Let's return to the initial problem where we want to plan a route, now by airplane, keeping always the same angle with the meridians. The problem of determining the angle becomes simple if we use a mapa obtained from Mercator's projection since in this map, loxodromes are represented as straight lines.
Using the following applet, you can choose two locations on the planet and verify which angle to pick and the travelled distance in each case. Notice that, in general, given two points, an infinity of loxodromes pass through these points. When doesn’t that happen?
- In the sphere are displayed two movable points \(A\) and \(B\). In order to move the points, you should press the right button of the mouse and, while pressing it, press key \(A\) or \(B\), respectively. Afterwards, release both. To fix one of the points, choose a location on the sphere and proceed in the same way, that is, press the right button of the mouse and, while pressing it, press key \(A\) or \(B\), respectively. Afterwards, release both.
- The map corresponds to Mercator's projection of the points with latitude between -80º and 80º. The x-axis (horizontal axis) is the projection of the Equatorial line; on this axis, the longitudes varying between -180º and 180º are highlighted and the meridians with longitude -180º coincides with the one with longitude 180º. The y-axis (vertical axis) is the image of Greenwich's meridian; in this axis, latitudes are highlighted.
- You can select some loxodromes with different angles passing through the points. The curves are ordered by increasing arc length of \(AB\).
- Observe that, in general, Mercator's projection of a loxodrome is a set of parallel straight lines, identifying points with the same latitude and whose longitudes differ by a factor of 360º, respectively.
