IA idea · Geometry, trigonometry & Voronoi diagrams
Voronoi diagrams in a grid city: Euclidean or taxicab distance?
Research question
How different are the Voronoi regions of the fire stations in [a grid-plan city] under Euclidean and taxicab (Manhattan) distance, and which better predicts the real nearest station by road?
Adapt it: change the place, the data or the comparison until the question is yours.
Why it makes a good exploration
In a city with a street grid, vehicles cannot drive diagonally. Taxicab distance changes the shape of Voronoi cells completely — and comparing both with real road distances is a genuinely original analysis.
The mathematics you'll need
- Euclidean and taxicab metrics
- Bisectors under the taxicab metric (piecewise lines)
- Voronoi construction by hand and technology
- Testing predictions with sampled points
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
Station coordinates from OpenStreetMap for a city with a grid layout; check nearest stations by road using an online map for 20 sample points.
- OpenStreetMap — Free map with exact coordinates of schools, hospitals, shops and stations; export or read off coordinates.
- 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
- Introduce the taxicab metric and its “circles”.
- Derive the shape of a taxicab bisector.
- Build both diagrams.
- Test 20 sample addresses against real road distances.
- Reflect on one-way streets, rivers and diagonal avenues.
Pitfalls that cost marks
- Choosing a city without a grid.
- Drawing taxicab bisectors as straight lines.
- Sampling points in a biased way.
Showing personal engagement
- Choose a city you have visited.
- Predict which metric will win and why.
- Draw a taxicab circle and explain what it means for response time.
See Criterion C: personal engagement for what examiners look for.
Taking it further
Consider a rotated grid or a city with major diagonals and define a hybrid metric.