<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) {
        long ans = LONG_MAX;  // Initialize the minimum area to a large value.
        map<vector<long>, multimap<long, vector<int>>> db;  // Database to store midpoints and vectors.
        int n = points.size();  // Number of points.

        for (int i = 0; i < n; i++) {
            for (int j = i + 1; j < n; j++) {
                long dx = points[j][0] - points[i][0];  // x-difference between points i and j.
                long dy = points[j][1] - points[i][1];  // y-difference between points i and j.
                long length = dx * dx + dy * dy;  // Calculate squared length of the vector.
                vector<long> mid = {points[i][0] + points[j][0], points[i][1] + points[j][1]};  // Compute the midpoint.

                auto &m = db[mid];  // Reference to the map entry for this midpoint.
                for (auto it = m.begin(); it != m.end(); ++it) {
                    if (it->first == length) {
                        // Calculate area using the cross product of vectors (dx, dy) and (it->second[0], it->second[1])
                        long area = abs(dx * it->second.at(1) - dy * it->second.at(0));
                        ans = min(ans, area);  // Update minimum area found.
                    }
                }
                m.insert({length, {int(dx), int(dy)}});  // Insert the vector and its length for the current midpoint.
            }
        }
        return ans == LONG_MAX ? 0 : ans / 2.0;  // If no rectangle found, return 0; otherwise, return the area.
    }
};

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