Which Map Shows the Shortest Route as a Straight Line? Only One
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Projecting the globe from its centre onto a flat sheet distorts almost everything and preserves one property exactly. That single property makes the map indispensable for planning long routes.
How the projection is made
Imagine a light at the exact centre of a transparent globe and a flat sheet touching the surface at one point. Every point on the globe casts a shadow onto the sheet along a straight line from the centre, and where that shadow falls is where the point is drawn. That construction is what defines the projection, and everything about its behaviour follows from it geometrically rather than from any formula chosen for convenience. It is among the oldest projections known, described in antiquity and used for sundials long before it was used for maps, which is where its name comes from.
What it does and does not preserve
The trade is unusually stark:
- •Every great circle appears as a straight line, exactly
- •So the shortest route between any two points is a straight line on it
- •Angles are wrong, so it cannot be used for steering a course
- •Areas are wrong, and badly so away from the centre
- •Shapes are wrong, increasingly with distance from the centre
- •Less than half the globe can be shown, and even that is extreme
Why only one property survives
The reason great circles come out straight is direct. A great circle is the intersection of the sphere with a plane through its centre, and every point on that circle therefore lies on a plane that also contains the projection centre, so all of their projected rays lie in that single plane. A plane intersecting the flat sheet produces a straight line, so the whole circle maps to one. Nothing else is preserved, because the projection stretches the surface enormously and unevenly as distance from the touching point increases, with points at ninety degrees from it projecting to infinity and points beyond that projecting backwards.
The related projections
Three projections share the construction of casting the globe onto a touching plane and differ only in where the light sits, which makes them easy to compare. Placing the light at the centre gives the one described here, with great circles straight and extreme distortion outward. Placing it at the point opposite the touching point gives the stereographic projection, which preserves angles and shapes locally and maps circles to circles, and is used in crystallography and for polar charts. Placing it infinitely far away gives the orthographic projection, which looks like a photograph of a globe from space and is used for illustration rather than measurement.
How navigators use it
The projection is used alongside another rather than instead of it, and the pairing is the practical point. A course of constant compass bearing appears as a straight line on the Mercator projection, which is what makes that projection useful for steering, but such a course is not the shortest route between distant points. So a navigator draws the shortest route as a straight line on this projection, reads off a series of points along it, transfers those points to a Mercator chart, and then sails or flies a sequence of constant-bearing legs between them, which approximates the shortest route using courses a compass can actually hold.
The takeaway
Projecting from the globe's centre onto a touching plane makes every great circle a straight line, because the circle's plane and the projection centre define a single plane that cuts the sheet in a line. Nothing else survives, since angles, areas and shapes are all wrong and less than half the globe fits. Navigators plot the shortest route on it and transfer points to a Mercator chart to steer.