Introduction to Perpendicular Bisectors and Voronoi Diagrams

Have you ever wondered how your phone knows which cell tower to connect to, or how a delivery app decides which pizza branch should handle your order? The answer lies in Voronoi diagrams. These are mathematical "maps" that divide a space into regions based on the distance to specific points. To understand them, we first need to master the perpendicular bisector, which is the "border" between any two points. This chapter is a perfect blend of coordinate geometry and real-world logic!

1. Perpendicular Bisectors

Before we can build complex diagrams, we need to know how to draw a fair boundary between two points. A perpendicular bisector is a line that divides a line segment into two equal halves at a \(90^{\circ}\) angle.

How to find the Equation of a Perpendicular Bisector

To find the equation of the perpendicular bisector between two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\), follow these three steps:

Step 1: Find the Midpoint
The bisector must pass through the exact center of the two points. Use the midpoint formula:
\(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)

Step 2: Find the Perpendicular Gradient
First, find the gradient (\(m\)) of the line segment \(AB\):
\(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Then, find the perpendicular gradient (\(m_{\perp}\)). Remember the rule: \(m_1 \times m_2 = -1\). In simple terms, flip the fraction and change the sign (the negative reciprocal):
\(m_{\perp} = -\frac{1}{m}\)

Step 3: Use the Point-Gradient Formula
Plug your midpoint \(M\) and your perpendicular gradient \(m_{\perp}\) into the straight-line equation:
\(y - y_1 = m(x - x_1)\)

Quick Example: Find the perpendicular bisector of \(A(2, 4)\) and \(B(6, 10)\).
1. Midpoint: \(M = (\frac{2+6}{2}, \frac{4+10}{2}) = (4, 7)\).
2. Gradient of \(AB\): \(m = \frac{10-4}{6-2} = \frac{6}{4} = 1.5\).
3. Perp Gradient: \(m_{\perp} = -\frac{1}{1.5} = -\frac{2}{3}\).
4. Equation: \(y - 7 = -\frac{2}{3}(x - 4)\).

Key Takeaway:

The perpendicular bisector represents all points that are equidistant (the same distance) from two specific points.

2. Introduction to Voronoi Diagrams

A Voronoi diagram is a collection of perpendicular bisectors that partition a plane into regions called cells.

Essential Terminology

  • Sites: The original points given in the problem (e.g., the locations of hospitals or shops).
  • Cells: The regions surrounding each site. Every point inside a cell is closer to its own site than to any other site.
  • Edges: The lines that form the boundaries between cells. These are segments of perpendicular bisectors.
  • Vertices: The points where three or more edges meet. A vertex is equidistant to the three (or more) closest sites.

Did you know? Voronoi diagrams are often called "Nearest Neighbor" diagrams because they tell you exactly which site is your closest neighbor at any given location.

3. Working with Voronoi Diagrams

In your IB exams, you will likely be asked to interpret a diagram or modify one.

Adding a Site to an Existing Diagram

If a new site (let's call it \(S_{new}\)) is added, the diagram must be updated:

  1. Identify which existing cells the new site is located in or near.
  2. Draw the perpendicular bisectors between \(S_{new}\) and the sites of the neighboring cells.
  3. Connect these new bisectors to form a new cell, erasing the old boundaries that are now "inside" the new cell.

Nearest Neighbor Interpolation

This sounds fancy, but it is very simple! If you have a piece of data (like rainfall or temperature) at specific sites, and you want to guess the value at a new location, you simply look at which cell that location falls into. You then assign it the value of the site in that cell.
Example: If Point X is in the cell of Site A, and Site A has a value of 10, then Point X is estimated to be 10.

4. The "Toxic Waste Dump" Problem

This is a classic IB application! Imagine you want to build something that is as far away as possible from all existing sites (like a toxic waste dump, a noisy factory, or even a rival business). Where should you put it?

The furthest point from all sites will always be located at:

  1. A Vertex of the Voronoi diagram (where edges meet).
  2. A point on the boundary of the area you are looking at.

Step-by-Step for the Toxic Waste Dump:

  1. Identify the coordinates of the vertices of the Voronoi diagram.
  2. Calculate the distance from each vertex to its nearest sites (the distance will be the same for at least three sites).
  3. If there is a boundary (like a fence or a map edge), check the corners of that boundary too.
  4. The point with the largest distance is your answer.

Common Mistake to Avoid: Don't just pick a vertex at random. You must calculate the distance using the distance formula \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\) to prove which one is the "most" distant.

Chapter Summary

1. Perpendicular Bisectors: Use the midpoint and the negative reciprocal gradient to find the "border" between two points.

2. Voronoi Anatomy: Sites are the points; Cells are the "nearest neighbor" zones; Edges are the boundaries; Vertices are the meeting points.

3. Logic: Every point in a cell is closer to that cell's site than to any other site on the map.

4. Applications: Use vertices to find the most "remote" locations (The Toxic Waste Dump problem) and use cells to estimate values for new points (Interpolation).

Don't worry if the diagrams look messy at first! Focus on one pair of sites at a time, find their perpendicular bisector, and the pattern will start to make sense.