Nonlinear Projections - Transformations and Projections in Computer Graphics - page 196

A small-scale map portrays a large area and a large-scale map portrays a small area
of the Earth. It is intuitively clear that a small region of a sphere is not much different
from a flat plane, which is why a large-scale map is not sensitive to the projection
algorithm. When mapping a small area of a sphere, practically any projection method
will produce a map where distances, areas, and angles are fairly accurate. The problem
of distortion arises when a large area of a sphere has to be mapped. In such a case, no
projection method will produce ideal results, and the algorithm used has to be selected
depending on the application at hand. One projection method may be suitable for
navigation, while another may produce maps useful for surveying.
The shortest path between any two points on a sphere is a great-circle arc, also
called a geodesic. Thus, a projection where great circles are displayed as straight lines
is ideal for measuring shortest paths. No sphere projection can generate such a map,
but the stereographic projection comes close to satisfying this requirement because it
preserves circles. Any circle on the sphere is mapped by this projection to a circle. In
particular, a great circle passing through the center of the projection is mapped to a
circle with infinite radius, a straight line. Thus, straight lines through the center of a
stereographic projection are great circles and indicate shortest paths. The downside is
that this projection can show only one hemisphere, which limits its use in air navigation
to short and medium distances. The gnomonic projection maps all great circles, not just
those passing through the central point, into straight lines, but this projection projects
even less than a hemisphere.
A map prepared especially for determining property taxes should allow for accurate
measurements of areas. If the scale of the map is s and if the area of a certain region
is A ,thentheareaoftheregionasmeasuredonthemapshouldbe A/s . Such a map
is termed equal-area and may distort the shapes of areas and display wrong distances
between points.
Even a quick glance at a Mercator map shows a huge Greenland about the same
size as all of Africa, obviously not to scale because the ratio of their areas is 1: 13 . 7. In
this projection, areas close to the poles appear bigger than they should. In contrast,
the Mollweide projection preserves areas.
It is easy to tell when a familiar shape becomes distorted or deformed. On the
other hand, it is not obvious how to measure distortion quantitatively. We are familiar
with the shape of the continents on Earth, so when a landmass gets distorted by a
projection, we recognize the deformation, but it took cartographers several centuries to
come up with a simple measure that shows the amount and direction of the distortion.
This measure was introduced by Nicolas Tissot in the 19th century and is known today
as Tissot's indicatrix. The idea is simply to add a grid of small circles to the globe area
being mapped. The circles are mapped with the other items in the area (land areas,
oceans, rivers, etc.), and a quick look at a circle shows the amount and direction of its
distortion. A circle may retain its shape and area, it may get scaled but keep its shape,
or it may become deformed.
Figure 4.45a shows the Tissot indicatrix for the sinusoidal projection. It is obvious
that distortion is minimal around the equator and increases toward the poles. Also,
the circles are distorted, but their area is preserved. In contrast, part (b) of the figure
indicates that the Mercator projection, which is conformal, does not distort shapes but