凸包

数学定义

• 平面的一个子集S被称为是凸的，当且仅当对于任意两点A，B属于S，线段PS都完全属于S
• 过于基础就不详细介绍了
  

凸包的计算

• github上找到了别人的代码，用4种方式实现了凸包的计算，把他放在这里
• 链接地址https://github.com/MiguelVieira/ConvexHull2D
• 先放这个代码的用法吧

int main() {
	vector<point> v = getPoints();
//1
	vector<point> h = quickHull(v);
	cout << "quickHull point count: " << h.size() << endl;
	print(h);
    
//2
	h = giftWrapping(v);
	cout << endl << "giftWrapping point count: " << h.size() << endl;
	print(h);
    
//3
	h = monotoneChain(v);
	cout << endl << "monotoneChain point count: " << h.size() << endl;
	print(h);
    
//4
	h = GrahamScan(v);
	cout << endl << "GrahamScan point count: " << h.size() << endl;
	print(h);

• 这是具体的代码，直接写在一个头文件里，包含一下就能在程序里实现上面的诸多用法了

#include <algorithm>
#include <iostream>
#include <vector>
#include<random>
	using namespace std;
namespace ch{
	struct point {
		float x;
		float y;

		point(float xIn, float yIn) : x(xIn), y(yIn) { }
	};

	// The z-value of the cross product of segments 
	// (a, b) and (a, c). Positive means c is ccw
	// from (a, b), negative cw. Zero means its collinear.
	float ccw(const point& a, const point& b, const point& c) {
		return (b.x - a.x) * (c.y - a.y) - (b.y - a.y) * (c.x - a.x);
	}

	// Returns true if a is lexicographically before b.
	bool isLeftOf(const point& a, const point& b) {
		return (a.x < b.x || (a.x == b.x && a.y < b.y));
	}

	// Used to sort points in ccw order about a pivot.
	struct ccwSorter {
		const point& pivot;

		ccwSorter(const point& inPivot) : pivot(inPivot) { }

		bool operator()(const point& a, const point& b) {
			return ccw(pivot, a, b) < 0;
		}
	};

	// The length of segment (a, b).
	float len(const point& a, const point& b) {
		return sqrt((b.x - a.x) * (b.x - a.x) + (b.y - a.y) * (b.y - a.y));
	}

	// The unsigned distance of p from segment (a, b).
	float dist(const point& a, const point& b, const point& p) {
		return fabs((b.x - a.x) * (a.y - p.y) - (b.y - a.y) * (a.x - p.x)) / len(a, b);
	}

	// Returns the index of the farthest point from segment (a, b).
	size_t getFarthest(const point& a, const point& b, const vector<point>& v) {
		size_t idxMax = 0;
		float distMax = dist(a, b, v[idxMax]);

		for (size_t i = 1; i < v.size(); ++i) {
			float distCurr = dist(a, b, v[i]);
			if (distCurr > distMax) {
				idxMax = i;
				distMax = distCurr;
			}
		}

		return idxMax;
	}


	// The gift-wrapping algorithm for convex hull.
	// https://en.wikipedia.org/wiki/Gift_wrapping_algorithm
	vector<point> giftWrapping(vector<point> v) {
		// Move the leftmost point to the beginning of our vector.
		// It will be the first point in our convext hull.
		swap(v[0], *min_element(v.begin(), v.end(), isLeftOf));

		vector<point> hull;
		// Repeatedly find the first ccw point from our last hull point
		// and put it at the front of our array. 
		// Stop when we see our first point again.
		do {
			hull.push_back(v[0]);
			swap(v[0], *min_element(v.begin() + 1, v.end(), ccwSorter(v[0])));
		} while (v[0].x != hull[0].x && v[0].y != hull[0].y);

		return hull;
	}


	// The Graham scan algorithm for convex hull.
	// https://en.wikipedia.org/wiki/Graham_scan
	vector<point> GrahamScan(vector<point> v) {
		// Put our leftmost point at index 0
		swap(v[0], *min_element(v.begin(), v.end(), isLeftOf));

