[LeetCode] Convex Polygon 凸多边形

Given a list of points that form a polygon when joined sequentially, find if this polygon is convex (Convex polygon definition).

Note:

1. There are at least 3 and at most 10,000 points.
2. Coordinates are in the range -10,000 to 10,000.
3. You may assume the polygon formed by given points is always a simple polygon(Simple polygon definition). In other words, we ensure that exactly two edges intersect at each vertex, and that edges otherwise don't intersect each other.

Example 1:

[[0,0],[0,1],[1,1],[1,0]]

Answer: True

Explanation:

Example 2:

[[0,0],[0,10],[10,10],[10,0],[5,5]]

Answer: False

Explanation:

这道题让我们让我们判断一个多边形是否为凸多边形，我想关于凸多边形的性质，我大天朝的初中几何就应该有所涉猎啦吧，忘了的去面壁。就是所有的顶点角都不大于180度。那么我们该如何快速验证这一个特点呢，学过计算机图形学或者是图像处理的课应该对计算法线normal并不陌生吧，计算的curve的法向量是非常重要的手段，一段连续曲线可以离散看成许多离散点组成，而相邻的三个点就是最基本的单位，我们可以算由三个点组成的一小段曲线的法线方向，而凸多边形的每个三个相邻点的法向量方向都应该相同，要么同正，要么同负。那么我们只要遍历每个点，然后取出其周围的两个点计算法线方向，然后跟之前的方向对比，如果不一样，直接返回false。这里我们要特别注意法向量为0的情况，如果某一个点的法向量算出来为0，那么正确的pre就会被覆盖为0，后面再遇到相反的法向就无法检测出来，所以我们对计算出来法向量为0的情况直接跳过即可，参见代码如下：

class Solution {
public:
    bool isConvex(vector<vector<int>>& points) {
        long long n = points.size(), pre = , cur = ;
        for (int i = ; i < n; ++i) {
            int dx1 = points[(i + ) % n][] - points[i][];
            int dx2 = points[(i + ) % n][] - points[i][];
            int dy1 = points[(i + ) % n][] - points[i][];
            int dy2 = points[(i + ) % n][] - points[i][];
            cur = dx1 * dy2 - dx2 * dy1;
            if (cur != ) {
                if (cur * pre < ) return false;
                else pre = cur;
            }
        }
        return true;
    }
};

参考资料：

https://discuss.leetcode.com/topic/70643/i-believe-this-time-it-s-far-beyond-my-ability-to-get-a-good-grade-of-the-contest

https://discuss.leetcode.com/topic/70664/c-7-line-o-n-solution-to-check-convexity-with-cross-product-of-adajcent-vectors-detailed-explanation/2

LeetCode All in One 题目讲解汇总(持续更新中...)