Transformations - Spatial Mathematics: Theory and Practice through Mapping - page 76

that the map projection affects not only the appearance of the buffers, but
how the buffers are computed in the first place. The distances that are used to
create them depend on the underlying map projection. This will be evident in
the next section as you change the scale and look at larger areas.
Zoom out until you see most of the Earth. Change the buffer distance to 100
miles or 100 kilometers. Draw a polyline near the Equator, and compare it
to a polyline that you draw near the North Pole. Next, draw a polyline from
northern Canada to the Equator. What do you observe? Why does the buffer
distance change with latitude? You should see that the buffer distance appears
to be wider as you approach the poles, and narrower as you approach the
Equator. The buffer distance is the same across the planet, but because the
map is cast onto the Web Mercator projection, in order to remain true to the
projection, the buffers have to widen to be shown accurately near the Poles.
Along with the buffers, you should notice that the underlying land masses
also appear to widen and become misshapen.
3.5.2 Geodesic versus Euclidean buffering
As we have discussed, like all spatial analytical tools, buffers fundamentally
depend on the mathematics behind them, and specifically, the shape of the
Earth. Geodesic buffers account for the actual shape of the Earth as an oblate
spheroid in their calculations. Euclidean buffers measure distances in a two-
dimensional Cartesian plane. Euclidean buffers work best when analyzing
distances around features that are concentrated in a relatively small area, in
a projected coordinate system. Using a web browser, access the Geodesic
Buffering web GIS application on: http://resources.arcgis.com/en/help/flex-
api/samples/index.html#//01nq0000002q000000.
Click anywhere on the map to generate a line. What shape do the resulting
buffers have, and why? The short line segments that you generate create
buffers that are oval in shape, because they represent shapes that enclose
areas within 1000 kilometers of the line segments. Figure 3.8 illustrates
geodesic and Euclidean buffers around Detroit, Michigan. Move closer to the
North Pole, redraw a line, and then repeat closer to the Equator. Why is the
geodesic buffer (shown in red) larger than the Euclidean buffer (shown in
blue)? The geodesic buffers are larger because they consider the Earth as a
curved object, whereas the Euclidean buffers treat the Earth as a flat plane.
Where are the two buffers closest in size to each other? Why? The two buf-
fers are most similar near the Equator and are quite different near the poles.
Interpretation problems can ensue when performing a Euclidean buffer on
features stored in a projected coordinate system where the map projection
distorts distances, angles, areas, and the shapes of features. The danger of
using the wrong kind of buffer will become obvious when we discuss a noto-
rious example regarding areas within range of a certain country's missiles
later in this topic.