凸包上的点（共12个点）并输出部分点的c++代码实现

凸包是指在一个点集中，所有的点都在凸包的边界上或内部的一个最小凸多边形。最常见的算法来计算二维点集的凸包是Graham扫描算法和Jarvis步进算法（也称为gift wrapping算法）。这里我将提供一个简化的Graham扫描算法实现部分点凸包的C++代码示例。

#include <iostream>
#include <vector>
#include <algorithm>
#include <cmath>

using namespace std;

struct Point {
    int x, y;
};

int orientation(Point p, Point q, Point r) {
    int val = (q.y - p.y) * (r.x - q.x) - (q.x - p.x) * (r.y - q.y);
    if (val == 0) return 0;  // colinear
    return (val > 0)? 1: 2; // clock or counterclock wise
}

bool compare(Point p1, Point p2, Point q) {
    int o = orientation(q, p1, p2);
    if (o == 0)
        return (dist(q, p1) >= dist(q, p2));
    return (o == 1);
}

void convexHull(vector<Point> &points, int n) {
    int ymin = points[0].y, min = 0;
    for (int i = 1; i < n; i++) {
        int y = points[i].y;
        if ((y < ymin) || (ymin == y && points[i].x < points[min].x))
            ymin = points[i].y, min = i;
    }

    int p = min, q;
    do {
        q = (p + 1) % n;
        for (int i = 0; i < n; i++) {
            if (orientation(points[min], points[i], points[q]) == 2)
                q = i;
        }
        swap(points[min], points[q]);
        p = min;
        min = q;
    } while (min != 0);

    int m = 0;
    for (int i = 1; i < n; i++) {
        if (orientation(points[0], points[i], points[m]) == 2)
            m = i;
    }
    swap(points[0], points[m]);

    sort(points.begin(), points.end(), [points](Point p1, Point p2) {
        return compare(p1, p2, points[0]);
    });

    vector<Point> hull;
    for (int i = 0; i <= m; i++) {
        while (hull.size() >= 2 && orientation(hull[hull.size()-2], hull[hull.size()-1], points[i]) != 2)
            hull.pop_back();
        hull.push_back(points[i]);
    }
    for (int i = m - 1; i >= 0; i--) {
        while (hull.size() >= 2 && orientation(hull[hull.size()-2], hull[hull.size()-1], points[i]) != 2)
            hull.pop_back();
        hull.push_back(points[i]);
    }

    // 输出凸包上的部分点
    for (int i = 0; i < hull.size(); i++) {
        cout << "(" << hull[i].x << ", " << hull[i].y << ")" << endl;
    }
}

int main() {
    vector<Point> points = {{0, 3}, {1, 1}, {2, 2}, {4, 4},
                            {0, 0}, {1, 2}, {3, 1}, {3, 3}};
    int n = points.size();
    convexHull(points, n);
    return 0;
}

上述代码中，我们首先找到纵坐标最小的点作为起始点，然后按照顺时针方向对所有点进行排序。之后使用Graham扫描算法找到凸包的顶点。最终我们输出构成凸包的点。