Calculus, by Smith and Minton

10.6 Surfaces in Space

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 two-dimensional graphs. Drawing curves and surfaces in three dimensions by hand or correctly interpreting computer-generated graphics is something of an art. After all, you must draw a two-dimensional 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 computer-generated 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 three-dimensional surface. When you see the word cylinder, you probably think of a right circular cylinder. For instance, consider the graph of the equation x²+y² = 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 x²+y² = 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 three-dimensional 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.

6.1

Sketching a Surface

Draw a graph of the surface z = y² in

³.

Notice that since there are no x 's in the equation, the trace of the graph in the plane x = k is the same for every k. This is then a cylinder whose trace in every plane parallel to the yz - plane is the parabola z = y². To draw this, we first draw the trace in the yz - plane (see Figure 10.53a) and then make several copies of the trace, locating the vertices at various points along the x - axis and finally, connect the traces with lines parallel to the x - axis to give the drawing its three-dimensional look (see Figure 10.53b). A computer-generated wireframe graph of the same surface is seen in Figure 10.53c. Notice that the wireframe consists of numerous traces for fixed values of x or y.

Figure 10.53a
Trace in the yz - plane.

Figure 10.53b
z = y².

Figure 10.53c
Wireframe of z = y².

6.2

Sketching an Unusual Cylinder

Draw a graph of the surface z = sin x ³.

Notice that once again, one of the variables is missing. In this case, there are no y 's and so, traces of the surface in any plane parallel to the xz - plane are the same. They all look like the two-dimensional graph of z = sin x. We draw one of these in the xz - plane and then make copies in planes parallel to the xz - plane, finally connecting the endpoints with lines parallel to the y - axis (see Figure 10.54a). In Figure 10.54b, we show a computer-generated wireframe plot of the same surface. In this case, the cylinder looks like a plane with ripples in it.

Figure 10.54a
The surface z = sin x.

Figure 10.54b
Wireframe: z = sin x.

Quadric Surfaces

The graph of the equation

ax²+by²+cz²+dxy+eyz+fxz+gx+hy+jz+k = 0

in three-dimensional space (where a, b, c, d, e, f, g, h, j and k are all constants and at least one of a, b, c, d, e or f is nonzero) is referred to as a quadric surface.

The most familiar quadric surface is the sphere:

(x-a)²+(y-b)²+(z-c)² = r²

of radius r centered at the point (a, b, c). To draw the sphere centered at (0, 0, 0), first draw a circle of radius r, centered at the origin in the yz - plane. Then, to give the surface its three-dimensional look, draw circles of radius r centered at the origin, in both the xz - and xy - planes, as in Figure 10.55. Note that due to the perspective, these last two circles will look like ellipses and will only be partially visible (we indicate the hidden parts of the circles with dashed lines).

Figure 10.55
Sphere.

A generalization of the sphere is the ellipsoid:

(Notice that when d = e = f, the surface is a sphere.)

6.3

Sketching an Ellipsoid

Graph the ellipsoid

To get an idea of what the graph looks like, first draw its traces in the three coordinate planes. (In general, you may need to look at the traces in planes parallel to the three coordinate planes, but the traces in the three coordinate planes will suffice, here.) In the yz - plane, x = 0, so we have the ellipse

which we graph in Figure 10.56a. Next, add to Figure 10.56a the traces in the xy - and xz - planes. These are

respectively, and are both ellipses (see Figure 10.56b). CASs have the capability of plotting functions of several variables in three dimensions. Many graphing calculators with three-dimensional plotting capabilities only

Figure 10.56a
Ellipse in yz - plane.

Figure 10.56b
Ellipsoid.

produce three-dimensional plots when given z as a function of x and y. For the problem at hand, notice that we can solve for z and plot the two functions

and

to obtain the graph of the surface. Observe that the wireframe graph in Figure 10.56c is not particularly smooth and appears to have some gaps. To correctly interpret such a graph, you must mentally fill in the gaps. This requires an understanding of how the graph should look, which we obtained drawing Figure 10.56b. As an alternative, many CASs enable you to graph the equation

using implicit plot mode. In this mode, the CAS numerically solves the equation for the value of z corresponding to each one of a large number of sample values of x and y and plots the resulting points. The graph obtained in Figure 10.56d is an improvement over Figure 10.56c, but doesn't show the traces that we used to construct Figure 10.56b.

The best option, when available, is often a parametric plot. In three dimensions, this involves writing each of the three variables x, y and z in terms of two parameters, with the resulting surface plotted by plotting points corresponding to a sample of values of the two parameters. (A more extensive discussion of the mathematics of parametric surfaces is given in section 14.6.) As we develop in the exercises, parametric equations for the ellipsoid are x = sin s cos t, y = 2sin s sin t and z = 3cos s, with the parameters taken to be in the intervals 0 s 2 and 0 t 2. Notice how Figure 10.56e shows a nice smooth plot and clearly shows the elliptical traces.

