec& b, const Vec& c) {
    if (eq(cross(b - a, c - a), 0)) return 0;
    if (lt(cross(b - a, c - a), 0)) return -1;
    return 1;
}

void sort_by_arg(std::vector<Vec>& pts) {
    std::ranges::sort(pts, [&](auto& p, auto& q) {
        if ((p.imag() < 0) != (q.imag() < 0)) return (p.imag() < 0);
        if (cross(p, q) == 0) {
            if (p == Vec(0, 0))
                return !(q.imag() < 0 || (q.imag() == 0 && q.real() > 0));
            if (q == Vec(0, 0))
                return (p.imag() < 0 || (p.imag() == 0 && p.real() > 0));
            return (p.real() > q.real());
        }
        return (cross(p, q) > 0);
    });
}
#line 5 "geometry/convex_hull.hpp"

std::vector<Vec> convex_hull(std::vector<Vec>& pts) {
    const int n = pts.size();
    if (n == 1) return pts;
    std::ranges::sort(pts, {}, [](const Vec& v) {
        return std::make_pair(v.imag(), v.real());
    });
    int k = 0;  // the number of vertices in the convex hull
    std::vector<Vec> ch(2 * n);
    // right
    for (int i = 0; i < n; ++i) {
        while (k > 1 && lt(cross(ch[k - 1] - ch[k - 2], pts[i] - ch[k - 1]), 0))
            --k;
        ch[k++] = pts[i];
    }
    int t = k;
    // left
    for (int i = n - 2; i >= 0; --i) {
        while (k > t && lt(cross(ch[k - 1] - ch[k - 2], pts[i] - ch[k - 1]), 0))
            --k;
        ch[k++] = pts[i];
    }
    ch.resize(k - 1);
    return ch;
}

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