What a perpendicular bisector represents

A perpendicular bisector is a straight line that crosses a line segment at its midpoint, forming a right angle. Any point on the bisector is the same distance from both endpoints of the segment. This equal-distance property is why perpendicular bisectors form the boundaries in a Voronoi diagram.

Starting from two points

Take the endpoints $A=(x_1,y_1)$ and $B=(x_2,y_2)$. Here, $A$ and $B$ are the endpoints, while $x_1$, $y_1$, $x_2$ and $y_2$ are their coordinates in the units used by the coordinate system.

Begin by finding the midpoint:

Now calculate the slope of the segment:

If the line is neither horizontal nor vertical, its perpendicular slope is

Use the midpoint in point–slope form:

Horizontal and vertical segments

Take care with the slope calculation when it would involve division by zero.

• If $AB$ is horizontal, its perpendicular bisector is the vertical line $x=h$.
• If $AB$ is vertical, its perpendicular bisector is the horizontal line $y=k$.

This approach is quicker and safer than trying to apply the negative-reciprocal rule to an undefined slope.

Starting from a line equation and its midpoint

Suppose a segment lies on

If its midpoint is $M=(h,k)$, the perpendicular bisector is

provided $m\ne0$. When $m=0$, the given segment is horizontal, so its perpendicular bisector is $x=h$. If the segment’s equation is vertical, its perpendicular bisector is $y=k$.

This is a direct application of straight-line equations: identify the midpoint, find a perpendicular direction and make the new line pass through the midpoint.

Structure of a Voronoi diagram

A site is a fixed point that acts as the centre of a nearest-location region. A Voronoi diagram partitions a plane into regions based on which site is nearest.

Every Voronoi cell contains all points that are at least as close to its own site as they are to any other site. A Voronoi edge is a boundary segment made up of points equally distant from the two sites on either side. A Voronoi vertex occurs where at least three Voronoi edges meet. In the usual non-degenerate case, that vertex is equally distant from three sites.

An edge separates points assigned to two different sites, so it lies on the perpendicular bisector of the segment joining them. However, only the relevant section of that bisector forms the edge; other sites may cut off the remaining parts. In an unrestricted diagram, outer cells can extend indefinitely. Real applications often add a boundary, such as a district or habitat.

Reading and calculating from the diagram

To find the site closest to a given point, first locate the cell that contains it. A point strictly inside a cell is nearest to that cell's site. On an edge, the point is tied between two sites; at a vertex, it is normally tied between three.

If an edge equation is needed, take the coordinates of the two sites separated by the edge and calculate their perpendicular bisector as in section 3.5. The required site coordinates will be supplied in an examination. You should be able to calculate boundary equations and interpret a provided diagram, but you won't be asked to complete a full ruler-and-compass construction of every bisector.

For bounded regions, find the area by splitting the cell into familiar shapes or using coordinate geometry. Read the question carefully: it may ask for a whole cell, a cell clipped by an external boundary, or several cells together.

Adding a new site

Adding a site affects only the region where that site becomes closer than every existing site. An incremental construction algorithm is a step-by-step process that inserts one new site into an existing diagram and updates the cells affected by it.

For each possible neighbouring site, consider the perpendicular bisector joining that site to the new one. Keep the sections that separate points nearer the new site from those nearer the existing site, then trim the old cells. The new cell is the intersection of all half-planes assigned to the new site. Edges that aren't affected stay unchanged.

This also provides a useful check. The new site must gain a cell containing itself, nearby cells may become smaller, and distant cells normally won't change.

Nearest-neighbour interpolation

Nearest-neighbour interpolation estimates the value at an unknown point by assigning it the recorded value of the nearest sampled site. On a Voronoi diagram, every point within a cell therefore receives the same estimated value as the site's value.

Suppose the sites are monitoring stations. Each cell takes the reading from its station. The model is piecewise constant: its estimate remains fixed within a cell, then jumps when an edge is crossed. This approach is simple and useful when only local observations are available, though it cannot show gradual change between sites.

The same mathematics appears in meteorological maps, ecological sampling, disease-spread monitoring and resource allocation. Urban planners can use it to model service areas. In geography and economics, it can represent the nearest facilities or markets; in biology, it can partition territories; and in computer science, it supports proximity searches. Such applications cross conventional subject boundaries, suggesting that the separation of knowledge into disciplines is organizational rather than a feature of the mathematics itself.

The toxic-waste-dump model

The toxic-waste-dump problem is an optimal-location problem where a new facility must be placed as far as possible from its nearest protected site. Those sites could represent settlements or environmentally sensitive locations. At any possible location, the limiting distance is its distance from the nearest site. The goal is to make this minimum distance as large as possible.

A Voronoi vertex is a natural candidate because it is equally distant from three sites. Moving away from the vertex will generally bring the location closer to at least one of them. In syllabus examination questions, the solution will occur at an intersection of three edges. Use this method:

1. identify the relevant Voronoi vertices;
2. determine the nearest-site distance at each candidate;
3. select the candidate with the greatest such distance, while respecting any stated boundary.

The same maximin reasoning applies to facilities that need to remain separated from hazards. Many practical location problems reverse the objective, however, and seek closeness instead. Translate the context before calculating: decide whether to maximize or minimize the distance to the nearest site.

Wider mathematical connections

A Delaunay triangulation is a network created by joining sites whose Voronoi cells share an edge. It is the geometric dual of the Voronoi diagram. Related enrichment applications include navigation for autonomous vehicles and visibility questions such as the art-gallery problem.

Natural-neighbour interpolation blends values from several nearby sites according to how inserting a new point would change neighbouring Voronoi cells. It gives smoother estimates than a method that assigns the value of just one site.

Standard Voronoi diagrams use ordinary straight-line distance. A Manhattan metric instead measures the total horizontal and vertical travel between two points, as though movement were limited to a rectangular street grid. A different distance rule produces different cell shapes, showing that a mathematical model depends on the assumptions selected for its context.