Figure 10.56c
Wireframe ellipsoid.

Figure 10.56d
Implicit wireframe plot.

Figure 10.56e
Parametric plot.

6.4

Sketching a Paraboloid

Draw a graph of the quadric surface

x²+y² = z.

To get an idea of what the graph looks like, first draw its traces in the three coordinate planes. In the yz - plane, we have x = 0 and so y² = z (a parabola). In the xz - plane, we have y = 0 and so, x² = z (a parabola). In the xy - plane, we have z = 0 and so, x²+y² = 0 (a pointthe origin). We sketch the traces in Figure 10.57a. Finally, since the trace in the xy - plane is just a point, we consider the traces in the planes z = k (for k > 0 ). Notice that these are the circles x²+y² = k , where for larger values of z (i.e., larger values of k ), we get circles of larger radius. We sketch the surface in Figure 10.57b. Such surfaces are called paraboloids and since the traces in planes parallel to the xy - plane are circles, this is called a circular paraboloid. Graphing utilities with three-dimensional capabilities generally produce a graph like Figure 10.57c for z = x² + y². Notice that the parabolic traces are visible, but not the circular cross-sections we drew in Figure 10.57b. The four peaks visible in Figure 10.57c are due to the rectangular domain used for the plot (in this case, -5

5 and -5

5).

An improvement to this can be made by restricting the range of z - values. With 0 z 15, you can clearly see the circular cross-section in the plane z = 15 in Figure 10.57d.

As in example 6.3, a parametric surface plot is even better. Here, we have x = scos t, y = ssin t and z = s² with -5 s 5 and 0 t 2. Figure 10.57e clearly shows the circular cross sections in the planes z = k, for k > 0.

Figure 10.57a
Traces.

Figure 10.57b
Paraboloid.

Figure 10.57c
Wireframe paraboloid.

Figure 10.57d
Wireframe paraboloid for 0 z 15.

Figure 10.57e
Parametric plot paraboloid.

Notice that in each of the last several examples, we have had to work hard to produce computer-generated graphs that adequately show the important features of the given quadric surface. We want to encourage you to use your graphing calculator or CAS for drawing three-dimensional plots, because computer graphics are powerful tools for visualization and problem solving. However, be aware that you will need a basic understanding of the geometry of quadric surfaces to effectively produce and interpret computer-generated graphs.

6.5

Sketching an Elliptic Cone

Draw a graph of the quadric surface

Be careful not to jump to conclusions. While this equation may look a lot like that of an ellipsoid, there is a significant difference. (Look where the z² term is!) Again, we start by looking at the traces in the coordinate planes. For the yz - plane, we have x = 0 and so,

so that y =

2z. That is, the trace is a pair of lines: y = 2z and y = -2z. We show these in Figure 10.58a. Likewise, the trace in the xz - plane is a pair of lines: x =

z. The trace in the xy - plane is simply the origin. (Why?) Finally, the traces in the planes z = k ( k

0 ), parallel to the xy - plane are the ellipses:

Adding these to the drawing gives us the double-cone seen in Figure 10.58b.

Since the traces in planes parallel to the xy - plane are ellipses, we refer to this as an elliptic cone.

Notice that one way to plot this with a CAS is to graph the two functions z = and In Figure 10.58c, we restrict the z - range to -10 z 10 to show the circular cross sections). Notice that this plot shows a large gap between the two halves of the cone. If you have drawn Figure 10.58b yourself, this plotting deficiency won't fool you. Alternatively, the parametric plot shown in Figure 10.58d, with and z = s with -5 s 5 and 0 t 2, shows the full cone with its circular and linear traces.

Figure 10.58a
Trace in yz - plane.

Figure 10.58b
Elliptic cone.

Figure 10.58c
Wireframe cone.

Figure 10.58d
Parametric plot.

6.6

Sketching a Hyperboloid of One Sheet

Draw a graph of the quadric surface

The traces in the coordinate planes are as follows:

(see Figure 10.59a),

and

Further, notice that the trace of the surface in each plane z = k (parallel to the xy - plane) is also an ellipse:

Finally, observe that the larger k is, the larger the axes of the ellipses are. Adding this information to Figure 10.59a, we draw the surface seen in Figure 10.59b. We call this surface a hyperboloid of one sheet.

To plot this with a CAS, you could graph the two functions and (See Figure 10.59c, where we have restricted the z - range to -10 z 10, to show the circular cross sections.) Notice that this plot shows a small gap between the two halves of the hyperboloid. If you have drawn Figure 10.59b yourself, this plotting problem won't fool you.

Alternatively, the parametric plot seen in Figure 10.59d, with x = 2cos s cosh t, y = sin s cosh t and z = sinh t, with 0 s 2 and -5 t 5, shows the full hyperboloid with its circular and hyperbolic traces.

