Mathematical Optimization in Graphics and Vision - Luiz Velho - Impa

1.2. MATHEMATICAL MODELING AND ABSTRACTION PARADIGMS 7
In order to represent the different length values (real numbers) we must look for
a discretization of the set of real numbers. A widely used tecnhique is the floating
point representation. Note that in this representation the set of real numbers is
discretized in a finite set of rational numbers. In particular this implies that the
concept of commensurability is lost when we pass from the mathematical to the
representation universe.
From the point of view of the implementation universe the floating point representation
can be attained by used the IEEE standard, which is implement in most
of the computers. A good reference for this topic is (Higham, 1996).
The previous example, although simple, illustrates the fundamental problem of
modeling in computational mathematics. Of particular importance in this example
is the loss of information we face when moving from the mathematical universe
to the represenation universe (from real numbers to their floating point representation),
where the concept of commensurability is lost. In general, passing from
the mathematical universe to the representation universe implies in a loss of information.
This is a very subtle problem that we must face when working with
modeling in computational mathematics, and, in particular, in computer graphics.
The modeling process consists in choosing an object from the physical world,
associate to it a mathematical model, discretize it and implement it. This implementation
should provide solutions to properly posed problems involving the
initial physical object. Note that the loss of information we mentioned above is a
critical factor in this chain.
We should remark that in the modeling process the association of an object
from an abstraction level (universe) to the other is the most generic possible. In
fact, the same object from one abstraction level may have distinct correspondents
in the next abstraction level (we leave to the reader the task of providing examples
for this). Instead, in the next two sections we will provide two examples which
show that different objects from the physical world may be described by the same
mathematical model.
1.2.1 Terrain representation
Consider the problem of representing a terrain in the computer (a mountain for
instance). In cartography a terrain may be described using a height map: we