The loxodrome and two projections of the sphere 
Curves and an angle
Imagine the planet Earth covered in water and three people located in different points on the planet. All three decide to start travelling, maintaining the same angle, previously fixed, with the meridians. Will their paths cross?
In these conditions, all three will describe loxodromes. Explore what might happen, using the following applet:
- Pick the angle that the loxodromes make with the meridians.
- Choose a position for each point. To move or fix a point, you should press the right button of the mouse and, while pressing it, also press \(A\), \(B\) or \(C\), respectively. Afterwards, release both.
- Notice that, by fixing the angle, two loxodromes with such angle of intersection with the meridians either coincide or are disjoint. Therefore, if none of the three people are over the loxodrome of another, they shall never meet.
