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Hack 30. Plot a Great Circle on a Flat Map | Mapping Your World
Hack 30. Plot a Great Circle on a Flat Map / Mapping Your World from Mapping Hacks
Hack 30. Plot a Great Circle on a Flat Map. Wherein our heroes discover that the shortest distance between two points on a globe is not a straight line, after all. What's so great about a great circle ? A great circle, technically speaking, is any circle that goes all the way around a sphere, with its center on the exact center point of the sphere. As it turns out, when a great circle connects two points on a sphere, the arc between them is always the shortest distance between those two points. Naturally, being able to take the shortest possible path is a matter of great financial and practical importance in this modern era of air travel. Familiar two-dimensional map projections don't give a good impression of great-circle distance. On a Mercator map, the straight line that seems to be the shortest route between San Francisco and London passes through Boston, but, in fact, due to the curvature of the globe, the actual shortest route runs nearer to the North Pole, passing over the south of Greenland. This hack makes use of Generic Mapping Tools, or GMT [Hack #28] to show how to plot segments of a great circle on many different cartographic projections, including those designed for marine and aerial navigation. We'll also show you how you can try this yourself with a quick Perl script. 3.10.1. Great Circles on a Mercator Projection. The Mercator projection was historically useful because it preserved navigational direction along lines of constant bearing, known as rhumb lines . One could draw a straight line to one's destination on the map, set off in the direction indicated by that line, and actually arrive at the intended destination sometime in the future. Navigating by a Mercator map therefore had the great advantage of simplicity, but the disadvantage was that the rhumb line between two points was often not the shortest path. Instead, the shortest path between two points follows a line of variable bearing, which turns out to be the arc of a great circle. This may seem counterintuitive, because a great-circle arc will usually end up looking curved on a flat map. Figure 3-31 depicts the great-circle arc connecting San Francisco and London on a Mercator projection of the world. Figure 3-31. Great-circle arc from SF to London on a Mercator projection. The following commands, using pscoast and psxy from GMT, were used to generate Figure 3-31: The call to pscoast draws the graticule (i.e., grid lines) and the base map of the continents. We recommend reviewing [Hack #28] to understand exactly how these particular pscoast options do their magic. The first call to psxy actually draws the great-circle arc into the same file. We give psxy the same projection parameters we did pscoast, along with a filename, points.txt. The points.txt file simply contains the following: The geographically savvy reader will recognize these as the longitude and latitude coordinates of San Francisco and London, respectively. The -W8 option to psxy tells it to make the great-circle arc 8 pixels thick. Finally, we call psxy one more time, with the same projection parameters and filename containing our points, but this time we give the -S option to request that symbols be drawn at each point, instead of a line connecting them. In this case, the -Sa.75c option draws us a star .75 centimeters wide at each end point, coloring each one black via the -G option. The shortest line between two points on any geometric figure is referred to as a geodesic , of which the great circle along a spherical surface is but an example. For this reason, the GRASS command for drawing great circles is called r.geodesic. The word geodesic, which comes from the Greek words for dividing the Earth," is also used to describe Buckminster Fuller's dome structures, in the way that they systematically divide the surface of a sphere (or hem




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