Mercator map: loxodrome or rhumb line in blue; part of a geodesic line or great circle in red 
However, that easy route would not be the most economical choice in terms of distance, as the geodesic line shows.
The same loxodrome and great circle in part of a polar azimuthal equidistant map 
Since there is a tradeoff:
The same great circle (this time covering 360°) and loxodrome in a "Lagrange" conformal map 
Moving circles between globe and map  
Suppose a set of concentric circles, with radiuses increasing in
1500km steps, centered on Campinas, Brazil. These are true
circles on Earth, which could demonstrate the theoretical range
of radio waves, airplanes, or missiles; they are represented here
in blue and are identical in all maps in this table. In each pair
of maps with blue shapes, the one on the left is a simple
aspect, while its right counterpart is usually
an oblique aspect centered on Campinas.
On the other hand, the orange lines were directly drawn as circles on a map on the left column; their true shape on the globe is presented on the right (they are the same curves only for each pair of maps). There is only one projection and aspect where both size and shape of circles is identical on globe and map. 

Like all azimuthal projections, the azimuthal equidistant preserves the shape of any circle centered on the map's center of projection, but not necessarily of others (above left); however, for those whose center does coincide with the map's, the scale is also preserved: on the map on the right, notice how the radiuses are linearly spaced. Given a proper aspect, this projection is the only correct tool for graphically finding ranges on a flat map.  
A Mercator map is conformal, thus preserving shapes locally but not globally. Scales change quickly towards the top and bottom of the map, especially vertically.  
Also conformal, an azimuthal stereographic projection preserves the shape of all circles, even those not centered on the map (left). Nevertheless, their scale is not preserved: they do not "grow" linearly, and those on the left, perhaps surprisingly, are not concentric.  
A very simple design, the Plate Carrée is a particular case of the equidistant cylindrical projection. The scale is the same on the Equator and all meridians (as measured on the equatorial aspect). Therefore the width and height are identical for the "circles" on the right only.  
Because all cylindrical projections exaggerate horizontal scales towards the top and bottom of maps, if circles are naively drawn with a pair of compasses on a Plate Carrée map, their correct shapes on Earth are pinched, as shown by the azimuthal equidistant map on the right.  
A particular case of Lambert's equalarea cylindrical projection, Gall's orthographic (also known as GallPeters or "Peters") projection preserves areas but strongly distorts shapes, with vertical scale changing very nonuniformly.  
Transferring circles drawn on a Gall's orthographic to the real globe yields shapes even farther from correct. Because scales are stretched between 45°N and 45°S, and compressed elsewhere, the true shapes are correspondingly deformed on the globe.  
Eckert's sixth projection is pseudocylindrical and equalarea; its poles are shown as lines, but with lesser horizontal exaggeration than in cylindrical projections. On the other hand, horizontal distortion depends on the longitude.  
Most interrupted maps are split at specific lines depending on the mapmaker's priorities; therefore changing the aspect usually either robs the map's purpose or demands a whole new set of interruptions. Here are normal aspects of Goode's interrupted homolosine (left) and Fuller's Dymaxion™ maps. 
An interesting shape to study is the circumference, the set of points at a fixed distance from a center, and the circle, the set of points it encloses. Since both scale and shape are often distorted, how faithfully are circles drawn on a globe translated to a map? Or, conversely, should one draw a circle on a map with a pair of compasses, which shape does it actually represent on Earth? The circle could represent the range of a radio station, or the autonomy of a vehicle. When rangefinding, correct usage of map projections could mean the difference between a successful trip and a mayday call.