Useful Map Properties: Directions
Are Directions Preserved?
A compass is the instrument of choice for following a
prescribed route. It shows the deviation from a standard
direction, called a bearing or
course (for our purposes, magnetic declination is
irrelevant).
- Problem: suppose one leaves home by plane, keeping a
constant course bearing. Which localities will be
visited?
- Problem: given two points on a map, which bearing must be
followed to travel between them?
The two problems are related and can not be easily solved for
every location in most maps, as general directions are seldom
preserved. In a (theoretically) perfect map, meridians
and parallels must cross at right angles in every point but the
poles.
|
| Loxodrome in Mollweide map |
 |
| Same line in equidistant cylindrical
map |
A loxodrome (or rhumb
line) is a line of constant bearing. It is the
easiest route between two points since a constant bearing is
enough to follow; any other path would require frequent changes
of direction. Loxodromes are an invention of
Pedro Nunes (ca. 1533), after suggestions by Martim Afonso
de Sousa.
The blue line
shows the loxodrome as a path starting near Campinas, Brazil,
with constant bearing 60° clockwise from true North.
The North Pole is reached after an infinite number of tighter
and tighter turns. The Mollweide map is
equal-area
but suffers from strong shape distortion near the poles.
Reading the loxodrome is a bit easier in the cylindrical
equidistant projection.
The orthographic
projection helps viewing the rhumb line's constant angle with
every parallel and meridian. In a stereographic map,
the rhumb line maps to a logarithmic spiral, the plane curve
which intersects every radius at a constant angle and looks the
same no matter how magnified.
Actually, any rhumb line is part of a curve which winds from
pole to pole, called a spherical helix.
|
 |
| Oblique orthographic map |
Polar
stereographic map |
 |
| Loxodrome in Mercator map, clipped at
85°N and 35°S |
Mercator's most
famous projection is unique:
every loxodrome is drawn as a straight line, making
trivial finding the bearing between any two points. However,
the Mercator map alone is not enough for general navigation.
Also, in this equatorial form, the polar regions can not be
included (here the loxodrome has constant slope and infinite
length; therefore the North Pole should be infinitely far
up).
 |  |  |  |  | | www.progonos.com/furuti September 21, 2002 |