Graphics Reference
In-Depth Information
CHAPTER 1
Introduction
1.1
Overview
This topic is about constructive geometry. Our object is to study geometry, all sorts of
geometry, and also to present a setting in which to carry out this investigation on a
computer. The adjective “constructive” is intended to imply that we will not be satis-
fied with an answer to a geometric problem unless we also are given a well-defined
algorithm for computing it. We need this if we are going to use a computer. However,
even though our goal is a computational one, our journey will take us through some
beautiful areas of pure mathematics that will provide us with elegant solutions to
problems and give us new insights into the geometry of the world around us. A reader
who joins us in this journey and stays with us to the end will either have implemented
a sophisticated geometric modeling program in the process or at least will know how
to implement one.
Figure 1.1 shows the task before us at a very top level. We have a number of rep-
resentation problems. Our starting point is the “real” world and its geometry, but the
only way to get our hands on that is to build a mathematical model. The standard
approach is to represent “real” objects as subsets of Euclidean space. Since higher
dimensional objects are also interesting, we shall not restrict ourselves to subsets of
3-space. On the other hand, we are not interested in studying all possible subsets. In
this topic, we concentrate on the class of objects called finite polyhedra. More exotic
spaces such as fractals (the spaces one encounters when one is dealing with certain
natural phenomena) will only be covered briefly. They certainly are part of what we
are calling geometric modeling, but including them would involve a large amount of
mathematics of a quite different flavor from that on which we wish to concentrate
here. Next, after representing the world mathematically, we need to turn the (contin-
uous) mathematical representations into finite or discrete representations to make
them accessible to computers. In fact, one usually also wants to display objects on a
monitor or printer, so there are further steps that involve some other implementation
and computation issues before we are done.
