## The loxodrome and two projections of the sphere (*)

### 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.

1. 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.
2. 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.
3. Find an angle such that the endpoints of the yellow arc are $$A$$ and $$B$$. Is there an unique angle in such conditions?
4. 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 reality, the magnetic poles do not coincide with the geographic poles and as a consequence, the compass does not point to the geographic north pole, but to the magnetic north pole. In the following pages, we assume that the angle corresponds the corrected angle given by the compass, taking into account the magnetic declination of the place (the magnetic declination of a place is the measure of the angle between the direction of Earth's magnetic north and the direction of the geographic north, determined by Earth's rotation axis).

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.

Translated for Atractor by a CMUC team, from its original version in Portuguese. Atractor is grateful for this cooperation.

This work integrates interactive components in CDF format prepared with the Mathematica program. To use these files, you must download them to your computer and access them with the CDF Player, which can be downloaded for free from 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