Geoscience Reference
In-Depth Information
Moreover, the Earth is not exactly a sphere. When land masses, weather pat-
terns, ocean currents, and other terrestrial phenomena are overlain on the
close-to-spherical Earth, the underlying geometry becomes even more com-
plex. Often it is helpful to look at the geometry, and the Earth itself, from the
synthetic as well as the analytic viewpoint.
The whole is often more than the sum of the considered individual parts.
Analysis begins with a global view and dissects that view to consider individ-
ual components, as they contribute to that whole. In contrast, synthesis often
begins with a sequence of local views and assembles a global view from these,
as a sum of parts. The assembled view may or may not fit “reality.” These two
different approaches to scholarship yield different results and employ different
tools. In this topic, we consider both approaches to looking at real world issues
that have mathematics as a critical, but often unseen, component. These two
approaches permit one, also, to learn the associated mathematics as it naturally
unfolds in real world settings, in a relatively “painless” and jargon-free manner.
The reader learns the mathematics required to consider the broad problem at
hand, rather than learning mathematics according to the determination of a
(perhaps) artificial curriculum. The underlying philosophy is different, as is
the topic format. The different format is required given the interactive and ani-
mated features that enliven the topic and motivate the reader to explore diverse
realms in the worlds of geography and mathematics and, especially, as they
combine and interact.
1.2 Theory: Earth coordinate systems
Consider the Earth modeled as a sphere. The Earth is not a perfect sphere, but
a sphere is a good approximation to its shape and the sphere is easy to work
with using the classical mathematics of Euclid and others.
• Consider a sphere and a plane. There are only a few logical possibili-
ties about the relationship between the plane and the sphere.
• The sphere and the plane do not intersect.
• The plane touches the sphere at exactly one point: The plane is
tangent to the sphere.
• The plane intersects the sphere ( Figure 1.1 ) .
And it does not pass through the center of the sphere: In that
case, the circle of intersection is called a small circle (Plane A ).
And it does pass through the center of the sphere: In that case,
the circle of intersection is as large as possible and is called a
great circle (Plane B ).
• The planes that contain great circles are called diametral planes,
because they contain a diameter of the sphere representing the Earth.
• Great circles are the lines along which distance is measured on a
sphere: They are the geodesics (shortest-distance path between two
points) on the sphere. Figure 1.2 shows a route along a small circle
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