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.