CHAPTER 10 Vectors and the geometry of space

The design of an offshore oil platform presents a significant engineering challenge. Underwater construction poses a unique set of constraints, and the high winds and waves that occur in storms at sea can damage or destroy even well-designed structures. In order to build a suitable platform, engineers must have an in-depth knowledge of three-dimensional geometry and be proficient in the analysis of stresses and strains throughout the structure. In this chapter, we take the first steps toward the fully three-dimensional calculus necessary to perform such analyses.

In designing an oil platform, an engineer's task is to find a structure that not only withstands large forces but also protects a crew of workers by remaining relatively motionless during storms. The hourglass-like shape shown in Figure 10.1 minimizes sway by transmitting lateral (sideways) forces through a series of reinforced struts directly into the seabed below. To analyze such a design, we first need a mathematical language for describing the size and direction of three-dimensional forces. The vectors developed in the first two sections of this chapter provide us with such a language. In the third and fourth sections of this chapter, we learn how to use vectors to split a lateral force into a set of component forces acting in different directions (for instance, along the struts of an oil rig). Engineers use such calculations to determine how much reinforcement is necessary for an oil rig to remain stable, even in the worst weather.

Figure 10.1
Offshore oil platform.

In the last two sections of this chapter, we introduce some of the fundamental structures of three-dimensional geometry. One of these is the hyperboloid of one sheet. The oil rig shown in Figure 10.1 has the shape of a hyperboloid, whose hyperbolic cross sections enable it to efficiently diffuse strong lateral forces. This geometric property makes the hyperboloid a natural choice for such an application. Hyperboloids are also used in other challenging design situations, such as the design of cooling towers for nuclear power plants.

This chapter represents a crossroads from the primarily two-dimensional world of first-year calculus to the three-dimensional world of many important scientific and engineering problems. The rest of the calculus we develop in this book builds directly on the basic ideas developed here.