Find the intersect area of two overlapping rectangles
/**
 * Returns intersecting part of two rectangles
 * @param  {object}  r1 4 coordinates in form of {x1, y1, x2, y2} object
 * @param  {object}  r2 4 coordinates in form of {x1, y1, x2, y2} object
 * @return {boolean}    False if there's no intersecting part
 * @return {object}     4 coordinates in form of {x1, y1, x2, y2} object
 */
const getIntersectingRectangle = (r1, r2) =&gt; {  
  [r1, r2] = [r1, r2].map(r =&gt; {
    return {
      x: [r.x1, r.x2].sort((a,b) =&gt; a - b),
      y: [r.y1, r.y2].sort((a,b) =&gt; a - b)
    };
  });

  const noIntersect = r2.x[0] &gt; r1.x[1] || r2.x[1] &lt; r1.x[0] ||
                      r2.y[0] &gt; r1.y[1] || r2.y[1] &lt; r1.y[0];

  return noIntersect ? false : {
    x1: Math.max(r1.x[0], r2.x[0]), // _[0] is the lesser,
    y1: Math.max(r1.y[0], r2.y[0]), // _[1] is the greater
    x2: Math.min(r1.x[1], r2.x[1]),
    y2: Math.min(r1.y[1], r2.y[1])
  };
};

/*  ↓  DEMO  ↓  */

const rectangle1 = { x1: 2, y1: 2, x2: 4, y2: 4 };
const rectangle2 = { x1: 3, y1: 3, x2: 6, y2: 2 };

console.log(getIntersectingRectangle(rectangle1, rectangle2));
// { x1: 3, y1: 2, x2: 4, y2: 3 }



First, both rectangles get transformed into object with keys x and y and sorted arrays of two of corresponding them coordinates as values.
r1 = {
  x: [2, 4], // x1, x2
  y: [2, 4]  // y1, y2
}
r2 = {
  x: [3, 6], // x1, x2
  y: [2, 3]  // Y2, Y1 !
}

That's because for this job it is important to know which of xs and ys is lesser and which is greater. Rather than Math.max() and Math.min() a single .sort() can be used.
Alternative would be to declare that (x1, y1) is assumed to always be top-left corner, but as I see in rectangle2 in your question, that is apparently not always the case.
Now, noIntersect is negated
!(a.left &gt; b.right || b.left &gt; a.right || a.top &gt; b.bottom || b.top &gt; a.bottom);

which tests if in both axes sides from one end of one figure exceed opposite-direction end side of the other figure, e.g. if left side of figure a is more to the right than the right side of figure b.
At the end we return false if there is no intersection of our rectangles, or an object with coordinates if there is. Intersecting part will always span from:
the greater of the 2 lesser xs of both rectangles
to
the lesser of the 2 greater xs of both rectangles.