Solution: Rectangle Area

Statement

You are given the coordinates of two axis-aligned rectangles in a 2D plane. Your task is to calculate the total area covered by both rectangles.

• The first rectangle is specified by the coordinates of its bottom-left corner (ax1, ay1) and top-right corner (ay1, ay2).

• Similarly, the second rectangle is defined by its bottom-left corner (bx1, by1) and top-right corner (bx2, by2).

Note: The rectangles may overlap.

Constraints:

• $-10^4 \leq$ ax1 $\leq$ ax2 $\leq 10^4$

• $-10^4 \leq$ ay1 $\leq$ ay2 $\leq 10^4$

• $-10^4 \leq$ bx1 $\leq$ bx2 $\leq 10^4$

• $-10^4 \leq$ by1 $\leq$ by2 $\leq 10^4$

Solution

When solving this problem, our first instinct might be to simply add the areas of the two rectangles. However, this approach overlooks a key detail: if the rectangles overlap, the overlapping region would be counted twice. To get the correct total area, we must subtract the overlapping portion from the sum of both individual areas.

To do this, we first calculate the area of each rectangle separately using the difference in their x and y coordinates. Then, we determine the dimensions of their overlapping region (if any) by finding the intersection along both the x-axis and y-axis. If both overlap dimensions are positive, we compute the overlapping area and subtract it from the total. Otherwise, if there’s no overlap, the sum of both areas remains unchanged.

Start by setting up variables representing each rectangle’s bottom-left and top-right corners. Use (ax1, ay1, ax2, ay2) for the first rectangle and (bx1, by1, bx2, by2) for the second. Calculate each rectangle’s area by multiplying its width and height:

• Rectangle A: (ax2 - ax1) * (ay2 - ay1)

• Rectangle B: (bx2 - bx1) * (by2 - by1)

Next, determine the horizontal overlap by identifying the rightmost left edge and the leftmost right edge. If the difference is positive, the rectangles share a valid horizontal span; otherwise, they do not overlap along the x-axis.

A similar approach applies to the y-axis. Identify the highest bottom edge and the lowest top edge. If the difference is positive, the rectangles overlap vertically; otherwise, they do not. If horizontal and vertical overlap exist, multiply their lengths to get the overlapping area. Finally, subtract this from the sum of the individual rectangle areas to avoid double counting.

Now, let’s look at the solution steps below:

1. Calculate individual areas:

   1. Rectangle A: areaA = (ax2 - ax1) * (ay2 - ay1)

   2. Rectangle B: areaB = (bx2 - bx1) * (by2 - by1)

2. Find x overlap:

   1. Left boundary: max(ax1, bx1) (Rightmost left edge)

   2. Right boundary: min(ax2, bx2) (Leftmost right edge)

   3. Overlap width: xOverlap = max(0, right - left) (if positive, rectangles overlap horizontally)

3. Find y overlap:

   1. Bottom boundary: max(ay1, by1) (Highest bottom edge)

   2. Top boundary: min(ay2, by2) (Lowest top edge)

   3. Overlap height: yOverlap = max(0, top - bottom) (if positive, rectangles overlap vertically)

4. Calculate overlap area:

   1. If both xOverlap and yOverlap are positive, their product is the overlapping area. Otherwise, it’s 0:
      overlapArea = xOverlap * yOverlap

5. Compute total area:

   1. total_area: areaA + areaB - overlapArea

Let’s look at the following illustration to get a better understanding of the solution: