80. Convex Checker

题目大意

思路讲解

原先的凸性检测没法检测出这样子的五角星。

// 不允许自交的多边形，顺序都可以，会把 poly 变为 CCW 顺序（逆时针）
// AC https://qoj.ac/submission/2002121
bool isConvex(vector<Point> &poly) {
    ll n = poly.size();
    Point p1 = poly[1] - poly[0], p2 = poly[2] - poly[0];
    if (sgn(cross(p1, p2)) <= 0) {
        reverse(poly.begin(), poly.end());
    }
    // 找到字典序最小的点
    ll pos_mn = 0;
    for (int i = 1; i < n; ++i) {
        if (poly[i] < poly[pos_mn]) {
            pos_mn = i;
        }
    }
    // Performs a left rotation on a range of elements
    // 向左循环移动，把 pos_mn 移动到第一个 [0]
    rotate(poly.begin(), poly.begin() + pos_mn, poly.end());
    // 在一个标准的凸多边形中，如果我们站在最左下角的点（poly[0]）往外看
    // 其他所有顶点 poly[1], poly[2], ..., poly[n-1] 应该按照逆时针方向整齐排列。
    // 从而排除星形（自交）多边形和乱序点集。
    for (int i = 1; i < n - 1; ++i) {
        if (sgn(cross(poly[i] - poly[0], poly[i + 1] - poly[0])) <= 0) {
            return false;
        }
    }
    // 判断单个内角是否 ≥ 180。
    for (int i = 0; i < n; ++i) {
        ll i1 = (i + 1) % n, i2 = (i + 2) % n;
        Vector a = poly[i1] - poly[i], b = poly[i2] - poly[i1];
        // 加上大于等于，就相当于排除所有的退化情况，不允许内角为 180 度
        if (sgn(cross(a, b)) <= 0) {
            return false;
        }
    }
    return true;
}

AC代码

AC
https://qoj.ac/submission/2002045

AC https://qoj.ac/submission/2002100

AC https://qoj.ac/submission/2002121

源代码

/**
 * Problem: Convex Checker
 * Contest: QOJ
 * Judge: Virtual Judge
 * URL: https://vjudge.net/problem/QOJ-7730
 * Created: 2026-02-03 18:38:37
 * Author: Gospel_rock
 *
 * Powered by AutoCp https://github.com/Pushpavel/AutoCp
 */

#include <bits/stdc++.h>
#define all(vec) vec.begin(),vec.end()
#define lson(o) (o<<1)
#define rson(o) (o<<1|1)
#define SZ(a) ((long long) a.size())
#define debug(var) cerr << #var <<":"<<var<<"\n";
#define cend cerr<<"\n-----------\n"
#define fsp(x) fixed<<setprecision(x)

using namespace std;

using ll = long long;
using ull = unsigned long long;
using DB = double;
// using i128 = __int128;
using CD = complex<double>;

static constexpr ll MAXN = (ll) 1e6 + 10, INF = (1ll << 61) - 1;
static constexpr ll mod = 998244353; // (ll)1e9+7;
static constexpr double eps = 1e-8;
const double pi = acos(-1.0);

ll lT;

/*
 *
 */

/*2026-2-3 增加编译器识别 Real 类型的功能，以使用不同类型的这个实现
 *
 */
using Real = ll;

int sgn(auto x) {
    if (-eps < x && x < eps) return 0;
    return x < 0 ? -1 : 1;
}

struct Point {
    Real x, y;
    ll idx;

    explicit Point(Real x_ = 0, Real y_ = 0) : x(x_), y(y_) {
    }

    // 只对左值生效的复合运算符
    Point &operator+=(Point p) & {
        x += p.x, y += p.y;
        return *this;
    }

    Point &operator-=(Point p) & {
        x -= p.x, y -= p.y;
        return *this;
    }

    Point &operator*=(Real t) & {
        x *= t, y *= t;
        return *this;
    }

    Point &operator/=(Real t) & {
        x /= t, y /= t;
        return *this;
    }

    Point operator-() const { return Point(-x, -y); }

    friend Point operator+(Point a, Point b) { return a += b; }
    friend Point operator-(Point a, Point b) { return a -= b; }
    friend Point operator*(Point a, Real b) { return a *= b; }
    friend Point operator*(Real a, Point b) { return b *= a; }
    friend Point operator/(Point a, Real b) { return a /= b; }

    friend bool operator<(Point a, Point b) {
        int sx = sgn(a.x - b.x);
        if (sx != 0) return a.x < b.x;
        return a.y < b.y;
    }

    friend bool operator>(Point a, Point b) { return b < a; }

    friend bool operator==(Point a, Point b) {
        return sgn(a.x - b.x) == 0 && sgn(a.y - b.y) == 0;
    }

    friend bool operator!=(Point a, Point b) { return !(a == b); }

    friend istream &operator>>(istream &is, Point &p) { return is >> p.x >> p.y; }
    // friend ostream &operator<<(ostream &os, Point p) { return os << "(" << p.x << ", " << p.y << ")"; }

    auto norm() const {
        if constexpr (is_integral_v<Real>) {
            return (__int128_t) x * x + (__int128_t) y * y;
        } else {
            return x * x + y * y;
        }
    }

