Suppose we need to plan the itinerary of a journey between two distant points like the cities of Campinas (Southeastern Brazil) and Tokyo (Japan). Assume we can travel by air, and simplify matters ignoring things to avoid, like bad weather and restricted airspaces. Obviously we'd like to reduce time and costs by choosing the shortest possible route, which on a flat surface would always be a simple straight line. We'd like to check our trajectory in order to plan for eventual refuelings and maybe stops for rest; it'd be useful for evaluating the risks of long stretches over ocean, desert, or other similarly inhospitable places.

So, how do we ascertain our path? Just pick up a ruler and a map, then draw a line joining the endpoints of our journey? It's not so simple, and choosing the right projection is essential if we want meaningful results.

Given two points *A* and *B* like Tokyo and
Campinas, plus the center of the Earth, a unique (unless *A* and
*B* are antipodes) plane is defined. The intersection of
this plane and the planet's surface is
the **great circle**
or **orthodrome**, also unique,
containing *A* and *B*; it is maximal, as its
radius and circumference are the same as the Earth's; here we
assume the planet to be a perfect sphere. Any surface circle
resulting from a plane which contains the planetary center is a
great circle, and any surface circle not doing so is
a **small circle**. Therefore, every meridian is a
great circle, while all parallels except the Equator are small
circles.

Points *A* and *B* split their great circle in two
arcs of which (except for antipodal *A* and *B*)
one is shorter than the other. The shorter arc,
called **geodetic**,
**great circle path**, **geodesic
curve** or
simply **geodesic**, is
actually the shortest *surface* path between *A*
and
*B*; the straight, truly shortest three-dimensional
path is underground and certainly not feasible with our
current technology.

So, ideally our map should present great circles as straight
lines, making easy drawing and measuring the geodesic.
Unfortunately, only a few projections can do so, and usually
only in special circumstances which limit their
general usage. Certainly no map projection can show the
geodesics between *any* two points as straight lines.
On projections like
the Robinson,
Mercator
and Mollweide,
commonly used in world maps,
great circles are drawn as complicated curves. Cylindrical projections,
still prevalent in wall maps, present special problems near
the poles, where horizontal scale is especially stretched.
Sadly, many cylindrical projections are probably chosen due to
their neat rectangular shape rather than any outstanding
cartographic property.

The
azimuthal class of projections present true directions
from a selected central point, the *azimuth*, which
usually coincides with the center of the map. In particular,
all great circles crossing the azimuth are drawn as straight
lines. The azimuthal
stereographic projection is *circle-preserving*,
as any circle upon the sphere (every geodesic, parallel and
meridian) is still mapped to a circle; geodesic lines crossing
the central point map to circles with infinite diameter,
i.e. straight lines. Unfortunately, it can show only one
hemisphere at a time. Another very important azimuthal
projection, the remarkable gnomonic projection maps into straight
lines *all* great circles, even those not passing through
the central point, but can present even less than one
hemisphere. It is the best tool for direct determination of
the great circle, but its usefulness is limited for distant
*A* and *B* due to extreme shape and scale
exaggeration far from the azimuth. The azimuthal
equidistant projection can include the whole world and
presents true direction *and* distance to any point from
the center while suffering from lesser distortion near the map
periphery.

Projection distortion and unfamiliar shapes could make difficult realizing that great circles on maps are as straight as they can be on spherical surfaces. The azimuthal orthographic projection clearly shows the Earth's sphericity as seen from a vantage point far away in space. This closely mirrors the practical expedient of finding a geodesic by applying a taut line against a globe.

Copyright © 1996, 1997, 2008 Carlos A. Furuti