Nonlinear Projections - Transformations and Projections in Computer Graphics - page 195

every projection of a sphere onto a flat plane must introduce distortions, and the prob-
lem of mapping a globe is to design and develop sphere
projections that eliminate or minimize certain distortions
(while perhaps increasing others). Thus, we can say that
cartography is the art and science of designing and choosing
the least inappropriate projection for a given application.
A map that preserves distances may be useful in certain ap-
plications even if it corrupts angles. Similarly, a map that
minimizes distortions around the equator may be ideal for
certain countries, such as Ecuador, even if it deforms the
shapes of regions close to the poles.
An important requirement in sphere projection is to preserve spatial relationships.
If a region A lies to the north of another region B on the globe, it should also appear
to the north of B on the projection (i.e., on the map resulting from the projection).
Other than preserving spatial relationships, any sphere projection is a compromise,
displaying some properties accurately while deforming others. Thus, when classifying
sphere projections, one attribute that should be considered is the extent to which a pro-
jection preserves or distorts certain properties. Following is a list and a short discussion
of the most important properties of maps. These properties are identified by answering,
for a given map, the following questions:
Can distances be accurately measured?
How easy is it to determine the shortest path between two points?
Are directions between points preserved?
Are shapes of geographical features preserved?
Are areas preserved to scale?
Which regions suffer the most distortion, and what kind of distortion?
These features are discussed here.
Scale. A map has to shrink the globe down to a convenient size that is determined
by the scale. In a 1: 10 , 000 scale map, points separated by two units on the map
represent geographical locations separated by 20,000 units on the sphere. However, no
map satisfies this condition perfectly. Scale on a flat map changes with location on the
map and the direction between the points. Measuring arbitrary distances on a map can
at best serve as an estimate of the real distances on the sphere. Recall that the shortest
distance between two points on a sphere is a great-circle arc, but such an arc is only
rarely represented by a straight line on a map.
However, some projections produce maps where certain lines are to scale. Dis-
tances measured along those lines are accurate. Such lines are called standard lines .In
a sinusoidal projection centered on the equator, all latitudes (parallels) are standard
lines. In an azimuthal equidistant map, all lines that pass through the central point
are standard. In a cylindrical equidistant map, the vertical lines (longitudes) and the
equator are standard lines.