Figure 10.59a
Trace in yz - plane.

Figure 10.59b
Hyperboloid of one sheet.

Figure 10.59c
Wireframe hyperboloid.

Figure 10.59d
Parametric plot.

6.7

Sketching a Hyperboloid of Two Sheets

Draw a graph of the quadric surface

First, notice that this is the same equation as in example 6.6, except for the sign of the y - term. As we have done before, we first look at the traces in the three coordinate planes. The trace in the yz - plane ( x = 0 ) is defined by

Since it is clearly impossible for two negative numbers to add up to something positive, this is a contradiction and there is no trace in the yz - plane. That is, the surface does not intersect the yz - plane. The traces in the other two coordinate planes are as follows:

and

We show these traces in Figure 10.60a. Finally, notice that for x = k, we have that

so that the traces in the plane x = k are ellipses for k² > 4 . It is important to notice here that if k² < 4 , the equation

has no solution. (Why is that?) So, if -2 < k < 2 , the surface has no trace at all in the plane x = k , leaving a gap which separates the hyperbola into two sheets. Putting this all together, we have the surface seen in Figure 10.60b. We call this surface a hyperboloid of two sheets.

Figure 10.60a
Traces in xy - and xz - planes.

Figure 10.60b
Hyperboloid of two sheets.

Figure 10.60c
Wireframe hyperboloid.

Figure 10.60d
Parametric plot.

We can plot this on a CAS by graphing the two functions

and

. (See Figure 10.60c, where we have restricted the z - range to -10

10, to show the circular cross sections.) Notice that this plot shows large gaps between the two halves of the hyperboloid. If you have drawn Figure 10.60b yourself, this plotting deficiency won't fool you.

Alternatively, the parametric plot with x = 2 cosh s, y = sinh s cos t and z = sinh ssin t, for -4 s 4 and 0 t 2, produces the right half of the hyper-boloid with its circular and hyperbolic traces. The left half of the hyperboloid has parametric equations and with -4 s 4 and 0 t 2. We show both halves in Figure 10.60d.

As our final example, we offer one of the more interesting quadric surfaces. It is also one of the more difficult surfaces to sketch.

6.8

Sketching a Hyperbolic Paraboloid

Sketch the graph of the quadric surface defined by the equation

z = 2y²-x².

We first consider the traces in planes parallel to each of the coordinate planes:

parallel to xy-plane (z = k): 2y² - x² = k (hyperbola, for k

0),

parallel to xz - plane (y = k) :z = - x² + 2k² (parabola opening down)

and

parallel to yz-plane (x = k) :z = 2y²-k² (parabola opening up).

We begin by drawing the traces in the xz - and yz - planes, as seen in Figure 10.61a. Since the trace in the xy - plane is the degenerate hyperbola 2y² = x² (two lines: x =

2y ), we instead draw the trace in several of the planes z = k. Notice that for k > 0, these are hyperbolas opening toward the positive and negative y - direction and for k < 0, these are hyperbolas opening toward the positive and negative x - direction. We indicate one of these for k > 0 and one for k < 0 in Figure 10.61b, where we show a sketch of the surface. We refer to this surface as a hyperbolic paraboloid. More than anything else, the surface resembles a saddle. In fact, we refer to the origin as a saddle point for this graph. (We'll discuss the significance of saddle points in Chapter 12.)

Figure 10.61a
Traces in the xz - and yz - planes.

Figure 10.61b
The surface z = 2y² - x².

Figure 10.61c
Wireframe plot of z = 2y²-x².

A wireframe graph of z = 2y² - x² is shown in Figure 10.61c (with -5

5 and -5

5 and where we limited the z - range to -8

12). Note that only the parabolic cross sections are drawn, but the graph shows all the features of Figure 10.61b. Plotting this surface parametrically is fairly tedious (requiring four different sets of equations) and doesn't improve the graph noticeably.

An Application

You may have noticed the large number of paraboloids around you. For instance, radiotelescopes and even home satellite television dishes have the shape of a portion of a paraboloid. Reflecting telescopes have parabolic mirrors which again, are a portion of a paraboloid. There is a very good reason for this. It turns out that in all of these cases, light waves and radio waves striking any point on the parabolic dish or mirror are reflected toward one point, the focus of each parabolic cross section through the vertex of the paraboloid. This remarkable fact means that all light waves and radio waves end up being concentrated at just one point. In the case of a radiotelescope, placing a small receiver just in front of the focus can take a very faint signal and increase its effective strength immensely (see Figure 10.62). The same principle is used in optical telescopes to concentrate the light from a faint source (e.g., a distant star). In this case, a small mirror is mounted in a line from the parabolic mirror to the focus. The small mirror then reflects the concentrated light to an eyepiece for viewing (see Figure 10.63).

Figure 10.62
Radiotelescope.

Figure 10.63
Reflecting telescope.