Overview

In the two preceding chapters we provided the reader with some guidelines for initiating and implementing the modeling process. Chapter 1 dealt with certain initial steps that can be taken to overcome what is often the greatest hurdle in modeling, getting off to a good start. We also alluded to the importance of concluding the process with an analysis of the results obtained and drawing the appropriate lessons. Chapter 2 provided a compendium of mathematical tools likely to be of use in modeling at the introductory level. Although we shall have occasion to use tools in addition to those presented there, our main working kit will comprise vector and matrix algebra, plus the essentials of ordinary differential equations. Supplementary topics will be addressed as they arise (see Chapter 8).

The present chapter inaugurates the modeling process in a more comprehensive manner. The common theme is the use of geometrical concepts, which we apply to a range of problems drawn from a wide variety of disciplines. The concepts referred to are not confined to pure geometry as it is commonly understood; rather, they encompass both plane and solid geometry, trigonometry, unusual modes of graphical representation, and certain simple topological concepts.

In Example 3.1 to 3.3, we start our deliberations at a deceptively simple level, but the reader will quickly note that even with these simplest of tools, the results obtained are not only highly useful but on occasion startling in the features they reveal and in the elegance of the...