    DB abs() const { return sqrt((long double) norm()); }
};

using Vector = Point;

auto cross(Point a, Point b) {
    // 在 Real 为整数的时候，使用 __int128_t 整型进行计算
    if constexpr (is_integral_v<Real>) {
        return (__int128_t) a.x * b.y - (__int128_t) a.y * b.x;
    } else {
        return a.x * b.y - a.y * b.x;
    }
}

// Real cross(Point p1, Point p2, Point p0) { // 叉乘 (p1 - p0) x (p2 - p0);
//     return cross(p1 - p0, p2 - p0);
// }
auto dot(Point a, Point b) {
    // 在 Real 为整数的时候，使用 __int128_t 整型进行计算
    if constexpr (is_integral_v<Real>) {
        return (__int128_t) a.x * b.x + (__int128_t) a.y * b.y;
    } else {
        return a.x * b.x + a.y * b.y;
    }
}

// 采用两点式
struct Line {
    Point p1, p2;

    Line() {
    }

    Line(Point a, Point b) : p1(a), p2(b) {
    }

    friend bool operator<(Line a, Line b) {
        if (a.p1 != b.p1) return a.p1 < b.p1;
        return a.p2 < b.p2;
    }

    friend bool operator==(Line a, Line b) {
        if (a.p1 != b.p1) return false;
        if (a.p2 != b.p2) return false;
        return true;
    }

    friend istream &operator>>(istream &is, Line &e) {
        Real a, b, c, d;
        is >> a >> b >> c >> d;
        e = Line(Point(a, b), Point(c, d));
        return is;
    }

    // friend ostream&operator<<(ostream &os, Line l) {
    //     return os << "<" << l.p << ", " << l.q << ">";
    // }
};

// 使用这个东西，Point 必须是 double 等小数类型
Point project(Point p, Line l) {
    Point vec = l.p2 - l.p1;
    //                        这个距离反正早除晚除都要除掉
    DB r = dot(vec, p - l.p1) / ((DB) vec.x * vec.x + vec.y * vec.y);
    return l.p1 + vec * r;
}

Point reflect(Point p, Line l) {
    Point proj = project(p, l);
    return proj + proj - p;
}

mt19937 rng(233);

#define ON_SEGMENT 0
#define COUNTER_CLOCKWISE 1
#define CLOCKWISE (-1)
#define ONLINE_BACK 2
#define ONLINE_FRONT (-2)

int CCW(Point p, Line l) {
    Vector a = l.p2 - l.p1, b = p - l.p1;
    if (sgn(cross(a, b)) > 0) {
        return COUNTER_CLOCKWISE;
    }
    if (sgn(cross(a, b)) < 0) {
        return CLOCKWISE;
    }
    if (sgn(dot(a, b)) < 0) {
        return ONLINE_BACK;
    }
    if (a.norm() < b.norm()) return ONLINE_FRONT;
    return ON_SEGMENT;
}

bool isOrthogonal(Line a, Line b) {
    Vector c = a.p2 - a.p1, d = b.p2 - b.p1;
    return !sgn(dot(c, d));
}

bool isParallel(Line a, Line b) {
    Vector c = a.p2 - a.p1, d = b.p2 - b.p1;
    return !sgn(cross(c, d));
}

bool isIntersect(Line a, Line b) {
    // 判断线段相交
    return CCW(b.p1, a) * CCW(b.p2, a) <= 0 && CCW(a.p1, b) * CCW(a.p2, b) <= 0;
}

// bool isIntersectStrict(Line a,Line b){    // 判断线段相交
//     return CCW(b.p1,a)*CCW(b.p2,a)<0 && CCW(a.p1,b)*CCW(a.p2,b)<0;
// }
// p+tv=q+sw
// (p-q) x w+tv x w = 0     (利用叉乘乘以自身为0，消掉sw)
// t=-(p-q)x w/(vxw)
// 传入的是方向式  拿到的是两直线交点
Point getintersect(const Line &a, const Line &b) {
    Point p = a.p1, v = a.p2 - a.p1, q = b.p1, w = b.p2 - b.p1;
    Point u = p - q;
    DB t = cross(w, u) / cross(v, w);
    return p + v * t;
}

// 也可以用 project 来算
DB distancePL(Point p, Line l) {
    auto val = cross(l.p2 - l.p1, p - l.p1);
    return sgn(val) * val / (l.p2 - l.p1).abs();
}

