"To project" means transferring features from Earth to a suitable
surface, like a tangent plane, a secant cylinder or a secant truncated
cone. Regions nearer the surface are usually better presented.
From ellipsoid or sphere (left) to
flat map (right). A conceptual intermediary surface (center) may be useful for
either actual construction or mere visualization, but the darker conversion
paths unavoidably incur in distortion.
An orange's peel provides a classic demonstration of distortion in maps: it
cannot be completely flattened unless compressed, stretched or
torn apart.
Maps, Globes and Projections
Any study in geography requires a reduced model of the Earth, like a
globe or map. Neither is
perfect: a globe is seldom practical, and flat maps are never free from errors.
Selecting or creating a good map involves interesting choices and
trade-offs.
What's a Projection?
A cartographical map projection is a formal process which converts (mathematically
speaking, maps) features between a
spherical or ellipsoidal surface and a projection surface, which is
often flat. Although many projections have been designed, just a few
are currently in widespread use. Some were once historically
important but were superseded by better options, several are useful only
in very specialized contexts, while others are little more than
fanciful curiosities.
Projection Surfaces
The map's support, the projection surface is usually created, i.e.,
developed, conceptually touching the mapped sphere in
one (the surface is tangent to the sphere) or more (the
surface is secant) regions. Intuitively, portions of the
surface nearer the touching regions depart less from the original
spherical shell; therefore, the corresponding portions on the map are
more faithfully reproduced. Some projections are actually composites,
fitting separate surfaces to different regions of the map: overall
error is reduced at the cost of greater complexity.
Sometimes a conceptual auxiliary surface
like a cone, open cylinder, ellipsoid or torus is employed: the sphere's
features are (often by perspective construction) transferred to that
surface, which is then flattened. Many projections are classified as
"cylindrical" or "conic"; however, for most of them, the naming is
just an analogy or didactic device, since they aren't actually
developed on an intermediary surface; rather, the
resulting map can be rolled onto a tube or a cone.
A few selected projections illustrate how the same spherical data
can be stretched, compressed, twisted and otherwise distorted
in different ways.
The azimuthal equidistant map has interesting
properties regarding directions and distance from the central pole,
but the outer hemisphere is greatly stretched: its pole becomes a circle.
Both poles become lines in the equal-area cylindrical map,
but it covers the same area as the original sphere; also, all octants
have identical shape. This particular star projection has unequal
octants and marked loss of continuity; however, it also preserves area.
In the Winkel Tripel map, octants have different shapes, area is
changed and poles are linear, but overall distortion is subjectively
smaller. Finally, the orthographic views, projections themselves, show
only part of the sphere.
All projections suffer from some distortion; none is "best" for
all purposes. Octants would assume even stranger shapes in oblique aspects.
No matter how sophisticated the projection process, the original
surface's features can never be perfectly converted to a flat map:
distortion, great or small, is always present in at
least one region of planar maps of a sphere. Distortion is a false
presentation of angles, shapes, distances and areas, in any degree or
combination.
Every map projection has a characteristic
distortion pattern.
An important part of the cartographic process is
understanding distortion and choosing the best combination of
projection, mapped area and coordinate origin minimizing it
for each job.
Cones and cylinders are developable surfaces with
zero Gaussian curvature (in a nutshell, at every point passes at least
one straight line wholly contained in the surface). Therefore, although
distortion always occurs when mapping a sphere onto a cone or cylinder,
their reprojection ("unrolling") onto a plane incurs in no further errors.
Another key feature of any map is the orientation, relative to
the sphere, of the conceptual projection surface.
A particular projection may be employed in several
aspects, roughly defined by the graticule lay-out
and the sphere's region nearest the conceptual projection surface,
commonly the center of a whole-world map (not
necessarily
the actual center, due to cropping or recentering):
a polar map aligns the north-south
axis with the projection system's, so it is useful when one of the poles must lie at the
map's conceptual center;
an equatorial (occasionally
known as meridian, or meridional) map is
centered on the Equator, which is set across one of the map's major
axes (mostly horizontally);
an oblique (seldom referred
to as a horizon) map has neither the polar axis nor the
equatorial plane aligned with the projection system.
Also, orthogonally,
the most "natural" aspect of a projection, called normal, conventional, direct or
regular, is ordinarily determined by geometric
constraints; it demands the simplest calculations and produces
the most straightforward graticule. The polar aspect is the normal one for the
azimuthal and
conic
groups of projections, while the equatorial is the normal
for
cylindrical
and
pseudocylindrical
groups. The normal graticule for azimuthal and conic projections
comprises exclusively straight lines and circular arcs; normal cylindrical
ones have altogether straight grid-like graticules
the transverse aspect
is created by rotating the polar axis by 90°: if the normal aspect
is centered on a pole, the transverse is centered somewhere on the Equator;
if the normal aspect is aligned with the Equator, the transverse is aligned
with a meridian, and so on.
(polar)
(equatorial)
(equatorial)
(polar)
(seldom used)
(seldom used)
(seldom used)
(seldom used)
Three (normal, transverse and oblique) aspects applied to four
(azimuthal equal-area,
Gall's stereographic
cylindrical,
Albers's conic and
"Lagrange"'s)
projections with different tangent projection surfaces in blue (just a
few of infinitely many possible oblique maps are presented). Some
projections like Gall's stereographic may actually be derived via
perspective geometry; for most, however, surfaces are only
illustrative: the map may be laid on a developable surface, but is not
calculated from it.
The distinctive graticules of some projection groups (radially
symmetric meridians in azimuthal and conic maps, rectangular grid in
cylindrical maps) are only realized using their simple, normal
aspects. Despite a common misconception, this classification is not
exclusive: most projections involve neither a cone nor a cylinder but
are not azimuthal either. Trivial rotations of the finished
map, like turning it sideways or upside-down, leave both aspect and
projection unchanged. On the other hand, modifying the aspect affects
neither represented area nor the shape of the whole map.
For both normal and transverse aspects, the only remaining choice is how much,
if at all, to rotate the Earth around the polar axis, determining the map's
central meridian. There are infinite choices for the two angles
of rotation determining oblique aspects.
Some authors consider a different definition of "aspect": it
determines whether the projection surface is secant or tangent to
the globe (this is of course a more limited meaning, since many
projections are not defined via an auxiliary surface). Still
others reserve "aspect" for one meaning and "case"
for the other, or even for distinguishing equations devised for
an ellipsoid versus the simpler sphere.
Theoretically, especially supposing a spherical Earth, any
projection may be applied in any aspect: after all, the
parallel/meridian system is a convention which might have origins
anywhere, although it is hard imagining others more useful than the
poles. However, many projections are almost always used in
particular aspects:
their properties may be less useful otherwise. E.g., many factors
like temperature, disease prevalence and biodiversity depend on
climate, thus roughly on latitude; for projections with constant
parallel spacing, on equatorial aspects latitude is directly
converted to vertical distance, easing comparisons.
several projections whose graticules in normal aspects are
comprised of simple curves were originally defined by geometric
construction. Since many non-normal aspects involve complex curves,
they were not systematically feasible before the computer age
(indeed, mapping was an important motivation for calculation
shortcuts like logarithms).
Even though oblique aspects are frequently useful,
in general calculations for the actual ellipsoid are
fairly complicated and are not developed for every projection.