Gnomonic vs Mercator: which straight line is shortest?
Two maps, two straight lines, and two different questions: shortest route versus constant compass bearing.
Straightness belongs to a projection, not a route
On Mercator, the straight line is a rhumb line: it crosses every meridian at the same angle. On a gnomonic map, the straight line is a great circle: the shortest route over the surface of a sphere. Neither map is lying; each answers a different question.
Why does Mercator straighten a constant bearing?
Gerardus Mercator presented his projection in 1569 on a large world map intended for navigation. The grid is built to preserve angles, so a line of constant compass bearing becomes straight. A sailor could set such a course with a ruler and a protractor.
Which is larger in reality: Africa or Greenland?
A rhumb line is usually longer than the great circle, though. The two have the same length only when they run along the Equator or along a meridian.
Why does the gnomonic map straighten great circles?
In the gnomonic projection, points are projected from the center of the Earth onto a plane touching the sphere. Every great circle lies in a plane through the Earth's center, so it meets the map plane in a straight line. In his USGS manual, John P. Snyder calls this the only truly useful feature of the projection.
The price is high. Away from the center, shapes, areas and scale are badly distorted, and points 90° from the center run off to infinity. A gnomonic map cannot show even a whole hemisphere. That is why our tool centers the view on the midpoint of the selected route.
A very old idea
The gnomonic is one of the oldest known projections. According to Snyder, Thales of Miletus already used it for star maps. The name comes from the gnomon, the pointer of a sundial: the meridians on such a map spread out like the hour lines on its face.
Experiment: from Warsaw to New York
Open the route lab. On the gnomonic map the great circle is straight. Switch to Mercator: the great circle bends north, and the straight rhumb line turns out to be longer. For the city points used in the tool the difference is about 495 km. We explain the details in Great circle vs rhumb line.
How can the two maps work together?
The USGS album of map projections notes that gnomonic maps helped navigators and aviators determine great-circle courses. A route drawn on such a map can be divided into waypoints, transferred to a Mercator chart and joined with legs of constant bearing. Each leg can be followed on a fixed course, while the whole chain stays close to the shortest route.