Projections at identical scale with
maximum angular deformation values represented by colors
Map in sinusoidal
(Sanson-Flamsteed) projection:
equal-area, but only the Equator and central meridian are
free of angular deformation
Mollweide's projection: the sinusoidal's "cross" of low deformation
is replaced by two oval "islands"
Mollweide's projection, in oblique aspect minimizing distortion on Southeastern Brazil and a corresponding area on the northern hemisphere
Mollweide's projection, interrupted with
four symmetric lobes. The size of each lobe limits the
distortion range, at the cost of distance discontinuities
Bromley's rescaled version of Mollweide's projection: the "islands"
of low distortion merge at the Equator
Interrupted Goode's homolosine projection (in a simplified
lobe arrangement with no repeated areas). Notice the discontinuity
at the two boundary parallels
Interrupted Boggs's eumorphic projection (in simplified
lobe arrangement, unbroken Eurasian lobe). Distortion is
continuous
Polyhedral maps are a particular case of interruption. On an
unfolded Waterman polyhedron, the gnomonic projection centered on faces is azimuthal, therefore the deformation pattern is radially symmetric.
Eckert's projection
I (neither equal-area nor conformal), with a sharp
direction break at the Equator
Eckert's projection
II. A comparison with
its predecessor shows that, as frequently
happens in cartography, adding a favorable property (areal
preservation) brings a drawback, in this case a smaller and
more irregular region of low shape distortion
Azimuthal stereographic
maps are conformal everywhere, but the point opposite the projection
center cannot be shown: they are customarily limited
to hemispheres
Lambert's azimuthal projection: equal-area, at the cost of pronounced distance and shape distortion near the periphery
Wiechel's pseudoazimuthal equal-area projection. Again, an improvement (correct scale along meridians) is offset
by a reduced area of lower distortion
Equidistant cylindrical (neither equal-area nor conformal) projection with standard parallel at Equator
Tissot's equations describe
how scaling factors are affected at each point in two
principal directions, along the meridian and along the
parallel which intersect there. On Earth, those directions
are of course orthogonal; on the projected map, the
transformed scaling factors provide:
the areal scaling factor, which expresses how dimensions are
stretched or compressed; it is 1 everywhere for equal-area projections
the maximum angular deformation, which
presents a deviation of directions and is null everywhere for
conformal
projections. It is conveniently calculated as a doubled angle
ranging from 0° to 180°. E.g., in an equatorial
world Mollweide
map, the highest maximum angular deformation occurs at the
poles, where the boundary meridians are exactly horizontal
instead of the correct vertical
The angular deformation has an orientation, clockwise or
counterclockwise, though in practice it is omitted, since the
absolute magnitude is much more relevant. Determining areal and
angular deformations is important for selecting a projection
and, once one is picked up, minimizing effects of distortion.
For some constrained groups of projections, like the normal azimuthal,
cylindrical
and conic projections, the two
distortions can be calculated almost directly from the
projection's equations. In the general case, they must be
evaluated numerically via partial differentiation. Either way,
their values may be represented as colors on a map, immediately
showing the places of greater and lesser distortion.
Deformation Patterns and Angular Distortion
Most projections conceptually based on a developable
surface present least distortion at points or lines of
tangency, which usually coincide with the center or axes of the
map. Similarly, the non-perspective sinusoidal
projection preserves areas everywhere but is free of angular
distortion (in the equatorial
aspect)
only along the Equator and the central meridian. In
contrast, the Mollweide projection, also equal-area, has zero
angular distortion in only two points, at the intersection of
the central meridian with two standard
parallels (about 40°N and 40°S); even though scale
is constant along the Equator and all meridians cross it
orthogonally, angular deformation along it is not zero due to the
exaggerated vertical scale. When choosing between these two
designs, it is worth considering that while the sinusoidal shows
a smaller area with maximum angular deformation below 20°,
its maximum deformation is only about 115° in contrast with
180° for Mollweide's.
Once a projection is decided, several approaches are available
for dealing with its limitations:
recentering,
which usually involves adopting an oblique
aspect, shifts regions of greater interest to areas of lesser
deformation. Like Tissot's indicatrices, the deformation
pattern is intrinsic to the projection and immune to aspect
changes. Nevertheless, recentering is not always feasible,
since it may disturb some desired functional aspect; e.g.,
horizontal straight parallels
interruptions: a spherical surface has no bounds, but any
conventional map creates at least one boundary and large
distance discontinuities at a point or line, predominantly
antipodal to the projection's center, which becomes part of
the map's border.
Interrupted maps add
even more boundaries; to make up for the additional
discontinuities, each interrupted region (lobe) may have its
own conceptual projection center, therefore limiting
deformation. Interrupting can take several forms, and
frequently lobes are shaped to roughly coincide with or
contain the relevant regions like continents or, conversely,
major oceans. Polyhedral
maps are a special case of interrupted projections
adjusting parameters: some projections can be generalized
or tuned by relatively elementary modifications, which again
may shift the deformation pattern. E.g., in Mollweide's
projection the standard parallels can be altered by
rescaling the map, and still keeping equivalence: Bromley's
version makes the Equator free of distortion. Likewise,
several authors proposed "new" equal-area
cylindrical projections by adjusting the standard parallels
of Lambert's
cylindrical design
combining methods: "hybrid" projections attempt to balance
features of two or more previous designs, either drawing each
relevant region with the most advantageous projection, or averaging
their equations. In the first case, whose classic example is Goode's
homolosine
projection (which juxtaposes a sinusoidal central
region between two Mollweide outer bands and is almost
always interrupted), the deformation pattern often suffers
from sharp discontinuities at the region boundaries;
sometimes transition zones are customized to smooth out the
abrupt changes, but generally at the cost of violating the
projection's major properties. Other examples are most
star
projections and the HEALPix
grid. In contrast, Boggs's eumorphic
(also equal-area), Eckert's V and
Winkel's tripel designs
are classic examples of averaging projections.
Equal-area projections are never conformal, and removing the areal
equivalence constraint can improve their range of angular deformation.
For instance, in Eckert's series of flat-polar
proposals, three pairs of projections look much alike, but one of
each pair is equal-area. Even though his projection I is not
conformal, its central area of lower deformation is larger and more
uniformly shaped than in the equal-area II.
