We have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes and spheres, which are examples of three-dimensional figures called surfaces, to explore a variety of other surfaces that can be graphed in a three-dimensional coordinate system.
The first surface we’ll examine is the cylinder. Although most people immediately think of a hollow pipe or a soda straw when they hear the word cylinder, here we use the broad mathematical meaning of the term. As we have seen, cylindrical surfaces don’t have to be circular. A rectangular heating duct is a cylinder, as is a rolled-up yoga mat, the cross-section of which is a spiral shape.
In the two-dimensional coordinate plane, the equation x 2 + y 2 = 9 x 2 + y 2 = 9 describes a circle centered at the origin with radius 3 . 3 . In three-dimensional space, this same equation represents a surface. Imagine copies of a circle stacked on top of each other centered on the z-axis (Figure 2.75), forming a hollow tube. We can then construct a cylinder from the set of lines parallel to the z-axis passing through circle x 2 + y 2 = 9 x 2 + y 2 = 9 in the xy-plane, as shown in the figure. In this way, any curve in one of the coordinate planes can be extended to become a surface.
Figure 2.75 In three-dimensional space, the graph of equation x 2 + y 2 = 9 x 2 + y 2 = 9 is a cylinder with radius 3 3 centered on the z-axis. It continues indefinitely in the positive and negative directions.
A set of lines parallel to a given line passing through a given curve is known as a cylindrical surface, or cylinder . The parallel lines are called rulings .
From this definition, we can see that we still have a cylinder in three-dimensional space, even if the curve is not a circle. Any curve can form a cylinder, and the rulings that compose the cylinder may be parallel to any given line (Figure 2.76).
Figure 2.76 In three-dimensional space, the graph of equation z = x 3 z = x 3 is a cylinder, or a cylindrical surface with rulings parallel to the y-axis.
Sketch the graphs of the following cylindrical surfaces.
Figure 2.77 The graph of equation x 2 + z 2 = 25 x 2 + z 2 = 25 is a cylinder with radius 5 5 centered on the y-axis.
Figure 2.79 The graph of equation y = sin x y = sin x is formed by a set of lines parallel to the z-axis passing through curve y = sin x y = sin x in the xy-plane.
Sketch or use a graphing tool to view the graph of the cylindrical surface defined by equation z = y 2 . z = y 2 .
When sketching surfaces, we have seen that it is useful to sketch the intersection of the surface with a plane parallel to one of the coordinate planes. These curves are called traces. We can see them in the plot of the cylinder in Figure 2.80.
The traces of a surface are the cross-sections created when the surface intersects a plane parallel to one of the coordinate planes.
Figure 2.80 (a) This is one view of the graph of equation z = sin x . z = sin x . (b) To find the trace of the graph in the xz-plane, set y = 0 . y = 0 . The trace is simply a two-dimensional sine wave.
Traces are useful in sketching cylindrical surfaces. For a cylinder in three dimensions, though, only one set of traces is useful. Notice, in Figure 2.80, that the trace of the graph of z = sin x z = sin x in the xz-plane is useful in constructing the graph. The trace in the xy-plane, though, is just a series of parallel lines, and the trace in the yz-plane is simply one line.
Cylindrical surfaces are formed by a set of parallel lines. Not all surfaces in three dimensions are constructed so simply, however. We now explore more complex surfaces, and traces are an important tool in this investigation.
We have learned about surfaces in three dimensions described by first-order equations; these are planes. Some other common types of surfaces can be described by second-order equations. We can view these surfaces as three-dimensional extensions of the conic sections we discussed earlier: the ellipse, the parabola, and the hyperbola. We call these graphs quadric surfaces.
Quadric surfaces are the graphs of equations that can be expressed in the form
A x 2 + B y 2 + C z 2 + D x y + E x z + F y z + G x + H y + J z + K = 0 . A x 2 + B y 2 + C z 2 + D x y + E x z + F y z + G x + H y + J z + K = 0 .
When a quadric surface intersects a coordinate plane, the trace is a conic section.
An ellipsoid is a surface described by an equation of the form x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 . x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 . Set x = 0 x = 0 to see the trace of the ellipsoid in the yz-plane. To see the traces in the xy- and xz-planes, set z = 0 z = 0 and y = 0 , y = 0 , respectively. Notice that, if a = b , a = b , the trace in the xy-plane is a circle. Similarly, if a = c , a = c , the trace in the xz-plane is a circle and, if b = c , b = c , then the trace in the yz-plane is a circle. A sphere, then, is an ellipsoid with a = b = c . a = b = c .
Sketch the ellipsoid x 2 2 2 + y 2 3 2 + z 2 5 2 = 1 . x 2 2 2 + y 2 3 2 + z 2 5 2 = 1 .
Start by sketching the traces. To find the trace in the xy-plane, set z = 0 : z = 0 : x 2 2 2 + y 2 3 2 = 1 x 2 2 2 + y 2 3 2 = 1 (see Figure 2.81). To find the other traces, first set y = 0 y = 0 and then set x = 0 . x = 0 .
Figure 2.81 (a) This graph represents the trace of equation x 2 2 2 + y 2 3 2 + z 2 5 2 = 1 x 2 2 2 + y 2 3 2 + z 2 5 2 = 1 in the xy-plane, when we set z = 0 . z = 0 . (b) When we set y = 0 , y = 0 , we get the trace of the ellipsoid in the xz-plane, which is an ellipse. (c) When we set x = 0 , x = 0 , we get the trace of the ellipsoid in the yz-plane, which is also an ellipse.
Now that we know what traces of this solid look like, we can sketch the surface in three dimensions (Figure 2.82).
Figure 2.82 (a) The traces provide a framework for the surface. (b) The center of this ellipsoid is the origin.
The trace of an ellipsoid is an ellipse in each of the coordinate planes. However, this does not have to be the case for all quadric surfaces. Many quadric surfaces have traces that are different kinds of conic sections, and this is usually indicated by the name of the surface. For example, if a surface can be described by an equation of the form x 2 a 2 + y 2 b 2 = z c , x 2 a 2 + y 2 b 2 = z c , then we call that surface an elliptic paraboloid . The trace in the xy-plane is an ellipse, but the traces in the xz-plane and yz-plane are parabolas (Figure 2.83). Other elliptic paraboloids can have other orientations simply by interchanging the variables to give us a different variable in the linear term of the equation x 2 a 2 + z 2 c 2 = y b x 2 a 2 + z 2 c 2 = y b or y 2 b 2 + z 2 c 2 = x a . y 2 b 2 + z 2 c 2 = x a .
