The Earth is not a drawing surface
Connect London and New York with a ruler on a Mercator map. The line looks direct, but straightness is a property of the drawing. On a sphere, a shortest path follows a great circle, whose plane passes through the centre of the Earth. For endpoints that are not antipodal, we choose its shorter arc.
The route lab draws both paths on Mercator and Equal Earth. Their endpoints and calculated distances stay the same. A curve’s length in screen pixels does not provide a reliable measure of journey distance.
Holding a bearing solves a different problem
A rhumb line crosses every meridian at the same angle, maintaining a constant compass bearing in the spherical model. Mercator draws it straight. A great-circle journey generally changes bearing along the way.
The routes coincide in special cases, such as an equatorial route or a meridian. Identical endpoints give zero distance in both calculations. Exactly antipodal endpoints admit multiple equally short great-circle arcs.
Try two contrasting journeys
Start with London and New York. Blue shows the shorter great-circle arc; dashed orange shows the constant-bearing route. Swap the cities: both lengths remain unchanged. Tokyo and New York make the excursion into high latitudes especially visible.
Quito and Singapore both lie near the equator, so their relative route-length difference is smaller. Check the numbers as well as the shapes. Equal Earth preserves relative areas; it does not preserve every distance or draw every shortest route straight.
The little map lab
An area on the move.
Compare the two patches within each panel. Both have equal area on the globe.
How the model works
We use approximate city centres rather than airport locations. D3 calculates the great circle’s central angle, which we multiply by a sphere radius of 6,371.0088 km. The rhumb calculation uses latitude difference and logarithmic Mercator latitude, choosing the shorter longitude interval across the date line.
Intermediate points draw the paths; the reported distances are calculated separately from the geometry. They are not measured from rendered pixels. Extra rhumb distance means only the difference between these two models.
Why this is not a flight plan
The Earth is better approximated by an ellipsoid than a perfect sphere. Precise ellipsoidal geodesics may differ from our results. The lab also excludes winds, altitude, airways, restricted airspace and operational constraints.
The line therefore does not establish an actual airline route or a particular fuel saving. It answers a narrower, reproducible question: how does projection change the appearance of two defined paths on a sphere?
