Transformations - Spatial Mathematics: Theory and Practice through Mapping - page 56

3.1 Transformations
Mathematics, like musical composition and other fine arts, is purely a human
creation. Without us, does it exist? This sort of “meta” question has long inter-
ested scholars with multidisciplinary interests. Indeed, does the societal cul-
ture and historical epoch in which the predominant mathematics is developed
and embedded influence the kind of mathematics that is developed? Again,
this question might be studied in many ways: We consider one case here—that
of the mathematical relation and selected real-world interpretations. These are
displayed in a number of visual formats not merely as curiosities but more
significantly for the suggestion they might offer as to why or why not certain
types of formal structures get created. It is important to attempt to understand
deeper processes such as these: The mathematics we use in the real-world
often influences the decisions we make (and vice versa). All data that we use
are created, modeled, and maintained based on mathematics. This includes
how data are classified, how they are symbolized, what map projection the
data are cast in, and even, as we will consider in this chapter, the transforma-
tions that are used on the data and the model that is used to represent them
on maps and in spatial databases.
Municipal authorities might use demographic forecasts based on curve fit-
ting to guide the direction of urban land use planning. A City Administrator
might use a rank-ordered set of priorities to decide how valuable taxpayer
funds will be allocated over a period of years to develop (or not develop)
infrastructure. The way in which the mathematics is used will influence the
outcome of the analysis and, therefore quite likely, the policy that is set in
place (Arlinghaus, 1995).
3.1.1 One-to-one, many-to-one, and one-to-many transformations
Much of modern mathematics considers functions as a primary form of math-
ematical transformation. A function , mapping a set X to a set Y permits an ele-
ment x in X to be sent to an element y in Y , or it permits a number of elements,
x 1 , x 2 , x 3 in X to be sent to a single element y in Y ( Figure 3.1 , left side). In the
former situation, the transformation is one-to-one and in the latter it is many-
to-one. Functions may be one-to-one or they may be many-to-one transforma-
tions. They are “single-valued.” Graphically, the idea is represented in Figure
3.1 (left side). There are also one-to-many transformations. Functions may not
be one-to-many transformations. Thus, one-to-many transformations between
two mathematical sets are an often neglected class of relationships. When one
element of X is permitted to map to many elements of Y , as in x mapping to
y 1 , y 2 , y 3 ( Figure 3.1 , right half) the associated mathematical transformation,
that is not a function, is often referred to as a relation .
In the Cartesian coordinate system, the same idea may be visualized ( Figure
3.2a-c ). In the case of a function, a vertical line cuts the graph of the
function no more than once ( Figure 3.2a shows a one-to-one function and