ATRACTOR

Imagine planet Earth covered with water and a navigator that always travels maintaining the same angle with the meridians. Will he return to the starting point of his trip?

In the following applet, you can simulate the path of such journey (a loxodrome).

1. You can move point \(A\). To accomplish this, press the right button of the mouse and, while pressing it, press also key \(A\). Afterwards, release both. In order to fix the point, choose a point on the sphere and proceed in a similar way: press the right button of the mouse and, while pressing it, press also key \(A\). Afterwards, release both.
2. You can select different loxodromes that pass through the point and adjust the angle of each curve with the meridians as needed. As particular choices, select angles with amplitude 0º and 90º.
3. You can notice that given a point in the sphere and an angle, there exists an unique loxodrome passing through that point and making that angle with all meridians. However, in general, there exist an infinite number of loxodromes passing through two given points.
4. Notice that:
   • if the angle is different from -90º, 90º and 0º the traveller never returns to a point where he has already passed; this means the curve does not intersect itself and it is not closed;
   • the traveller never reaches the poles, spiralling around them (when the angle is different of -90º, 90º and 0º);
   • the loxodrome has finite length given by \(\frac{\pi}{|\cos\alpha\,|}r\) where \(\alpha\neq\) -90º, 90º is the angle that the loxodrome makes with the meridians and \(r\) is the radius of the sphere.