Perpendicular bisectors and Voronoi diagrams

Maths: Applications and Interpretation · Geometry and trigonometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

The perpendicular bisector of two points is the line through their midpoint at right angles to the line joining them. Every point on it is the same distance from both points.

A Voronoi diagram starts with a set of points called sites. It divides the plane into cells, one for each site. A cell contains all the points that are closer to its site than to any other site. The edges between cells are parts of perpendicular bisectors, and a vertex is where edges meet.

The diagram answers questions such as which shop is nearest, what value to estimate at a place, and where the point furthest from every site lies.

Rules to remember

Perpendicular bisector: passes through the midpoint ((x₁ + x₂)/2, (y₁ + y₂)/2) with gradient −1 ÷ (gradient of the line joining the points).
Edge: every point on it is equidistant from the two sites on either side.
Vertex: equidistant from the three (or more) nearest sites, where their cells meet.
Adding a site creates one new cell; only the cells next to it get smaller and the rest of the diagram does not change.
Nearest neighbour interpolation: a point takes the value of the site whose cell it lies in.
Toxic waste dump problem: the point furthest from all sites is at a vertex (inside the region); compare the vertices and choose the one with the greatest distance to its nearest sites.

Common mistake

Students give the perpendicular bisector the same gradient as the line joining the two points. Its gradient is the negative reciprocal, and it must pass through the midpoint, not through either point.

Worked example

Find the equation of the perpendicular bisector of A(1, 3) and B(7, −1).

Midpoint = ((1 + 7)/2, (3 + (−1))/2) = (4, 1).
Gradient of AB = (−1 − 3)/(7 − 1) = −4/6 = −2/3, so the perpendicular gradient is 3/2.
y − 1 = (3/2)(x − 4), so y = 1.5x − 6 + 1 = 1.5x − 5.
Check with (6, 4): distance² to A = 25 + 1 = 26 and to B = 1 + 25 = 26, so it is equidistant.

Answer: y = 1.5x − 5 (or 3x − 2y = 10)

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Topic 3: Geometry and Trigonometry

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