		// Sort the rest of the points in counter-clockwise order
		// from our leftmost point.
		sort(v.begin() + 1, v.end(), ccwSorter(v[0]));

		// Add our first three points to the hull.
		vector<point> hull;
		auto it = v.begin();
		hull.push_back(*it++);
		hull.push_back(*it++);
		hull.push_back(*it++);

		while (it != v.end()) {
			// Pop off any points that make a convex angle with *it
			while (ccw(*(hull.rbegin() + 1), *(hull.rbegin()), *it) >= 0) {
				hull.pop_back();
			}
			hull.push_back(*it++);
		}

		return hull;
	}


	// The monotone chain algorithm for convex hull.
	vector<point> monotoneChain(vector<point> v) {
		// Sort our points in lexicographic order.
		sort(v.begin(), v.end(), isLeftOf);

		// Find the lower half of the convex hull.
		vector<point> lower;
		for (auto it = v.begin(); it != v.end(); ++it) {
			// Pop off any points that make a convex angle with *it
			while (lower.size() >= 2 && ccw(*(lower.rbegin() + 1), *(lower.rbegin()), *it) >= 0) {
				lower.pop_back();
			}
			lower.push_back(*it);
		}

		// Find the upper half of the convex hull.
		vector<point> upper;
		for (auto it = v.rbegin(); it != v.rend(); ++it) {
			// Pop off any points that make a convex angle with *it
			while (upper.size() >= 2 && ccw(*(upper.rbegin() + 1), *(upper.rbegin()), *it) >= 0) {
				upper.pop_back();
			}
			upper.push_back(*it);
		}

		vector<point> hull;
		hull.insert(hull.end(), lower.begin(), lower.end());
		// Both hulls include both endpoints, so leave them out when we 
		// append the upper hull.
		hull.insert(hull.end(), upper.begin() + 1, upper.end() - 1);
		return hull;
	}


	// Recursive call of the quickhull algorithm.
	void quickHull(const vector<point>& v, const point& a, const point& b,
		vector<point>& hull) {
		if (v.empty()) {
			return;
		}

		point f = v[getFarthest(a, b, v)];

		// Collect points to the left of segment (a, f)
		vector<point> left;
		for (auto p : v) {
			if (ccw(a, f, p) > 0) {
				left.push_back(p);
			}
		}
		quickHull(left, a, f, hull);

		// Add f to the hull
		hull.push_back(f);

		// Collect points to the left of segment (f, b)
		vector<point> right;
		for (auto p : v) {
			if (ccw(f, b, p) > 0) {
				right.push_back(p);
			}
		}
		quickHull(right, f, b, hull);
	}

	// QuickHull algorithm. 
	// https://en.wikipedia.org/wiki/QuickHull
	vector<point> quickHull(const vector<point>& v) {
		vector<point> hull;

		// Start with the leftmost and rightmost points.
		point a = *min_element(v.begin(), v.end(), isLeftOf);
		point b = *max_element(v.begin(), v.end(), isLeftOf);

		// Split the points on either side of segment (a, b)
		vector<point> left, right;
		for (auto p : v) {
			ccw(a, b, p) > 0 ? left.push_back(p) : right.push_back(p);
		}

		// Be careful to add points to the hull
		// in the correct order. Add our leftmost point.
		hull.push_back(a);

		// Add hull points from the left (top)
		quickHull(left, a, b, hull);

		// Add our rightmost point
		hull.push_back(b);

		// Add hull points from the right (bottom)
		quickHull(right, b, a, hull);

		return hull;
	}

	vector<point> getPoints() {
		vector<point> v;
		std::default_random_engine e(std::random_device{}());
		std::uniform_real_distribution<double> dist_x(0.05, 0.95);
		std::uniform_real_distribution<double> dist_y(0.05, 0.95);


		for (int i = 0; i < 30; ++i) {
			v.push_back(point(dist_x(e), dist_y(e)));
		}

		return v;
	}
}

可视化实现

这个exe小程序我放在主页了，大家可以免费下载，之后会创建一个仓库把代码公开出来，大家就可以在我的基础上实现更多的几何算法。