Display the shortest distance between two points on the map:

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Why is a curve the shortest route? Why not just draw a straight line? The answer is this: The earth is not a flat map, but a complicated body that can be described to the best approximation as a flattened sphere, a so-called rotational ellipsoid or spheroid. The map shown is a so-called projection of this ellipsoid. It is called a Mercator projection. The best way to explain this is to use the degree grid:

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Source: wikimedia.org

If you now imagine that all meridians, i.e. all vertical lines, actually meet at the top and bottom at one point, the poles, and that the left and right sides also touch, then you’ll get the body of the earth. In principle, therefore, all the apparently perfectly straight lines on the map are actually curved. If you were to draw an apparently straight connecting line on the map in the same way, it would of course also be curved when it is transformed into the body of the earth.

Conversely, a curved line on the ellipsoid may suddenly appear straight and direct:

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Source: wikimedia.org

In precisely this case, we are dealing with a so-called geodesic or orthodrome, which represents the actual shortest path on the surface of a body. The geodesic lies on a so-called great circle, a circle around the entire body on which both the start and end points lie.

Nevertheless, we have now implemented in version 3.1 of the GC Wizard the option to display this direct connection between two points (blue):

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This is actually not the shortest real connection, but a so-called rhumb line or loxodrome. In contrast to orthodromes or geodesics, it has the property of being “angularly stable”. What does that mean? If you look at the geodesic, it always changes direction at every point on its curve. In our example, the red line starts in the west with a bearing of 36° to the east. However, if you were still following the course of 36° halfway along the route, you would end up somewhere completely different:

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With a geodesic, the direction is therefore always corrected with regard to north. This is also the reason why you cannot simply turn the direction by 180° if you want to return from the destination to the start. In our example, this would not be 36° + 180° = 216°, but the geodesic actually starts in the return direction with a bearing of 305°.

The rhumb line, on the other hand, does not have this problem, or is even defined in such a way that it never changes its bearing:

What looks absolutely straight on the map, because every perpendicular meridian on the Mercator map has the same angle to the rhumb line, leads in reality to a spiral on the ellipsoid.

Today, rhumb lines play a subordinate role in navigation. As a really significant deviation is only really noticeable over large distances, they were only of interest for seafaring. On large maps, a ruler could simply be applied and a course was set NNW and held permanently – even if this meant not taking the shortest route. But you didn’t have to constantly calculate a new course. Today’s ships and airplanes always go as efficiently as possible along geodesics, thanks to GPS (given no other factors, such as politics or weather, play a role).

For geocachers, however, this function is actually not entirely uninteresting:

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Suppose you know the north part of your coordinate (exactly 51° in the picture), but not the east part. So you want to draw a connecting line on the map and use it to see what it intersects. The map would actually show you the shortest route, but not a real interpolation line. In this extreme example, you can clearly see that, for example, in the middle between the two end points, the north value differs significantly from what you have already determined:

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The north value can therefore change on the geodesic, which is not what your interpolation is for. Instead, a connection as a rhumb line is recommended here:

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It also becomes interesting if you need a real projection but only barely know the distance. So you know: From point X, the target is at 45°. You could now project a very distant point at 45° onto the map and see what crosses the line again:

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Due to the angular stability, the rhumb line (blue) can actually detect all points at any distance at a bearing of 45° from your point X, whereas the geodesic (red) differs. However, to be honest, the deviation in our playing areas is rather theoretical at a few 100 to a maximum of 1000 meters, but it is nice to know that the GC Wizard does not lack a certain accuracy here either. And who knows, maybe the global mystery of historical seafaring will come along one day 😉

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And how can you display these rhumb lines? As of version 3.1, you can simply click on your line in the map view, click on the edit icon and then change the line type in the line editor window. Of course, in the pop-up menu of the line, the length of the line will then be displayed as a rhumb line instead of the normal geodesic length. Of course, there will also be two new tools in the coordinates section which, like the familiar tools for calculating bearings and distances (based on geodesics), can also perform these calculations directly for rhumb lines beside the map.