### Introduction

Imagine a navigator during the Discoveries period who wants to travel from Portugal to Brazil. How can such a trip be accomplished using solely a compass as a navigation tool?

One way of fulfilling this task is to fix an angle with the meridians and travel keeping that angle constant, using the compass*. But which angle should be used? Is there more than one possible angle? Does the trajectory obtained correspond to the shortest path between the points of origin and destination? What would happen if, once the destination was reached, the navigator continued his course with a constant angle? Would he return to the point of origin?

Using the following applet, you can perform a simulation of such a trip and look for answers to the previous questions.

- On Earth**, points \(A\) and \(B\) are marked. To make point \(A\) or \(B\) movable/fixed, press the right button of the mouse and, while pressing it, the key \(A\) or \(B\), respectively. Afterwards, release both.
- In yellow is presented an arc of curve making always the same angles with all meridians it intersects. This angle can be changed using the cursor.
- Find an angle such that the endpoints of the yellow arc are \(A\) and \(B\). Is there an unique angle in such conditions?
- Selecting the option
*Solution*, the arc \(AB\) (in red), which has the same angle with all meridians, is displayed. This arc has the smallest length possible, in such conditions.

In the end, you can check the solution of this problem taking into account the difference between both magnetic and geographic poles.

**In this work, we consider a spherical model of planet Earth.

`http://wolfram.com/cdf-player`

(*) This work was carried out under the guidance of Professor Samuel Lopes from the University of Porto, under a grant by the FCT - Fundação para a Ciência e Tecnologia to develop a project for the promotion of Mathematics in Atractor.

Difficulty level: Upper Secondary, University