// AC https://qoj.ac/submission/2001668
DB distancePS(Point p, Line l) {
    // 避免 distanceSS 调用的时候发生这个退化（即线段退化为了点）
    if (l.p1 == l.p2) {
        return (p - l.p1).abs();
    }
    Vector t = l.p2 - l.p1;
    if (sgn(dot(t, p - l.p1)) < 0) {
        return (p - l.p1).abs();
    }
    if (sgn(dot(t, p - l.p2)) > 0) {
        return (p - l.p2).abs();
    }
    return distancePL(p, l);
}

// AC https://qoj.ac/submission/2001588
DB distanceSS(Line a, Line b) {
    if (isIntersect(a, b)) return 0;
    return min({distancePS(a.p1, b), distancePS(a.p2, b), distancePS(b.p1, a), distancePS(b.p2, a)});
}

DB polygonArea(const vector<Point> &poly) {
    DB res = 0;
    for (int i = 0; i < poly.size(); ++i) {
        res += cross(poly[i], poly[(i + 1) % SZ(poly)]) / 2;
    }
    res = abs(res);
    return res;
}

// 不允许自交的多边形，顺序都可以，会把 poly 变为 CCW 顺序（逆时针）
// AC https://qoj.ac/submission/2002121
bool isConvex(vector<Point> &poly) {
    ll n = poly.size();
    Point p1 = poly[1] - poly[0], p2 = poly[2] - poly[0];
    if (sgn(cross(p1, p2)) <= 0) {
        reverse(poly.begin(), poly.end());
    }
    // 找到字典序最小的点
    ll pos_mn = 0;
    for (int i = 1; i < n; ++i) {
        if (poly[i] < poly[pos_mn]) {
            pos_mn = i;
        }
    }
    // Performs a left rotation on a range of elements
    // 向左循环移动，把 pos_mn 移动到第一个 [0]
    rotate(poly.begin(), poly.begin() + pos_mn, poly.end());
    // 在一个标准的凸多边形中，如果我们站在最左下角的点（poly[0]）往外看
    // 其他所有顶点 poly[1], poly[2], ..., poly[n-1] 应该按照逆时针方向整齐排列。
    // 从而排除星形（自交）多边形和乱序点集。
    for (int i = 1; i < n - 1; ++i) {
        if (sgn(cross(poly[i] - poly[0], poly[i + 1] - poly[0])) <= 0) {
            return false;
        }
    }
    // 判断单个内角是否 ≥ 180。
    for (int i = 0; i < n; ++i) {
        ll i1 = (i + 1) % n, i2 = (i + 2) % n;
        Vector a = poly[i1] - poly[i], b = poly[i2] - poly[i1];
        // 加上大于等于，就相当于排除所有的退化情况，不允许内角为 180 度
        if (sgn(cross(a, b)) <= 0) {
            return false;
        }
    }
    return true;
}

// 2 是包含，1是在线上，0是出界
int isInPolygon(Point p, const vector<Point> &poly) {
    ll res = 0;
    Line l(p, p + Point(2e5, rng()));
    for (int i = 0; i < poly.size(); ++i) {
        int i1 = (i + 1) % SZ(poly);
        Line seg = Line(poly[i], poly[i1]);
        int ccw = CCW(p, seg);
        if (ccw == ON_SEGMENT) {
            return 1;
        }
        if (isIntersect(l, seg)) {
            ++res;
        }
    }
    // 奇偶法则，多边形内点的射线一定只会经过奇数条边。
    if (res & 1) return 2;
    return 0;
}

// 定义获取区域的辅助函数，辅助
int get_part_of_point(const Point &p) {
    if (p.y < 0) return 0; // 下半平面 (-pi, 0)
    if (p.y == 0 && p.x >= 0) return 1; // 角度为 0 的部分 (含原点)
    return 2; // 上半平面 (0, pi]
}

// 你需要将所有点按照 atan2(yi,xi)进行排序，也即从 x 轴负半轴（不含）开始进行逆时针排序。
// AC https://qoj.ac/submission/2001485
void polar_angle_sort(vector<Point> &poly) {
    sort(poly.begin(), poly.end(), [](const Point &a, const Point &b)-> bool {
        int part1 = get_part_of_point(a), part2 = get_part_of_point(b);
        if (part1 != part2) {
            return part1 < part2;
        }
        return sgn(cross(a, b)) == 1;
    });
}

void Solve() {
    ll N;
    cin >> N;
    vector<Point> poly(N);
    for (int i = 0; i < N; ++i) {
        cin >> poly[i];
    }
    // polar_angle_sort(poly);
    if (isConvex(poly)) {
        cout << "Yes\n";
    } else {
        cout << "No\n";
    }
}

signed main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);
    cout.tie(nullptr);

    // cin >> lT;
    // while(lT--)
    Solve();
    return 0;
}

/*
AC
https://qoj.ac/submission/2002045

*/

心路历程（WA，TLE，MLE……）