Should bigger hospitals get bigger regions? Weighted Voronoi diagrams

Why it makes a good exploration

A standard Voronoi diagram treats a small clinic and a major hospital the same. Weighting sites by capacity is a real planning idea with new geometry (curved boundaries) — ideal for an ambitious AI student.

The mathematics you'll need

• Standard Voronoi diagrams
• Multiplicatively weighted distance and Apollonius circles (new — explain)
• Estimating population per region
• Comparing load per bed

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Hospital locations from OpenStreetMap and bed numbers from official health-service publications; population from census data.

• OpenStreetMap — Free map with exact coordinates of schools, hospitals, shops and stations; export or read off coordinates.
• Office for National Statistics (UK) — UK population, earnings, inflation, births and deaths time series.
• GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

1. Explain the weighted distance d/w and why it changes boundaries.
2. Derive the boundary between two weighted sites (a circle).
3. Construct both diagrams for 4–6 hospitals.
4. Estimate population per bed for each.
5. Reflect on specialities, emergency departments and travel time.

Showing personal engagement

• Use hospitals in your own region.
• Discuss with a health professional how catchments are actually decided.
• Compare additive and multiplicative weighting.

See Criterion C: personal engagement for what examiners look for.

Taking it further

Test a power diagram (additively weighted) and compare with the multiplicative version.