<p>You are given an array of points in the <strong>X-Y</strong> plane <code>points</code> where <code>points[i] = [x<sub>i</sub>, y<sub>i</sub>]</code>.</p>

<p>Return <em>the minimum area of any rectangle formed from these points, with sides <strong>not necessarily parallel</strong> to the X and Y axes</em>. If there is not any such rectangle, return <code>0</code>.</p>

<p>Answers within <code>10<sup>-5</sup></code> of the actual answer will be accepted.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/21/1a.png" style="width: 398px; height: 400px;" />
<pre>
<strong>Input:</strong> points = [[1,2],[2,1],[1,0],[0,1]]
<strong>Output:</strong> 2.00000
<strong>Explanation:</strong> The minimum area rectangle occurs at [1,2],[2,1],[1,0],[0,1], with an area of 2.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/22/2.png" style="width: 400px; height: 251px;" />
<pre>
<strong>Input:</strong> points = [[0,1],[2,1],[1,1],[1,0],[2,0]]
<strong>Output:</strong> 1.00000
<strong>Explanation:</strong> The minimum area rectangle occurs at [1,0],[1,1],[2,1],[2,0], with an area of 1.
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/22/3.png" style="width: 383px; height: 400px;" />
<pre>
<strong>Input:</strong> points = [[0,3],[1,2],[3,1],[1,3],[2,1]]
<strong>Output:</strong> 0
<strong>Explanation:</strong> There is no possible rectangle to form from these points.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= points.length &lt;= 50</code></li>
	<li><code>points[i].length == 2</code></li>
	<li><code>0 &lt;= x<sub>i</sub>, y<sub>i</sub> &lt;= 4 * 10<sup>4</sup></code></li>
	<li>All the given points are <strong>unique</strong>.</li>
</ul>

class Solution {
public:
    double minAreaFreeRect(vector<vector<int>>& points) {
        constexpr double kIlegalArea = 1e11; // larger than max (4*10^4)^2
        double minRectangleArea = kIlegalArea + 1.;
        // center without div 2, len is square, [x1, y1, x2, y2]
        map<pair<int,int>, unordered_map<int, vector<tuple<int,int,int,int>>>> centerLenPoint;
        for (int i = 0; i < points.size(); ++i) {
            for (int j = 0; j < i; ++j) {
                int dx = points[j][0] - points[i][0];
                int dy = points[j][1] - points[i][1];
                int len = (dx*dx) + (dy*dy);
                pair<int,int> center = {points[j][0] + points[i][0], points[j][1] + points[i][1]};
                for (auto [x1,y1,x2,y2] : centerLenPoint[center][len]) {
                    int v1x = (x1 - points[j][0]);
                    int v1y = (y1 - points[j][1]);
                    int v2x = (x2 - points[j][0]);
                    int v2y = (y2 - points[j][1]);
                    double area = sqrt(1. * (v1x*v1x + v1y*v1y) * (v2x*v2x + v2y*v2y));
                    minRectangleArea = min(minRectangleArea, area);
                }
                
                centerLenPoint[center][len].push_back({points[j][0], points[j][1], points[i][0], points[i][1]});
            }
        }
        return minRectangleArea > kIlegalArea ? 0 : minRectangleArea;
    }
};
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