
In Chapter 3, you expended considerable effort learning how to use the calculus to draw graphs of a wide range of functions in two dimensions. Now that we have discussed lines and planes in ^{3}, we continue our graphical development by drawing more complicated objects in three dimensions. Don't expect a general theory like we developed for twodimensional graphs. Drawing curves and surfaces in three dimensions by hand or correctly interpreting computergenerated graphics is something of an art. After all, you must draw a twodimensional image that somehow represents an object in three dimensions. Our goal here is not to produce artists, but rather to leave you with the ability to deal with a small group of surfaces in three dimensions. For our presentation over the next several chapters, you will want to have at your disposal a small number of familiar surfaces. You will need to recognize these when you see them and have a reasonable facility for drawing a picture by hand. We also urge you to learn to produce and interpret computergenerated graphs. Follow our hints carefully and work lots of problems. In numerous exercises in the chapters that follow, taking a few extra minutes to draw a better graph will often result in a huge savings of time and effort.
Cylindrical Surfaces
We begin with a simple type of threedimensional surface. When you see the word cylinder, you probably think of a right circular cylinder. For instance, consider the graph of the equation x^{2}+y^{2} = 9 in three dimensions. Your first reaction might be to say that this is the equation for a circle, but you'd only be partly correct. The graph of x^{2}+y^{2} = 9 in two dimensions is the circle of radius 3, centered at the origin, but what about its graph in three dimensions? Consider the intersection of the surface with the plane z = k, for some constant k. Since the equation has no z 's in it, the intersection with every such plane (called the trace of the surface in the plane z = k ) is the same: a circle of radius 3, centered at the origin. Think about it: whatever this threedimensional surface is, its intersection with every plane parallel to the xy  plane is a circle of radius 3 , centered at the origin. This describes a right circular cylinder, in this case one of radius 3, whose axis is the z  axis (see Figure 10.52).
Figure 10.52
Right circular cylinder.
More generally, the term cylinder is used to refer to any surface whose traces in every plane parallel to a given plane are the same. With this definition, many surfaces qualify as cylinders. 