Partitioning of Data - Spatial Mathematics: Theory and Practice through Mapping - page 138

definition of regions in the absence of regional information. All rely on the
concept of partition. We look here at a small selection of possibilities.
6.5.1 Isolines; contours
Thiessen polygons offer one way of defining regions in the absence of regional
information. Still another way to partition the plane involves isolines. An
isoline is a curve or a line on a map along which there is a constant value.
It partitions a set of values as all the points on one side of the line are less
than the constant value and all the points on the other side are greater than
the constant value. When that value is based on topographic elevation, the
isoline is called a contour. Isolines appear commonly on television weather
stations representing temperature (isotherms), precipitation (isohyets), baro-
metric pressure (isobars), and other weather phenomena.
From the standpoint of the mathematical connection, students of calculus will
recognize isolines as so-called level curves—as in a set of cross sections at dif-
ferent levels of a surface. Slicing a cone with a circular base and apex directly
over the center of the circular base yields circular cross sections of different
sizes at different levels (heights above the base). When these are projected
back into the plane, they form a set of concentric circles, similar in appear-
ance to contours. Real-world contours representing a mountain on a map
could be created by finding level curves of the mountain. To do so, however,
would require knowing the equation of the surface/volume of the mountain.
Mountains are not cones or any other straightforward three-dimensional math-
ematical structure. What is mapped, however, is a two-dimensional represen-
tation of a two-dimensional surface—the third dimension present in the real
world is lost in the transformation. The purely mathematical approach, which
affords accuracy, is difficult.
A lunar data viewer offers an interesting way to look at cross-sections. It can
be accessed at http://target.lroc.asu.edu/da/qmap.html . Pick a crater. Draw
a line across it and a nice cross-sectional graph will appear. How wide and
deep are some of these craters? The largest ones are just over 500 kilometers
wide and 4000 meters deep. Use caution when interpreting the cross sections
from this or any website, paying attention to the horizontal and vertical units
of measurement. Speculate on angle of impact, relative age of the crater, and
size of the meteor! Spatial thinking lends one to the conclusion that the newer
impacts are those that are superimposed on other, older, landforms. The lunar
maria, formed from basalt flows from volcanoes, have in many locations oblit-
erated the former surface, so despite their old age, they too are younger than
the landscape underneath.
In simple plane geometry, there are other strategies to approximate contour
location that are easier to implement than finding level curves and that rely
only on simple partitioning of data. The example below suggests the general
strategy.