Introduction

In computational geometry and computer graphics, determining the union of rectangles is a common operation. This article delves into creating an algorithm to obtain the union of multiple rectangles and to determine whether the resulting union can be considered a single rectangle. We'll explore the steps involved, technical examples, and conditions necessary for the union to remain a rectangle.

Mathematical Representation of Rectangles

A rectangle in a 2D plane can be described using two defining points, typically the bottom-left corner ((x1, y1) ) and the top-right corner ((x2, y2) ). This convention simplifies calculations:

• Width: $x2 - x1$
• Height: $y2 - y1$

Union of Rectangles

The union of two or more rectangles involves identifying a new boundary that encompasses all input rectangles. Mathematically, this means:

• The left edge of the union will be the minimum of all $x1$ values.
• The right edge of the union will be the maximum of all $x2$ values.
• The bottom edge of the union will be the minimum of all $y1$ values.
• The top edge of the union will be the maximum of all $y2$ values.

This leads to the bounding rectangle that covers all given rectangles. Here’s a simple algorithmic representation:

Algorithm to Find the Union

• Rectangle A: (1, 2, 4, 5)
• Rectangle B: (3, 3, 6, 6)
• $x1 = \min(1, 3) = 1$
• $y1 = \min(2, 3) = 2$
• $x2 = \max(4, 6) = 6$
• $y2 = \max(5, 6) = 6$
• No Spaces Between: Ensure that there are no spaces or gaps between rectangles.
• Aligned Boundaries: All adjacent edges of rectangles must align exactly where overlap occurs.
• Rectangle C: (1, 1, 3, 3)
• Rectangle D: (3, 2, 5, 5)