Geometry of the Sphere - Spatial Mathematics: Theory and Practice through Mapping - page 17

nearby cell phone towers. Of course, if no cell phone towers are nearby, the
positional accuracy on a smartphone will be compromised.
1.5 Other Earth models
The discussion in this chapter has been focused on a spherical Earth. However,
due to the centripetal force of the Earth's rotation, the Earth is not a perfect
sphere. It is an oblate spheroid, with an equatorial diameter that is 42.72 kilo-
meters (26.54 miles) longer than its diameter measured through the poles.
Because we do not have a tape measure that we can wrap around the Earth to
determine its exact shape and measurements, these all have to be mathemati-
cally calculated. Geodesy is the science that studies the size and shape of the
Earth and its gravitational field. Further complicating the situation is that the
shape, rotation, and revolution of the Earth changes over time.
When modeling the Earth in geography and in mathematics, three concepts
must be simultaneously addressed: The physical Earth, the geoid, and the ref-
erence ellipsoid. The physical Earth is the actual shape and size of the planet;
it is what we try to measure and model. Gravity is not even across the Earth's
surface, due to the Earth's rotation, altitude, and differences in rock density.
The geoid represents a surface of constant gravity. This is idealized by the
notion of mean sea level, which is a theoretical level of the average height of
the ocean's surface. It does not, however, correspond to the actual level of the
sea at any given point at any instant in time, but is the halfway point between
mean high and low tides. The geoidal surface is irregular, but smoother
than the physical Earth's surface. The reference ellipsoid is a mathematically
defined shape that is a “best-fit” to the geoid, most often for a continent, but
potentially for the entire globe. One way to view goodness-of-fit is to consider
the degree of separation between in the geoid and ellipsoid (the value N in
Figure 1.14 ). The reference ellipsoid is smoother than the geoid, and it is the
surface upon which coordinate systems are defined ( Figure 1.14 ) .
Another fundamental concept in spatial mathematics is the concept of a
datum. We have already discussed the importance of a zero point when
calculating horizontal position such as latitude and longitude. This concept
is also critical with respect to other common mapping coordinate systems
(such as Universal Transverse Mercator (UTM) coordinates, or state plane
coordinates). However, where is the “zero point” in terms of elevation, or
the “ z ” coordinate, given the complications in the shape of the Earth as
described above? Datums help solve this problem. Datums are reference
surfaces; that is, mathematically derived, against which position measure-
ments are made. They define the location of zero on the measurement scale.
Datums form the basis of coordinate systems. Hundreds of locally developed
reference datums exist around the world usually referenced to convenient
local reference points. Modern datums are based on increasingly accurate
measurements of the shape of the Earth, however, and tend to cover larger