How much do two rectangles overlap?

Calculating the area of the overlapping region is a standard method for determining the overlap ratio. The following steps outline how to compute the overlap ratio between two rectangles:

1. Understanding the Representation of Rectangles

Typically, a rectangle is defined by the coordinates of its bottom-left and top-right corners. Suppose there are two rectangles A and B, represented as:

• Rectangle A: from (Ax1, Ay1) to (Ax2, Ay2), where (Ax1, Ay1) is the bottom-left corner and (Ax2, Ay2) is the top-right corner.
• Rectangle B: from (Bx1, By1) to (Bx2, By2), using the same convention.

2. Calculating the Coordinates of the Overlapping Region

The bottom-left corner of the overlapping region is determined by the maximum of the bottom-left coordinates of A and B, and the top-right corner is determined by the minimum of the top-right coordinates of A and B. Specifically:

• Bottom-left corner: (max(Ax1, Bx1), max(Ay1, By1))
• Top-right corner: (min(Ax2, Bx2), min(Ay2, By2))

3. Checking if Rectangles Overlap

The rectangles overlap only if both coordinates of the overlapping region are valid, meaning the bottom-left coordinates are strictly less than the top-right coordinates. This condition is expressed as:

• If max(Ax1, Bx1) < min(Ax2, Bx2) and max(Ay1, By1) < min(Ay2, By2), then the rectangles overlap.

4. Calculating the Area of the Overlapping Region

If the rectangles overlap, the area of the overlapping region is computed using the formula:

• Overlap area = (min(Ax2, Bx2) - max(Ax1, Bx1)) * (min(Ay2, By2) - max(Ay1, By1))

5. Calculating the Overlap Ratio

The overlap ratio is typically defined as the ratio of the overlapping area to the sum of the areas of the two rectangles. It is given by:

• Overlap ratio = overlap area / (area A + area B - overlap area)

Where area A and area B are:

• Area A = (Ax2 - Ax1) * (Ay2 - Ay1)
• Area B = (Bx2 - Bx1) * (By2 - By1)

Example

Suppose two rectangles A and B have the following coordinates:

• A: from (1, 1) to (3, 4)
• B: from (2, 3) to (5, 6)

1. Calculating the coordinates of the overlapping region:

   • Bottom-left corner: (max(1, 2), max(1, 3)) = (2, 3)
   • Top-right corner: (min(3, 5), min(4, 6)) = (3, 4)

2. Checking for overlap:

   • Since 2 < 3 and 3 < 4, the rectangles overlap.

3. Calculating the overlap area:

   • Overlap area = (3 - 2) * (4 - 3) = 1

4. Calculating the areas of the two rectangles:

   • Area A = (3 - 1) * (4 - 1) = 6
   • Area B = (5 - 2) * (6 - 3) = 9

5. Calculating the overlap ratio:

   • Overlap ratio = 1 / (6 + 9 - 1) = 1 / 14 ≈ 0.0714 or 7.14%

Therefore, the overlap ratio between rectangles A and B is approximately 7.14%.