Why are gnomonic projections useful to navigators in plotting used in air travel?

Answer 1

The gnomonic projection of a presumed spherical Earth is formed by imagining rays from the point at the center of the Earth projected onto a plane that is tangent to the Earth at some specified point. The geographic features would be plotted onto this plane.

The gnomonic projection has one very interesting feature: all great circles on the spherical Earth or globe map onto straight lines. Thus, every meridian (line of longitude) maps into a straight line, the Equator (if it appears in your map at all--see below...) is a straight line. And, the minimum distance route from _any_ point to _any_ other appearing on the map is simply the straight line between them. This is why it would be useful to navigators--even before air travel it would have been useful for long-distance ocean navigation, but especially with air travel, where pilots can in principle choose to go "straight" for their objectives without too much concern about obstacles.

(In real life of course, even modern high-speed airplanes are still concerned with taking winds into account, especially the jet stream; they have to take political boundaries into account as well, and over densely populated or heavily trafficked regions, local air traffic control will assign air routes rather than let every pilot fly on whatever track they feel like.)

However, one reason the gnomonic projection is rather obscure is that it really doesn't have a lot of other virtues. To begin with, the area it can map is limited--consider how it is formed; if we chose our center to be the North Pole, for instance, the Equator would be mapped onto infinity! No gnomonic projection can even map an entire hemisphere, no matter how it is oriented and centered. Related to this is that distortion becomes severe pretty soon. You can find some examples of gnomonic projections of large sections of a hemisphere on the internet, and there you can see that the places distant from the projection center are badly misshapen.

In fact all flat maps of any section of the globe must fail in accuracy in some way--typically there is a trade-off between conformality--or local accuracy in portraying angles and hence overall shapes--distance, and area. The gnomonic projection conserves _none_ of these features!

Thus, if you find your great circle route on your gnomonic map, you can follow it if you rely on simply looking down and flying over the features identified on the map. But if you want to rely instead on compass bearings (say you are crossing the ocean and have no landmarks, or simply can't rely on landmarks because it is dark or cloudy below) you will be badly misled if you simply look at the angles the straight route cuts across the _graticule_ (grid of latitude and longitude) you may have marked on the map. This is because the gnomonic projection is not conformal; the angle on the map between the route and the meridian you are crossing will _not_ be the same as the angle your airplane needs to be turned from North in order to stay on the route!

(The graticule on a gnomomic projection can quickly demonstrate this, if you choose centers of projection other than the Poles--all meridians will always be straight lines, but the only line of latitude that will be straight would be the Equator as it is the only one that is a great circle. The other parallels would map as ellipses or hyperbolae, and they would generally _not_ make right angles with the meridian lines when they cross them. If you choose a Pole for the center the parallels will be circles and the meridians will be evenly spaced rays and they will cross at right angles, but in general a straight line's angles across the meridian-rays will still not correspond to the corresponding great circle's angle across the real meridians).

Relatedly, the distances you have to cross in the air will not relate directly to the distances on the map--if you look at some of the gnomonic projections available on line you will see that distortion of distant regions puts them at incredibly far apparent distances, as is also obvious when you reflect that the great circle 90 degrees away from the projection center maps to infinity.

In order to use gnomonic projections for practical navigation, one would either need formulaes to convert the particular angles found on the map at particular locations to the proper bearing from North, or else one would use the gnomonic projection to find the route, and then replot it on some other kind of projection that has more useful properties. For navigating relative to true North for instance, one would want to re-map the route onto a conformal map--either the Mercator projection for the lower latitudes, or a stereographic projection centered on either North or South pole for the appropriate high latitudes. There, your route will _not_ be a straight line but a curve, but the angles it makes with the meridians _will_ match your necessary heading at that point.

Another difficulty is that gnomonic maps, as I have said, cannot possibly map an entire hemisphere, and practically speaking are rarely drawn out to more than 60 degrees from their centers, so there is a good chance that your origin and destination cannot be plotted on the same map. (And if you rely on a set of prepared charts, the chances are even greater they won't be on the same one). In order to find the direct route between points on different gnomonic charts, one has to use some sophisticated techniques! The ones I have thought of involve computing the necessary route directly using vector graphics on a sphere, and enough skill in that method would render the whole project of using gnomonic projections unnecessary.

Today, we have cheap, fast, reliable computers, and it is a fairly elementary exercise in three-dimensional vector algebra to come up with formulae that can take your current location, given in latitude and longitude, your destination given the same way, and compute both the distance you have to travel and the proper bearing you'd have to have relative to north to stay on that route. As you progress you just keep inputting your new location and it will give you the best heading and the distance you still have to go. With GPS systems your current location is given automatically and so such systems can guide you continuously. With such technology available, there is little practical use today for gnomonic projections in navigation.

Gnomonic projections continue to have other uses, for geological mapping for instance.

And I think they are very interesting, in that they reveal where the great circles are easily to the eye. They are also good for mapping the Earth onto polyhedrea such as cubes, because the great circles that form the boundaries of the segments map onto straight lines.

And a local map of a limited region might as well be gnomonic as just about any other projection; on a small scale all maps have small distortion.