IA idea · Geometry, trigonometry & Voronoi diagrams

Why do flight paths curve on the map?

AA SLAA HLAI SLAI HL Ambitious Also in: Networks & graphs, Pure maths

Research question

How much shorter is the great-circle route between [two cities] than the route that appears straight on a Mercator map, and how does the saving depend on latitude?

Adapt it: change the place, the data or the comparison until the question is yours.

Why it makes a good exploration

Long-haul flights seem to curve towards the poles on maps. Showing, with vectors or spherical trigonometry, that these routes are actually the shortest is a beautiful exploration of 3-D geometry.

The mathematics you'll need

Latitude/longitude to 3-D coordinates
Vectors and the angle between them (HL) or the spherical law of cosines
Arc length
Comparing with a rhumb line (constant bearing)

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

City coordinates from OpenFlights; optionally real flight distances from airline schedules.

OpenFlights data — Airports (with latitude/longitude) and airline routes as CSV.

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

Convert coordinates to 3-D vectors.
Find the angle between vectors and the great-circle distance.
Compute the rhumb-line distance for comparison.
Repeat for routes at different latitudes.
Reflect on winds, airspace and the flat-map illusion.

Pitfalls that cost marks

Mixing degrees and radians.
Using the haversine formula as a black box.
Comparing with a ruler distance on a Mercator map without explaining the projection.

Showing personal engagement

Use a flight you have taken.
Plot both routes on a globe with string.
Explain why Europe–North America flights pass near Greenland.

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

Taking it further

Derive the maximum latitude reached by a great-circle route between two cities.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).